The Pennsylvania State University The Graduate School SONIC BOOM POSTPROCESSING FUNCTIONS TO SIMULATE ATMOSPHERIC TURBULENCE EFFECTS

Transcription

1 The Pennsylvania State University The Graduate School SONIC BOOM POSTPROCESSING FUNCTIONS TO SIMULATE ATMOSPHERIC TURBULENCE EFFECTS A Dissertation in Acoustics by Lance L. Locey 2008 Lance L. Locey Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2008

2 The dissertation of Lance L. Locey was reviewed and approved by the following: Victor W. Sparrow Professor of Acoustics Dissertation Advisor, Chair of Committee Karl M. Reichard Assistant Professor of Acoustics Kenneth S. Brentner Professor of Aerospace Engineering Anthony A. Atchley Professor of Acoustics Head of the Graduate Program in Acoustics Signatures are on file in the Graduate School.

3 Abstract Two methods of representing the effects of atmospheric turbulence on sonic boom propagation are developed. The first method involves estimating a finite impulse response (FIR) filter which converts a sonic boom measurement made at altitude into an estimate of the same sonic boom when measured on the ground, after it has propagated through a portion of the atmosphere. The filter functions model the linear propagation between the measurement points in space and time. The filters can be applied as post processing functions to sonic boom waveforms based on computational fluid dynamics (CFD) predictions to simulate realistic turbulent propagation. Three experimental data sets are used to estimate turbulent filter functions. The second method for developing filter functions employs a numerical algorithm to propagate an N-wave. The propagation algorithm used is the nonlinear progressive wave equation, or NPE. The NPE was modified to include a statistical representation of turbulence based on the method of Fourier mode summation, using a modified von Kármán spectrum. The results from the numeric propagation can be used to estimate filter functions in the same way that the experimental data sets were used. The filters are then applied to various shaped sonic boom waveforms. The shaped sonic booms are the result of CFD calculations for aircraft specifically designed to produce low level sonic booms, or low-booms. When the various filters are applied to the low-booms results are presented in terms of Steven s Mk VII Perceived Level (PL) of noise, which is a metric which describes the perceived loudness of a sonic boom. It is shown that the PL values are usually decreased by turbulence, but the PL values sometimes increase. iii

4 Table of Contents List of Figures Acknowledgments viii x Chapter 1 Introduction Overall Approach What Causes a Sonic Boom? Sonic Boom Shape Theory Shaped Boom Waveforms Potential Market for a Small Supersonic Jet How Loud is Too Loud? Atmospheric Turbulence and the Sonic Boom Dissertation Outline Chapter 2 Loudness and Annoyance Loudness & Stevens Mk VII Perceived Level (PL) Stevens MK VII Implementation Turbulence and Human Perception of Sonic Boom Chapter Summary Chapter 3 The Atmosphere as a Transfer Function The Atmosphere and the Planetary Boundary Layer Transfer Function Concept Nonlinear Acoustics iv

5 3.2.2 Transfer Function Limitations Transfer Function Based on Matlab s DECONV Function Transfer Function Based on Matrix Deconvolution Filter Function Verification Filter Limitations Applying Filter Functions Chapter Summary Chapter 4 Filters from Experimental Data Sets The Shaped Sonic Boom Experiment (SSBE) SSBD Sailplane Waveforms F5E Waveforms Meteorological Conditions Waveform Preparation Doppler Correction Filter Estimation by Matrix Deconvolution SSBE Filter Agreement Applying the SSBE Filters NASA Dryden Sailplane Waveform Preparation Altitude Pressure Correction Low Frequency Microphone Correction Doppler Correction Waveform Truncation Ground waveform preparation Low Frequency Microphone Correction Waveform Truncation Filter Estimation by Matrix Deconvolution Filter Agreement Applying the 2006 Filters NASA Dryden Meteorological Conditions Sailplane Waveform Preparation Altitude Pressure Correction Low Frequency Microphone Correction Doppler correction Waveform Truncation Ground waveform preparation Low Frequency Microphone Correction v

6 Waveform Truncation Filter Estimation by Matrix Deconvolution Filter Agreement Applying 2007 Filters Filter estimation evolution Chapter Summary Chapter 5 The Nonlinear Progressive Wave Equation (NPE) NPE Overview NPE Initial and Boundary Conditions Turbulence Incorporating Turbulence into the NPE Turbulence Spectrum Turbulence Parameter Selection Refractive-Index Temperature Structure Parameter Running the NPE Converting NPE Waveforms NPE Filter Agreement Applying NPE Filters Chapter Summary Chapter 6 Summary and Conclusions Summary Implications of the Assumptions Made Contributions of this Dissertation Future Work Bibliography 98 Appendix A Compiling and running the NPE 103 A.1 LionXC A.1.1 Logging In A.1.2 Transferring Files A.1.3 Compiling Code A.1.4 Submitting Code to the Queue on LionXC A.1.5 Necessary Files A.2 RTJones vi

7 A.2.1 Logging in to SFE A.2.2 Logging In to RTJones A.2.3 Transferring Files to dmzfs1 from a PSU Computer Appendix B Matlab Code 108 B.1 Matrix Deconvolution Code B.2 PLdB Code B.3 SSBE Code B.4 Dryden 06 Code B.5 Dryden 07 Code B.6 NPE Code B.7 Misc. Code vii

8 List of Figures 1.1 Sonic Boom Evolution Mach Cone Mach Angle Sonic Boom Carpet Loudness of Shaped Booms F5 Variants A Shaped Sonic Boom Measurement Shaped Sonic Boom Demonstrator Aircraft A Shaped Sonic Boom Waveform Six Theoretical Low-boom Waveforms Measured N-waves Rippled Shock Front D Shock Front After Propagation Loudness vs. Annoyance A Symmetric Sonic Boom Waveform An Asymmetric Sonic Boom Waveform The Atmosphere as a Filter SSBE Microphone Locations Sailplane Measurement of SSBD Sonic Boom SSBE Temperature and Relative Humidity SSBE Wind Speed Profile Sailplane Measurement of F5E Sonic Boom SSBE Filter Error Filter Agreement for SSBE Microphone Location B Closeup of SSBE Filters Applied to a Low-boom Waveform Effects of SSBE Filters on a Low-boom Dryden 2006 Ground Microphone Background Pressure Variation Removal, Dryden viii

9 Sailplane Waveform Low Frequency Correction Dryden 2006 Sailplane Waveform Truncation Sailplane and Ground Waveform Closeup Dryden 2006 Ground Waveform Truncation Dryden 2006 Filter Error Dryden 2006 Filter Agreement Turbulized Low-boom Waveform Using Dryden 2006 Filters PL Values Based on 2006 Filters Dryden 2007 Microphone Tower SSBE Temperature and Relative Humidity SSBE Wind Speed Profile Dryden 2007 Filter Error Dryden 2007 Filter Agreement Five Turbulized Low-boom Waveforms PL Values Based on 2007 Filters The Atmosphere as a Filter NPE Grid NPE Boundary Conditions Periodic Turbulence Turbulence Spectra NPE Based Normalized Structure Parameter m Wide NPE Wavefront NPE Booms NPE Filter Error Low-boom PL Values Based on NPE Filters B.1 A Symmetric Sonic Boom Waveform B.2 An Asymmetric Sonic Boom Waveform ix

10 Acknowledgments I would like to acknowledge the hand of God in this work. I have a firm belief in the existence of God and am convinced that He has been a part of this work. Next, I would like to acknowledge my wife Wendy, for her love and support as I have worked toward the completion of my degree. My son, Lucas, has also been a source of joy since his birth. I would also like to acknowledge the following individuals: Victor W. Sparrow My advisor and the person responsible for allowing me the privilege and opportunity to conduct this research. Brenda Sullivan Brenda has provided code, constructive criticism, and a healthy dose of humor to this work. Ken Plotkin Ken supplied me with assistance in the form of his code PCBoom, the SSBE data set, as well as many helpful comments and recommendations along the way. Ed Haering Ed s insights helped with identifying problems with preboom noise in the ground waveform/sailplane waveform. Ed also provided GPS and sailplane data along with many helpful suggestions. Tom Gabrielson Dr. Gabrielson provided sailplane data, the low frequency microphone correction technique, and the sailplane descent correction technique. Andy Piacsek Andy supplied the source code for the NPE, without which, this research would not have been possible. Andy also collaborated on much of the turbulence work. Peter Coen Peter provided access to NASA s RTJones computer. Committee Anthony A. Atchley Kenneth S. Brentner x

11 Karl M. Reichard I would also like to thank the following organizations: USAF Test Pilot School PARTNER Center of Excellence FAA & NASA- Financial Support. Industry Partners: Wyle Laboratories Boeing Commercial Airplanes Cessna Aircraft Company Lockheed/Martin Aeronautics Gulfstream Aerospace Corporation xi

12 Chapter 1 Introduction Supersonic overland flight is currently prohibited within the United States of America, except for small supersonic corridors designated for military training purposes. The prohibition on supersonic flight is the result of public objection to the sonic boom caused by large supersonic aircraft. 1 In recent years, advances in computational power have led to new designs for quieter aircraft and hence a renewed interest in supersonic travel. 2 4 This renewed interest is focused on small supersonic business jets, as opposed to larger aircraft such as the Concorde. Five years after the Concorde s final flight, there are plans for a new generation of supersonic aircraft. One such aircraft is the Quiet Super Sonic Transport (QSST) designed by Supersonic Aerospace International (SAI). 5 SAI s proposed jet could transport business executives from New York to Los Angeles in under three hours. Aerion Corporation is another manufacturer who claims that their aircraft would have the same operating costs as a traditional business jet. 6 Before these new generation supersonic aircraft would be permitted to fly overland, engineers must come up with a way to reduce the sonic boom they create. One such proposed technique is the Quiet Spike 4 proposed by Gulfstream Aerospace Corporation. Gulfstream has been collaborating with NASA to verify the Quiet Spike concept. Gulfstream claims that their design would create a sonic boom that was so quiet you would hardly notice if the aircraft were to pass overhead. If supersonic overland flight becomes a reality for small supersonic business jets, there is speculation within the sonic boom research community that the return

13 2 of commercial supersonic flight could follow within twenty years. The market for small supersonic business jets is on the order of 450 aircraft at a price of million USD each. 2 That is roughly the price of a commercial subsonic airliner. 1.1 Overall Approach The overall goal of this dissertation is to find a way to incorporate the effects of realistic atmospheric turbulence into sonic boom waveforms. Industry is currently able to come up with sonic boom waveforms based on proposed aircraft geometry using computational fluid dynamics (CFD) codes and nonlinear propagation codes. The catch is that such techniques do not include the effects of a realistic, turbulent atmosphere, and therefore do not accurately predict noise on the ground. Once a technique for representing turbulence is developed, that technique can be applied to baseline waveforms of proposed aircraft. The turbulized waveforms can then be passed on to researchers involved in perception and subjective testing to determine public response to the proposed aircraft. Eventually, the results from the subjective testing will provide the FAA with the information it needs to make informed decisions about revising existing supersonic flight regulations. The ultimate goal of this research is to provide a tool which can be used to better understand how turbulence alters a sonic boom as it propagates through the atmosphere. 1.2 What Causes a Sonic Boom? A sonic boom occurs any time an object travels faster than the speed of sound. Sonic booms are typically associated with supersonic aircraft. However, a sonic boom can also be produced by a high speed bullet or the tip of a whip. When an aircraft flies at supersonic speeds, sound waves build up around the nose and other protrusions from the aircraft, such as the tail, wings, and engine inlets. The sound waves created by the aircraft geometry cannot propagate away from the aircraft without coalescing, or stacking up on each other, because the aircraft is moving faster than the local speed of sound. In the near-field region close to the aircraft, shocks form. Figure [1.1] shows how shocks coalesce in the

14 3 plane that intersects the long axis of the aircraft and the ground. Eventually, the waveform that is measured on the ground resembles an N and is often referred to as an N-wave because of its shape. Figure 1.1: Shocks near the supersonic aircraft coalesce to produce an N-wave. Taken from Ref. 7. Figure [1.1] shows what happens to a sonic boom in a plane. In reality, this process occurs in three dimensions, the result being a cone which forms around the aircraft. This cone is referred to as the Mach cone. Figure [1.2] shows the full 3D representation of the Mach cone. Figure 1.2: The 3D Mach cone created by supersonic flight. Taken from Ref. 8. The angle of the cone with respect to the direction of travel of the aircraft is

15 4 referred to as the Mach Angle µ = sin 1 ( 1 M ) (1.1) where µ is the Mach Angle and M is the Mach number, the aircraft velocity divided by the local speed of sound. Figure [1.3] shows a diagram of Mach Angle. Figure 1.3: A diagram of Mach Angle. 9 sound is c, and the Mach Angle is µ. The aircraft velocity is v, the speed of Figure [1.3] only shows a two dimensional slice of the cone created by the supersonic aircraft. Figure [1.4] shows the area affected by a supersonic aircraft passing overhead. The primary boom carpet is associated with the area where the Mach cone intersects the ground. It is approximately 30 miles (48 km) wide for an aircraft flying at 60,000 feet (18,288 m). The secondary carpet is associated with sonic boom rays that are diffracted by atmospheric effects. The secondary boom carpet is approximately 100 miles (160 km) wide.

16 5 Figure 1.4: A diagram of the area affected by the passing of a supersonic aircraft. The primary boom carpet is approximately 30 miles (48 km) wide and the secondary boom carpet is approximately 100 miles (160 km) wide, assuming a nominal flight altitude of 60,000 feet (18,288 m). Taken from Ref Sonic Boom Shape Theory In the late 1960 s, George and Seebass 11 developed a theory which suggested that it would be possible to design an aircraft which would generate a sonic boom waveform (when measured on the ground) that would be something other than an N-wave. They hypothesized that it would be possible to generate either a ramp or flat-top sonic boom waveform. (Figure [1.5] shows an example of three ramp type booms, where the acoustic pressure quickly jumps from ambient and then increases linearly to a maximum. Flat top booms can be described as booms which have an initial shock followed for a short time by a constant pressure. Please note that throughout this work, unless explicitly stated otherwise, pressure always refers to acoustic pressure, which is a small perturbation about atmospheric pressure.) This type of sonic boom could best be described as a shaped, or engineered, boom. The implication was that a shaped sonic boom could potentially be less annoying than a traditional N-wave. At the time, engineers did not have sufficient computational resources to design such an aircraft.

17 6 Figure 1.5: Loudness levels of a traditional N-wave and three shaped booms. All four waveforms have the same initial rise time of 1 ms and a peak pressure of 1 psf. The three shaped booms have varying intial shock strengths. Figure courtesy Brenda Sullivan of NASA Langley Research Center. It is worth noting here that there may be other ways to reduce sonic boom. One such way is by flying just over Mach 1 but slower than the cutoff Mach number, where the aircraft s ground speed is slower than the sound speed on the ground. A recent paper by Plotkin et al. reports on this technique. 12 Aerion Corporation is pushing the FAA to permit its supersonic aircraft to fly at Mach cutoff. One of the difficulties associated with this technique is the need for extremely accurate weather information in order to determine the sound speed along the ground. Another difficulty with Mach cutoff is that it has an upper limited in the region of Mach 1.2. In 2000 the Defense Advanced Research Projects Agency (DARPA) set out to prove or at least validate George and Seebass theory. One part of their work included a demonstration aircraft, designed to produce a shaped flat-top sonic

18 7 boom instead of a traditional N-wave. According to George and Seebass theory, a shaped boom demonstration aircraft would require a nose unlike any existing aircraft. Rather than design a whole new aircraft, engineers had to choose a baseline aircraft which would lend itself to extensive nose modification. A Northrop Grumman F5E was selected as the baseline aircraft to be modified. It was chosen because several variants of the plane already existed, including a two seat trainer and a single seat version which had an elongated nose. Engineers needed an aircraft that had a variant with a long nose so that pilots familiar with the existing longer aircraft would be able to safely fly the proposed shaped boom aircraft. Figure [1.6] shows several variants of the Northrop Grumman F5. Figure 1.6: 4 variations of the Northrop Grumman F5 fighter aircraft, including a two seat trainer version. Note the variation of the nose geometry of the different aircraft in the inset. Taken from Ref. 13. Engineers were then tasked with modifying the nose of the regular F5E, based on CFD predictions provided by George and Seebass theory. The resulting aircraft, the Shaped Sonic Boom Demonstrator (SSBD) made history on 27 August 2003 when the first ever flat-top sonic boom was recorded on the ground, propagating through a real atmosphere, validating George and Seebass theory. 13 Figure [1.7] shows the first ever measurement of a shaped sonic boom after propagating through a real atmosphere. Figure 1.8 shows the SSBD flying over NASA Dryden Flight Research Center. Proof of George and Seebass theory greatly increased aircraft

19 8 manufacturers interest in small supersonic business jets. The SSBD aircraft did as its name indicates, it demonstrated that George and Seebass theory was correct. It was not, however, a true shaped boom aircraft, because the aircraft was not originally designed according to George and Seebass theory. Other waveforms, which represent shaped boom aircraft designed specifically to that end, are necessary if the FAA is ever to permit overland supersonic flight. Figure 1.7: The first ever measurement of a shaped sonic boom, with a traditional sonic boom for comparison. The nominally flat top shaped boom of the SSBD confirmed George and Seebass approach. Taken from Ref Shaped Boom Waveforms Several simulated shaped boom signatures were used in the present study. These waveforms are the result of 1D nonlinear propagation codes which take near-field CFD results and then propagate the waveforms to the ground. The propagation

20 9 Figure 1.8: The Shaped Sonic Boom Demonstrator aircraft. Note the two pressure waveforms painted on the side of the aircraft- in red is a traditional N-wave and in black is the hypothesized shaped boom. models used to generate these waveforms include nonlinear steepening, stretching and atmospheric absorption, but do not include the effects of turbulence. Figure [1.9] shows an example of a shaped boom after it has propagated through a quiescent atmosphere. This particular waveform was provided by Gulfstream Aerospace Corporation. It is based on Gulfstream s Quiet Spike concept. 3, 4 Notice how this waveform has a very gentle rise, indicating that it has very little highfrequency energy. Figure [1.10] shows an example of six additional shaped sonic boom waveforms. These waveforms are also the results of CFD predictions and 1D propagation codes. They are for a single aircraft operating under various flight conditions, such as accelerating, flying at level cruise, etc. These waveforms all have a short initial rise

21 Pressure (psf) Time (s) Figure 1.9: A shaped sonic boom waveform, based on the Quiet Spike concept by Gulfstream Aerospace Corporation. Notice the gentle change in pressure due to the bow and tail shocks. time (the time it takes to go from 10 of the shock peak to 90 of the shock peak) and peak pressure values on the order of 0.5 psf, or pounds force per square foot. (Please note that in the aerospace industry within The United State of America, psf is the standard unit of measure for reporting sonic boom pressure values psf = Pa) The short rise times of these waveforms indicate that they contain high-frequency energy. 1.5 Potential Market for a Small Supersonic Jet One consideration worth noting is the financial implications of overland supersonic flight. The success of the SSBD prompted a number of companies to evaluate

22 Pressure (psf) Time (milliseconds) Figure 1.10: Six shaped sonic boom waveforms. Notice the short rise times associated with the bow shocks. The waveforms correspond to different flight conditions for one particular proposed aircraft design, based on CFD predictions. the market for small, shaped boom supersonic business jets (SBJs). A number of market studies were conducted. A paper by Henne of Gulfstream Aerospace Corporation 2 suggested that the worldwide market for SBJs to be between 250 and 450 units, with a price between $50M and $100M per unit. Such aircraft would fly between Mach 1.4 and Mach 1.8. With the possibility of overland supersonic flight, new travel concepts come into play. Within the United States, coast to coast commuting becomes possible. A flight leaving New York at 7 a.m. Eastern time could arrive in Los Angles at 7 a.m. Pacific time. A return flight, 8 hours later, would take off at 3 p.m. Pacific time and touch down in New York at 10 p.m. Eastern time. The total time spent for the trip would be 14 hours, as opposed to two days given a conventional business

23 12 jet (or three days in the case of commercial airlines). In another example, such an aircraft could leave New York and conceivably get anywhere in the world in 10 hours with just one fuel stop. These new concepts in business travel have created excitement among aircraft manufacturers. One question remains- How loud will these aircraft really be? 1.6 How Loud is Too Loud? Immediately following the introduction of the Concorde, the FAA was inundated with complaints which eventually led to the current ban on overland supersonic flight. Conversely, shaped boom aircraft have the potential to fly at supersonic speeds without generating loud, annoying sonic booms. 14 However, before such shaped boom aircraft are permitted to fly, the FAA must be certain that the sonic booms they generate will be acceptable to the public. The question that must be answered, then, is How loud is too loud? In order to address that question, a metric which quantifies the loudness of sonic booms is necessary. One such metric exists- Stevens Mk VII Perceived Level of noise, or PL, for short. Research is ongoing to determine if other metrics might better describe the loudness of shaped sonic booms. In particular, the author is aware of a time varying loudness metric under development at Purdue University. 15 However, at the time of this research, Stevens Mk VII PL is the best available metric for quantifying loudness of shaped sonic booms. 16 The answer to the question How loud is too loud? is not yet known. Initial shaped boom results suggest that the waveforms generated by shaped boom aircraft will be substantially quieter than traditional N-waves. 17 Figure [1.5] shows a comparison of a traditional N-wave with three shaped booms. All four booms have the same initial rise time of 1 ms. The three shaped booms have varying initial shock strengths followed by a gradual pressure increase (the ramp ) to the same peak pressure. As the initial shock strength goes down, the perceived level of loudness goes down, suggesting that a shaped boom would be much quieter than a traditional N-wavefor the same peak pressure. The implication is that shaped booms could best be described as low-booms. Throughout the rest of this work, shaped booms will be referred to as low-booms because of this result. The focus of

24 13 this work is not to address the loudness question, rather it is to assess the impact of turbulence on sonic boom propagation and how that impact alters the loudness level of the sonic boom. It should be noted that low-booms need not be limited exclusively to shaped booms. It is possible to generate a low-boom by means of special manoeuver which yields a low amplitude traditional N-wave, which could also be termed a low-boom. For details on this manoeuver, the reader is directed to the work by Klos. 18 However, for the purpose of this dissertation, a low-boom will refer to a shaped boom. 1.7 Atmospheric Turbulence and the Sonic Boom So why is turbulence important? Figure [1.11] shows the influence of turbulence on traditional N-waves. These sonic booms were measured in the Chicago area in In the left column are N-waves measured on days with low wind velocities and therefore low levels of atmospheric turbulence. The waveforms all show essentially the same N-wave shape. There is very little variation between the booms. In the right column, the N-waves show great variability in terms of rise time and peak pressure. The top waveform has a very sharp initial rise and a peak pressure nearly twice what was measured in the left column. Other waveforms on the right have multiple peaks and or rounding effects. All of these distortions contribute to variations in the perceived level of these sonic booms, as compared to the booms measured on days with low wind velocities. It should be noted that these waveforms were recorded over 40 years ago and were not available to the author in a reliable digital format for computing metrics. Sonic booms have energy that is concentrated in the hertz frequency range. 20 Turbulence amplifies and de-amplifies individual portions of the sonic boom wavefront, and this causes spiking and rounding of the shocks. This focusing does concentrate and deconcentrate the energy of the sonic boom waveform. Although the total energy of the wave does not change (except for atmospheric absorption and cylindrical spreading effects), the energy in the wavefront is not uniform anymore (as it would be for a homogeneous atmosphere). On a local level, one will see amplitudes increase and decrease due to turbulence as one follows a particular

25 14 Figure 1.11: N-waves measured on days with low and high amounts atmospheric turbulence. Taken from Ref. 19. portion of the wavefront. Piacsek explained this concept. 21 Figure [1.12] shows an example of a plane wave that passes through a chain of vortices (representing turbulence). The resulting wavefront is rippled. As the rippled wavefront propagates, the region of the wavefront that was concave leads to focusing which in turn can lead to the addition of high frequencies. The region of the plane wave that was convex leads to defocusing. Turbulence does not directly add energy to the wavefront. Rather, the turbulence creates conditions which can lead to either focusing or defocusing. Piacsek numerically propagated a step shock through vortices to simulate a plane wave propagating through turbulence. Figure [1.13] shows some of the results from Piacsek s work. The figure shows that focusing can lead to spiking which yields increases in high frequency energy. Currently, there is no way to predict beforehand the turbulence present in the atmosphere. Turbulence can be modeled statistically. A method to incorporate the effects of realistic turbulence into low-boom waveforms is needed so that realistic waveforms can be created which represent what would be heard by people outdoors. This subject will be covered in greater detail in Chapter 2.

26 15 Figure 1.12: Looking down on a plane wavefront which becomes rippled after passing through a chain of vortices (which simulate turbulence). Taken from Ref Dissertation Outline The next chapter describes in greater detail how loudness is quantified. Following that is a chapter discussing the atmosphere and how it could be modeled as a transfer function. Three experimental data sets are then presented, along with how the data sets are used to develop filter functions. Results are presented which show the outcome when the filter functions are applied to low-boom waveforms. The next chapter presents the Nonlinear Progressive Wave Equation (NPE) propagation algorithm, with enhancements including a new turbulence representation based on temperature fluctuations. Filter functions are derived from the NPE and are applied to low-boom waveforms. The last chapter is a discussion of the results obtained from both the experimental data sets and the NPE.

27 Figure 1.13: (a) A step shock wavefront after propagating numerically the equivalent of 1.6 s. The initial condition was a rippled wavefront, similar to Figure [1.12]. (b) The evolution of the shock profile along the axis of focus. Taken from Ref

28 Chapter 2 Loudness and Annoyance The overall goal of this dissertation is to find a way to incorporate the effects of realistic atmospheric turbulence into sonic boom waveforms. Once that is accomplished, a future task would be to quantify the loudness and public acceptability of the resulting turbulized waveforms. This would require human subjective testing of the waveforms to assess public acceptability. The work described here is to provide the waveforms necessary for subject testing. Though no subjective testing is performed, an existing metric is used as a starting point for quantifying loudness. Ideally, the metric used would be sufficiently robust such that subjective testing would not be necessary. However, given the existing ban on overland supersonic flight, government agencies are likely to permit overland supersonic flight only when provided with sufficient compelling evidence to do so. Such compelling evidence must include results from extensive subjective testing. The rest of this chapter will cover loudness, a metric for quantifying loudness, and subjective testing of sonic booms. 2.1 Loudness & Stevens Mk VII Perceived Level (PL) When describing how loud a sonic boom is, there are many words that a person could use to describe his or her experience. Often the words loud or annoying

29 18 are used to quantify sonic booms. Research by McCurdy suggests that subjective response to sonic boom is the same whether expressed in terms of loudness or in terms of annoyance. 22 Figure [2.1] shows the results from McCurdy s research. As one can see there is a high correlation between the subjective loudness and annoyance subjective measures. Figure 2.1: Comparison of mean annoyance scores with mean loudness scores by McCurdy. 22 There are many metrics that could be used to quantify the loudness of low-boom waveforms such as sound exposure level (SEL), A-weighted sound pressure level, or Stevens MK VII Perceived Level of noise. NASA Langley Research Center recently conducted a study with the specific intent of assessing human response to candidate low-boom waveforms (that did not include the effects of turbulence). The study also included waveforms based on classic N-waves. The results of the study indicated that Stevens MK VII Perceived Level of noise (PL) was found to be the best predictor of loudness of low-boom waveforms. 16 Consequently, Stevens MK VII Perceived Level of noise, abbreviated as PL, will be used throughout this work to quantify loudness. It should be noted that the term loudness captures subjective perception whereas the term level attempts to assign an objective number to an event. According to Sullivan, PL is the best metric for predicting the loudness of a low-boom sonic boom.

30 19 The NASA study also found that asymmetry effects due to differences between the front and rear shocks of the boom were insignificant for the booms used in their study. 16 Asymmetry will be discussed in more detail later in this chapter. It is worth noting that there are no readily available models for quantifying the loudness and annoyance of low level sonic booms heard indoors, although this is currently being studied. For the purposes of this work, Stevens Mk VII will be used when attempting to quantify the loudness of sonic boom waveforms with the assumption that such booms would be heard outdoors. Stevens Mk VII Perceived Level uses a set of frequency weighted contours that are experimentally based. The reference sound is a 1/3 octave band centered at 3150 Hz. The PL grows as the 2/3 power of sound pressure. This causes the perceived magnitude to double with each increase in level of 9 db. The frequency weighting contours extend down to 50 Hz, making it possible to calculate perceived 23, 24 levels of sonic booms and other noises. Figure [1.5] compares the PL of a traditional N-wave with three low-boom waveforms. According to the figure, the low-boom on the right would have a PL of approximately one quarter of the perceived magnitude of the traditional N-wave. The major difference between the two waveforms is the rise time and the initial shock maximum pressure. 2.2 Stevens MK VII Implementation This research made use of Fortran code written to determine Stevens Mk VII PL values from stored waveforms. The code outputs PL values on a decibel scale with units of db. The code was provided by Sullivan of NASA Langley Research Center. 16 It was complied on the local PC used in this research. A Matlab wrapper function was written to call the compiled Fortran code and pass the input waveform and parameters to the code. For additional details about the implementation of the Stevens Mk VII PL, see Appendix [B]. Figure [2.2] is an example of a symmetric sonic boom waveform. Figure [2.3] is an example of an asymmetric waveform. The results of Sullivan 16 indicate that asymmetric effects due to differences between the front and rear shocks of low-boom sonic booms were found to be not significant.

31 20 Figure 2.2: A symmetric sonic boom waveform. The author went through several iterations with Sullivan to verify that the compiled PL code was working properly by comparing the output of the author s instance of the code with the output of Sullivan s code. This consisted of running the same code with the same input waveform and parameters on different machines with different operating systems and hardware architectures. One of the waveforms that was passed back and forth for comparison was the measurement made at microphone location K from flight 21 of the Shaped Sonic Boom Experiment. The Shaped Sonic Boom Experiment will be discussed in more detail in Chapter 4. It was discovered that the two versions of the code (the author s and Sullivan s) initially had slightly different parameters used when defining the 1/3 octave bands. Once this difference was corrected, it was found that different hardware and compilers gave slightly different results. The differences were on the order of.001 db or smaller and thus were deemed insignificant.

32 21 Figure 2.3: An asymmetric sonic boom waveform used in a human perception study. Taken from Ref Turbulence and Human Perception of Sonic Boom One of the goals of this research is to provide researchers with low-boom waveforms that include realistic turbulence effects. These waveforms would then be used in human subjective testing to determine acceptability of the low-boom waveforms and therefore acceptability of the proposed supersonic aircraft. As mentioned above, subjective studies have already been conducted using low-boom waveforms. 16 Subjective studies will benefit from this research by being able to present more realistic waveforms to listeners than are currently available. Subjective testing usually is conducted in specially designed sonic boom simulators such as Gulfstream s Supersonic Acoustic Signature Simulator II. 25 Subjective testing requires accurate reproduction of sonic boom signatures including frequency range, amplitude, and phase. Most sonic boom simulators can reproduce frequencies down to 6 Hz or lower, well below normal hearing limits (f> 20 Hz). On the high end, sonic boom simulators reproduce frequencies up to between 6 and 7 khz. The Nyquist-Shannon sampling theorem states that the sampling rate must be at least 2 times the highest frequency of interest. 26 In the case of sonic boom simulator playback, signals must be sampled at a minimum of 14 khz.

33 Chapter Summary This chapter has discussed the metric used in quantifying the loudness of sonic boom waveforms as they would be perceived outdoors. In addition, a detailed description is given for how the metric is implemented along with the input parameters that are necessary for the calculations. The next chapter will provide a discussion on how the atmosphere might be modeled as a transfer function.

34 Chapter 3 The Atmosphere as a Transfer Function The purpose of this chapter is to introduce the atmosphere, the effects of the atmosphere as a transfer function, and to discuss specific methods of representing such a transfer function. A detailed description of turbulence will be provided in Chapter The Atmosphere and the Planetary Boundary Layer As mentioned in Chapter 1, the atmosphere can add spiking and rounding effects to sonic boom waveforms, and this is the primary way that the atmosphere acts on sonic booms. In fact most of the distortions occur in the portion of the sonic boom propagation path that is closest to the ground. The two primary mechanisms for creating turbulence in the atmosphere are wind and temperature fluctuations, and these are both caused by interaction with the ground surface. The wind must slow down near the surface due to the no slip condition, and it is also disturbed by objects (trees, buildings, etc.) on the ground. Thus the wind profile near the ground is not uniform, and this is one cause for turbulence. Secondly, the sun shining on the ground causes it to heat up. The ground in

35 24 turn heats the air closest to the ground. The warm air rises creating a convection condition. This thermal effect is another contributor to atmospheric turbulence. Wind and thermal effects in the atmosphere are concentrated near the earth s surface, and hence, the turbulence is strongest nearest the ground. The lowest layer of the atmosphere near the ground is called the planetary boundary layer (PBL), and this is where sonic booms distort the most. The thickness of the PBL changes during the day and is dependent on the weather. On a typical, dry sunny afternoon in the summer near Edwards, CA, where many sonic boom field experiments have been undertaken, the PBL is thinnest at sunrise when it is only a few hundred meters thick. As the day progresses and solar radiation increases, the PBL can grow to a thickness of some 10,000 feet (3048 m) by early afternoon. Overnight, the PBL shrinks back to being much thinner. This trend is seen in sonic boom recordings at Edwards. Typically one sees very little distortion in sonic boom waveforms measured in the early morning when the PBL is at its thinnest, and large distortions in the late morning and afternoon when the PBL is thickest. Pierce was one of the first to hypothesize that the distortions seen in sonic boom waveforms were caused by turbulence. 27 Another primary contributor to the relation between turbulence and sonic booms was Plotkin. 28, 29 It was later shown by Gionfriddo at Penn State that the amount of sonic boom distortion was, in fact, related to the thickness of the PBL. 30 Hence, we will consider the effects of atmospheric turbulence to be concentrated in the PBL for the remainder of this dissertation. 3.2 Transfer Function Concept Now, let s focus on ways to incorporate turbulence into clean sonic boom waveforms. One way to investigate the effect of turbulence on a sonic boom would be to numerically propagate the waveform of interest through the PBL. This would require a detailed description of the atmosphere and would require a large amount of computational resources. Such a possibility may exist in the future and will be discussed in the final chapter of this work. Another possibility would be to investigate the relationship between a sonic boom measurement made above the

36 25 PBL and the same sonic boom measurement made on the ground. Conceptually, it would be possible to estimate a transfer function which converts a sonic boom waveform measured at-altitude into the waveform measured on the ground, after it has propagated through turbulence. Figure [3.1] shows a diagram of this idea where the transfer function is a filter. Once the transfer function has been estimated, it can then be applied to any waveform to simulate the effects of propagation, provided the target waveform meets certain criteria. Figure 3.1: A transfer function which approximates the input-output relationship between a sonic boom measurement made above the planetary boundary layer and the same waveform measured on the ground after propagating through a turbulent atmosphere. The transfer function is a linear time-invariant approximation of all the processes affecting the waveform. This concept relies on several assumptions. The first assumption is that the method used to estimate the transfer function would be able to capture, or at least approximate, all the underlying physics of propagation. The next assumption is that all the pertinent turbulence would occur below the measurement altitude. Another

37 26 assumption is that the energy lost between the waveform measured at-altitude and the waveform measured on the ground is negligible. Lastly, this technique would have to assume that the sonic boom measurement made at-altitude corresponded to the same sonic boom that is measured on the ground. These assumptions provide the framework for the transfer function concept. The transfer function used in this research has some limitations based on the assumptions made above. These limitations are described below. Prior to that discussion, a brief description of nonlinear acoustics provides insight into sonic boom propagation Nonlinear Acoustics The linear wave equation is derived from the equations of state, momentum, and continuity. The second- and higher-order terms from these initial equations are neglected in the derivation process. As the acoustic pressure or particle velocity increase, the neglected terms have increasing significance and their omission leads to increased error. As stated by Gee, the amplitude at which the second-order terms, for example, start to become important is dependent on a number of variables, such as frequency, type of geometrical spreading present, and the properties of the medium. 31 If the second order terms of the original equations are carried through the derivation, the result is a nonlinear wave equation. In the case of lossless, one-way propagation of a finite-amplitude sound wave with acoustic particle velocity, u, it can be shown [32] that the (nonlinear) speed of sound can be represented in the absence of shocks as c = c 0 + βu (3.1) where c 0 is the small-signal (ambient) sound speed and β is the (nondimensional) coefficient of nonlinearity of the medium (β = for air). Again from [32], it can be shown that the instantaneous rate at which a discontinuity advances, dv, is dv = β p. (3.2) 2ρ 0 c 0

38 27 Here is it assumed that the discontinuity is a jump from a zero value of acoustic pressure to a peak value of p. Here c 0 is the small-signal sound speed, β is the (nondimensional) coefficient of nonlinearity of the medium, and ρ 0 is the ambient density. In the case of a sonic boom, the bow shock and the tail shock advance in opposite directions because the tail shock travels slower than the ambient speed of sound. This difference in the speed of sound causes the sonic boom to stretch as it propagates. One consequence of Eq. [3.1] is that the amplitude-dependent speed of sound causes the waveform to distort as the waveform propagates nonlinearly. Figure [1.1 shows this process of distortion. The three waveforms depict various states in the sonic boom evolution. Close to the aircraft, the waveform has many peaks. These peaks travel faster than the ambient speed of sound and stack up on each other, or coalesce. The middle waveform has just two peaks, as the other peaks have all coalesced into just the two. By the time the waveform reaches the ground, the traditional N-wave shape is observed Transfer Function Limitations One key limitation lies in the way the transfer function represents the change between the two waveforms. The amount of turbulent propagation is small compared to the overall propagation length. The waveform is initially generated at an altitude of approximately 32,000 ft (9,753 m). We are considering the propagation path of the last several thousand feet, from an altitude of approximately 5,000 ft (1,524 m). Taking in to account the Mach Angle, the propagation distance of interest is approximately 8,000 ft (2,438 m) as compared to the total propagation distance, which is approximately 45,000 ft (13,716 m). Although the propagation is nonlinear, we approximate the turbulent propagation as linear because it is short ( 17) compared to the total propagation distance. One of the consequences of this assumption is that the input waveforms (measurements at-altitude) used to generate the transfer functions must have rise times which are shorter than or equal to the rise times of the output waveforms (measurements on the ground). In terms of frequency, the transfer function cannot add high-frequency energy to a waveform unless there is already some high frequency

39 28 energy present in the initial waveform. Another consequence is that the waveforms to which the transfer functions are applied must have rise times similar to the rise times of the output waveforms used when generating the transfer functions. Another limitation of the transfer function technique is the assumption that all the turbulence is below the measurement made at-altitude. This is probably not the case, as some turbulence exists within the atmosphere at 35,000 feet (10,668 m) above ground level and beyond. Anyone who has ever flown in an aircraft for an extended period has experienced atmospheric turbulence to some degree. However, the turbulence of interest in this research is limited to the turbulent PBL which can range from 100 m to several km in height above the surface of the earth. 33 Though turbulence above the PBL does exist, it does not have as large an impact on sonic boom distortions as the turbulence within the PBL, as indicated by sonic booms measured by gliders. 34 The last assumption touches on the notion that the sonic boom measured on the ground be the same sonic boom as was measured at-altitude, the only difference being that the second measurement was made after the waveform has propagated through turbulence. This assumption requires that the geometry of the associated measurements line up so that the same sonic boom is measured at both points. One way to think of the sonic boom wavefront is as though it were a ray of light. The ray leaves the supersonic aircraft, propagates down toward the ground, intersects the measurement point at-altitude, and then intersects the ground measurement point. If for some reason the ray does not intersect either the measurement point at-altitude or the measurement point on the ground, then the transfer function will not be an accurate representation of the turbulent propagation. Three techniques for estimating the transfer function between the at-altitude measurement and the ground measurement were investigated. The first technique was based on Welch s averaged periodogram method, but this method was quickly abandoned since the method assumes periodicity in a signal. The second technique was based on a built-in Matlab function which uses polynomial long division in the z domain. The third technique was developed as part of this research and is referred to as Matrix Deconvolution. Each of these techniques will be discussed later in this chapter. There may exist other ways to estimate transfer functions which represent

40 29 the transformation from a sonic boom waveform measured without the effects of turbulence to a sonic boom waveform measured with the effects of turbulence. Possibilities include a Hilbert Transform, an active and or nonlinear transfer function, or an Infinite Impulse Response (IIR) function although these techniques were not considered in the present work due to time constraints. 3.3 Transfer Function Based on Matlab s DECONV Function The second method considered to estimate turbulence transfer functions was based on Matlab s 35 built in function deconv. The Matlab function implemented a direct form II transposed implementation of the standard difference equation, which performs a z-transform of the signals and then estimates a transfer function in the z-domain using polynomial long division. The technique did not work because the polynomial long division did not converge. A similar example would be the outcome of dividing 1 by 3. The result is a repeating decimal,. 3, which does not ever converge. The polynomial division did not converge, so the technique estimated ever increasing coefficients. The largest coefficients were on the order of A more detailed description of z-transforms and the standard difference equation can be found in Oppenheim and Schafer Transfer Function Based on Matrix Deconvolution Matrix Deconvolution is the process of estimating a transfer function in the time domain. No averaging is done. Each transfer function represents the change between a waveform measurement made at-altitude and a corresponding waveform measurement made on the ground. (There is a time delay between the measurement made at-altitude and the measurement made on the ground.) The transfer function creation process is best described as matrix based convolution in reverse, or matrix deconvolution.

41 30 Equation (3.3) gives the definition for the convolution sum, y[n] = N 1 k=0 h[k]x[n k], (3.3) where x and h are sampled input signals, N is the number of samples in the signal h, and y is the sampled signal output. One implicit assumption in Equation (3.3) is that values of x with negative indices are zero. Equation (3.4) shows the first four terms of Equation (3.3) written out term by term. y[0] = h[0]x[0] y[1] = h[0]x[1] + h[1]x[0] y[2] = h[0]x[2] + h[1]x[1] + h[2]x[0] y[3] = h[0]x[3] + h[1]x[2] + h[2]x[1] + h[3]x[0]. (3.4) Equation (3.4) can be rewritten in matrix vector form as x x 1 x x 2 x 1 x x 3 x 2 x 1 x x 4 x 3 x 2 x 1 x 0 0 x 5 x 4 x 3 x 2 x 1 x x M N x M N x M N x M N x M N. h 0 h 1 h 2. h N 1 = y 0 y 1 y 2 y 3 y 4 y 5. y M x M N (3.5) The matrix has dimensions M by N, where M is the length of the vector y

42 31 and N is the length of the vector h. The vector x has a length of M-N+1 and is zeropadded to have a total length of M to match the length of the vector y. When computing the convolution sum, the known quantities are the vectors x and h. In the case of matrix deconvolution, the known quantities are the vectors x and y. The unknown quantity is the vector h. The measurement at-altitude, x could be taken by a microphone mounted to a sailplane, as is used later in this dissertation, or it could be made by a microphone attached to a tethered or free floating balloon. Even a tall tower could be used. The measurement made on the ground is y. The vector h is determined by a two step process. The first step is to determine the pseudo-inverse by using a least squares fit to the data represented by (3.5). In the second step, the pseudo-inverse is multiplied by the ground measurement y, yielding the vector h which has a length, N. The vector h, when convolved with the at-altitude measurement x, yields an estimate of the ground measurement y and provides a way to check the accuracy of the transfer function. Transfer function verification will be discussed later in this chapter. The elements of the vector h could be described as the coefficients of a finite impulse response (FIR) filter. The filter is a set of weights for scaled and delayed copies of the original at-altitude sonic boom. Another way to view the filter coefficients is as though each coefficient represents the arrival of the original waveform delayed in time due to a unique propagation path that has caused a unique scale factor. The filter coefficients can be applied to suitable low-boom waveforms to approximate the effects of the atmosphere. The filter would approximate the atmosphere that existed between the two measurements points at that particular time and space propagation path when the original waveforms, used to generate the filter, were measured. Applying the filter to a low-boom waveform would approximate the propagation as though the low-boom waveform had propagated through that portion of the atmosphere, on that exact propagation path, on that day. Linearity is the primary assumption in developing these filters. In reality some nonlinear steepening and stretching will still be occurring as the waveform propagates through the lowest (turbulent) portion of the atmosphere. Assuming the waveform propagates at nominally 1,125 ft/s (343 m/s) for a distance of 8000 ft (2,438 m), then the waveform will stretch for about 7.1 s. Peak N-wave amplitudes

43 32 measured by the sailplane were on the order of.5000 psf, or Pa. Given these values and Eq. [3.2], the waveform will stretch approximately.5 m as it propagates through the last portion of the atmosphere. The separation distance between peaks at the time the sailplane measures the waveform is about 30 m. This work assumes that the additional stretching between the at-altitude microphone position and the ground may be neglected or absorbed into the linear approximation. 3.5 Filter Function Verification The purpose of generating filter functions is to approximate the effects of the atmosphere on sonic booms. Once the filters are estimated, they can be applied as post processing functions to low-boom waveforms to create turbulized waveforms which would be more realistic representations of the waveforms which would reach the ground. The turbulized waveforms would then be used in subjective testing to evaluate human response to the proposed low-boom aircraft, as discussed in Chapter 2. Perceived loudness (PL) is the metric by which the turbulized low-boom waveforms will be evaluated in this work. It therefore makes sense to use this metric to also evaluate the effectiveness of the filters being estimated. The efficacy of each filter will be determined by comparing the PLdB of the ground measurement with the PLdB of the ground measurement estimate. The ground measurement estimate is obtained by convolving the filter with the measurement made at-altitude. 3.6 Filter Limitations Approximating the atmosphere as an FIR filter has several limitations. One of these is that the FIR filter is strictly a passive system that does not add high-frequency energy to the waveform unless it is already present. In reality, turbulence moves energy by focusing and defocusing the wavefront which can lead to a localized increase in high frequency energy. 21 Another limitation is that the FIR filters are created using N-waves which have short rise times. If the filters are then applied to low-boom waveforms which do not have similar rise times, no high-frequency variations will be seen in the output waveforms because no high-frequency energy

44 33 existed in the original input waveform. 3.7 Applying Filter Functions One of the assumptions made above is that the energy lost by a waveform as it propagates through the turbulent portion of the atmosphere is negligible. When a filter is applied to a low-boom waveform, that same assumption holds. Consequently, whenever a filter is applied to a low-boom waveform, the energy of the new waveform is computed and compared with the energy of the original low-boom waveform. The filter coefficients are then scaled by the square root of the ratio of the original low-boom waveform energy divided by the turbulized low-boom waveform energy. This process ensures that the filter preserves the total energy in the sonic boom waveform, although it is known that the focusing and defocusing due to turbulence can slightly increase or decrease the local waveform energy. 3.8 Chapter Summary This chapter began with a description of the planetary boundary layer, and an explanation as to why one could ascribe the effects of turbulence to this portion of the sonic boom propagation path. The transfer function concept was then discussed both conceptually and practically. Techniques for estimating such an appropriate FIR filter were discussed. Lastly, a method for validating the filter was presented along with a discussion of some of the limitations of such a filter. The next chapter discusses three data sets used in the generation of filters and the results obtained when the filters are applied to various low-boom waveforms.

45 Chapter 4 Filters from Experimental Data Sets Three experimental data sets were used to generate filter functions: The Shaped Sonic Boom Experiment (SSBE), a data set taken at NASA Dryden in 2006, and a data set taken at NASA Dryden in In each case, supersonic aircraft flew a predetermined flight track in which sonic booms were generated and then measured by both a sailplane and on the ground. This chapter will discuss the three data sets and the process by which filters were estimated. The filters will then be applied to several low-boom waveforms and PL results will be presented. The SSBE data set will be presented first. The Dryden 2006 and 2007 data sets will then be presented. These three experiments were conducted for a variety of reasons, one of which was to collect data which could be used to generate filter functions, as described previously in Chapter 3. In each experiment, a US Air Force Test Pilot School Super Blanik L-23 sailplane was fitted with one or two Brüel and Kjær (B&K) 4193 microphones and each microphone was fitted with a bullet nose attachment and windscreen to reduce wind noise. The microphone was attached to the port wing tip of the sailplane for the Shaped Sonic Boom Experiment. 34 For the 2006 and 2007 experiments, one microphone was fitted to the nose of the sailplane and a second microphone was attached to the wing. 36 The results presented in this dissertation used only data taken from the wing mounted microphone from the SSBE and the nose mounted microphone for the 2006 and 2007 data sets. (The wing mounted microphones for the 2006 and 2007 data sets were configured to record low amplitude N-waves and did not yield useable measurements of traditional N-waves. 18 )

46 35 All three experiments were conducted within the supersonic corridor associated with NASA Dryden Flight Research Center (DFRC) at Edwards Air Force Base (EAFB) in Edwards, CA. The airspace in the vicinity of EAFB is one of the few designated supersonic corridors within the continental US where military aircraft are permitted to fly at supersonic speeds. Sonic booms are a common occurrence for the people who live and work near EAFB. 4.1 The Shaped Sonic Boom Experiment (SSBE) The SSBE was a series of 21 data flights conducted at NASA Dryden Flight Research Center, Edwards Air Force Base, California, in January of 2004 to determine if it was possible for a shaped sonic boom to persist to the ground through a realistic atmosphere, and thus better understand the generation, propagation, and impact of a shaped sonic boom. One specific task of the experiment was to fly the SSBD followed 30 seconds later by a regular F5E in an effort to gain insight into the effects of turbulence on sonic booms generated by both aircraft. 13 In addition to the supersonic aircraft, a sailplane was part of the experiment. The purpose of the sailplane was to record the sonic booms above the PBL. The sailplane microphone data was sampled at 8.33 khz and low pass filtered at 2.2 khz. The sailplane pilot was given a portable GPS unit and instructed to be at a specific way point when signaled by the supersonic aircraft. The sonic boom measurements made on the ground are described below. Ground measurements were made using 4 different data acquisition systems, ranging in sampling frequency from 8.33 khz up to 50 khz. Measurements were made at 500 foot (152 m) intervals along an old road at NASA Dryden Flight Research Center. 37 Figure [4.1] shows the microphone positions. The microphone sites were named A (Alpha) through Z (Zulu) from east to west, in the order they would be overflown by the supersonic aircraft. There were also several extra microphones in the center of the microphone array.

47 36 Figure 4.1: Measurement locations made at NASA Dryden Flight Research Center during the Shaped Sonic Boom Experiment. The microphones were spaced 500 feet apart and ran parallel to the North Base runway. Taken from Ref SSBD Sailplane Waveforms Sailplane measurements of SSBD sonic booms were multi-stepped. The waveforms were still experiencing nonlinear steepening when measured at altitude. Figure [4.2] shows the raw (non-doppler corrected) SSBD boom measured by the sailplane during flight 21. The step function at.04 s is a consequence of the nonlinear boom shaping. Similar step functions were present in all sailplane measurements of SSBD booms. The multi-step nature of the SSBD booms made it impossible for the filter method to converge. The method did not converge because the rise time of the ground measurement was shorter than the rise time of the at-altitude measurement. Put in terms of frequency, the input waveform (at-altitude measurement) had less high-frequency content than the output waveform (ground measurement). The linear filter method, which is an approximation based on the least squares fit, cannot add high-frequencies to the input waveform and consequently wasn t able to converge on a solution F5E Waveforms Even though SSBD waveforms were not suitable for filter generation, sailplane measurements of the traditional F5E N-waves proved to be useful in estimating

48 Pressure (psf) Time (s) Figure 4.2: Raw sonic boom measurement of the SSBD made by the sailplane during flight 21, without Doppler correction. Note the step function at approximately.035 s. filter functions. As mentioned above, the SSBE consisted of a number of supersonic flights. This work focuses on the data obtained for flight 21. Flight 21 was investigated for several reasons: 1. Flight 21 had the most complete data available, including ground data, sailplane data, and global positioning system (GPS) data. 2. Flight 21 occurred at 10 a.m. PST on January 15, 2004, when turbulence was present but not excessive. Earlier flights experienced little turbulent distortion. Later flights exhibited large distortions. The rest of this section makes use of F5E waveforms from flight 21 of the SSBE.

49 Meteorological Conditions Meteorological data taken during the SSBE was limited to weather balloons, which were periodically launched from Dryden Flight Research Center. A weather balloon was launched on the day of flight 21 at 10 a.m. PST. Figure [4.3] shows temperature and relative humidity measured by the balloon as a function of altitude. Figure [4.4] shows the wind profile as a function of altitude. Unfortunately, weather conditions were not measured at each microphone location, so there was no way to correlate the weather data to the filter functions Atmospheric Conditions Altitude (meters) Temperature Relative Humidity Temp ( C), R. H. () Figure 4.3: Temperature and relative humidity as a function of altitude at Dryden Flight Research Center. Measured on 15 January 2004 at 10:00 a.m. PST by a weather balloon.

50 Alt. (m) 39 Wind speed (m/s) North South West East Figure 4.4: Wind speed profile, in m/s, at Dryden Flight Research Center on 15 January 2004 at 10:00 a.m. PST, as measured by a weather balloon. The black arrow points north Waveform Preparation Prior to estimating the filter functions, the at-altitude and ground measurements were preprocessed. The sailplane waveform was corrected for the Doppler shift associated with the motion of the sailplane, with the technique as described below. The sailplane waveform was then scaled by a factor of two to account for the fact that the ground waveforms experience pressure doubling due to the presumed hard wall boundary at the ground. Then the first.0394 s of the waveform was insignificant pre-boom noise and was removed. The ground waveform was down sampled to match the sampling frequency of the sailplane (8330 Hz) if necessary. Then the first.0394 s of the waveform were

51 40 removed. Then the ground waveform was zero padded to include an additional one second of zeros. This was done to increase the length of the ground waveform Doppler Correction Next, the sailplane waveform was modified using a Doppler correction into what would have been measured by a stationary microphone at the location and altitude of the sailplane. Several simplifying assumptions were made when determining the Doppler correction for the SSBE data set. They include the following: The supersonic aircraft was flying straight and level. The sailplane was flying straight and level, directly underneath the supersonic aircraft, at the time it recorded the boom. The sonic boom exhibited straight ray propagation. The Mach Angle µ was computed based on the flight velocity of the F5E. The sonic boom ray velocity vector v ray, was then computed as 1 c 2 µ v ray = 3 c 2 µ, (4.1) c cos(sin 1 ( 1 )) µ where µ is the Mach Angle and c was the ambient speed of sound, taken to be 343 m/s. (The speed of sound was taken to be constant because the data provided by the weather balloon was not near the ground microphones. This value is within 2 of the speed of sound based on the measurements made by the balloon.) The vector components of v ray are based on the north, east, and down velocities in m/s. The sailplane velocity, v g was then projected onto v ray, yielding the projection of the sailplane velocity onto the ray velocity, ( ) vray v g v gproj = v v ray v ray T ray (4.2) where T indicates transpose. The waveform was then resampled based on ratio = ( v ray v gproj ) ( v ray v gproj ) T c. (4.3)

52 41 The ratio was less than unity for the SSBE, so the waveform was down-sampled, to account for the fact that the sailplane was flying in the same heading as the F5E. Figure [4.5] shows the sailplane measurement of the F5E sonic boom after the Doppler correction before after Pressure (psf) Time (s) Figure 4.5: Sailplane measurement of the F5E sonic boom recorded at approximately 10 a.m. PST on January 15, The red waveform is the raw data (before Doppler correction). The blue waveform has been corrected for motion Filter Estimation by Matrix Deconvolution The filter function coefficients were estimated after the at-altitude and ground waveforms were properly prepared, as described above. The length of the filter to be estimated was determined by the difference between the ground measurement

53 42 length and the at-altitude measurement length. The ground measurement was zeropadded, which increased the number of elements in the filter. Each filter estimate represented the transfer function between the sailplane measurement and the measurement made on the ground. The same sailplane measurement was used for each filter estimate SSBE Filter Agreement Once the filters were estimated, they were checked to see how well they were able to convert the input signal (the at-altitude measurement) into the output signal (the ground measurement). The filters were checked by convolving the filter with the input measurement, computing the PL value for the resulting waveform, and then comparing that PL value of the original ground waveform with the PL value of the ground estimate. Figure [4.6] shows the difference in PL between the estimated ground waveform (based on the filter and the input signal) and the PL value of the actual ground measurement. The significance of Figure [4.6] is that it shows that most of the filters were able to reproduce the ground waveform to within.2 db. The filter created from microphone location E is an outlier, with a PL value of nearly 1.4. The reason for the poor agreement is due to a pressure discontinuity in the start of the waveform. Overall, the filters do a good job of recreating the ground waveform, as differences in PL of.2 would be imperceptible. (A noticeable difference would be 1 or 3 db, depending on how the measurement was made. 38 ) It should be noted that the filters generated here were limited because of the 2.2 khz low pass filter installed in the data acquisition system of the sailplane. Figure [4.7] shows an example of how well one of the SSBE filters is able to reproduce the ground measurement. The output of the filter begins to diverge from the ground measurement in the vicinity of 25 ms, which is where the filter was truncated. The filter was truncated at 25 ms. This number was chosen because it was within the temporal window of the ear based on the law of first wavefronts, as discussed by Blauert. 39 The implication is that the filter coefficients are roughly analogous to scaled and delayed arrivals of the original waveform. Arrivals within the temporal window of the ear would be perceived as the same event. Later in this

54 43 PL, db ML02 M L I J K M206 M205 M204 M203 M202 M201 H G F E D C B Z Y X W V U T S R Q204 Q P O N MR03 microphone Figure 4.6: Difference between the PL values of the ground measurements and the PL values of the ground estimates. Positive values indicate the filter estimate yielded a waveform which had a larger PL value than the actual measurement and negative values indicate the filter estimate yielded a waveform which had a lower PL value. research, a value of 50 ms was used, as that is closer to the dividing line between two signals being perceived as one continuous event or two separate events. It should be noted that the temporal window of the ear is only an estimate. The impulse response of a typical filter function can be characterized as a primary spike that quickly decays. The point where the filter is truncated is therefore somewhat arbitrary as the filter coefficients oscillate about zero (with a small amplitude) after the initial spike. Notice that the minimum of the filter output is greater in magnitude than the

55 Microphone location B ground measurement filter output Pressure (psf) Time (s) Figure 4.7: Filter agreement for SSBE microphone location B, flight 21, using 25 ms of filter coefficients. ground measurement, i.e. the signature s most negative pressure is lower for the filter output. This is a likely consequence of using a linear representation of a nonlinear process. Additionally, the rear shock of the filter output occurs prematurely because the linear approximation does not account for nonlinear waveform stretching. The turbulent fluctuations after the rear shock of the filter output are similar to the ground measurement turbulent fluctuations Applying the SSBE Filters Figure [4.8] demonstrates three examples of incorporating turbulent effects into a low-boom waveform supplied by Gulfstream Aerospace Corporation, which appeared as Figure [1.9] in Chapter 1. The filters in Figure [4.8] were created based on measurements at microphone locations B, E, and X. The turbulized low-boom

56 45 waveform is band limited between 0.1 Hz and 2,200 Hz, making it inadequate for subjective studies, i.e. listening tests. It is important to remember that while the sailplane measured data at 8.33 khz, a low pass filter was installed in between the microphone and the recording device, effectively reducing the upper bandwidth frequency limit from 4 khz down to 2.2 khz. 37 Included in Figure [4.8] is the filter estimate based on microphone location E. This filter demonstrated the poorest agreement between the actual and estimate ground waveform. The poor performance was due to a small discontinuity in the ground waveform. The implication is that even though the agreement was poor, the variation in the low-boom waveform was minimal Lowboom vs. Turb. Lowboom Ground Microphone Locations B, E, & X lowboom filter B filter E filter X 0.4 Pressure (psf) Time (s) Figure 4.8: Closeup of 25 ms of filter coefficients applied to the low-boom waveform, filters created from measurements made at SSBE microphone locations B, E, and X. Figure [4.8] shows one of the limitations of the filter function technique. In this

57 46 case, the low-boom waveform that has been turbulized has a long rise time and consequently very little high frequency energy. When the filters are applied to this low-boom waveform, the filters only cause the waveform to become more rounded, resulting in lower PL values because there is no high-frequency energy in the initial waveform to excite the high-frequency portion of the filter. If the waveform were to have a shorter rise time, it is likely that at least some of the filters would result in an increase in Perceived Level. Figure [4.9] shows the effects of all the SSBE filters when applied to the lowboom waveform provided by Gulfstream Aerospace Corporation. The figure shows the difference in PL between the CFD based low-boom waveform that would be heard on the ground without the effects of turbulence and the CFD based low-boom waveform that would be heard on the ground with the effects of turbulence, for each of the microphone locations where data was collected and a filter was estimated. The significance of Figure [4.9] is to reaffirm what was stated earlier. Low-boom waveforms that have little high-frequency energy, prior to being turbulized by applying turbulent filters, are not likely to have increased levels of Perceived Level. 4.2 NASA Dryden 2006 Sonic boom measurements were made at a house near NASA Dryden Flight Research Center on June 22, The ground measurements were made by NASA Langley Research Center personnel at a ranch style home. 18, 40 The house was instrumented with nearly 300 microphones and accelerometer sensors to determine the incident pressure and the structural response of the building. One microphone was set up in a field adjacent to the house. This lone microphone provided the waveforms used in this data set. Figure [4.10] shows a picture of the ground microphone. For information regarding the meteorological data taken during the 2006 experiment, the reader is directed to Appendix I of Klos work. 18 Source aircraft for the sonic booms were NASA Dryden s two research F18s- 860 and 862. Again a sailplane was fitted with a microphone. The sailplane instrumentation was enhanced to sample at a rate of 24 khz. Ground measurements were sampled at 25.6 khz. For this experiment, the F18s were flying 381 m/s at a heading of 270 degrees. The file numbers corresponding to the sonic boom

58 PL, db ML02 M L I J K M206 M205 M204 M203 M202 M201 H G F E D C B Z Y X W V U T S R Q204 Q P O N MR03 microphone Figure 4.9: Difference between the PL value of the turbulized low-boom and the original low-boom waveform. The filters all yielded negative values, indicating that the PL values were all reduced by the filters. measurements that were investigated were: 603, 606, 609, 610, and 612. For simplification, these file numbers will be referred to as flight numbers throughout the rest of this dissertation. The five flights mere made over a period of approximately 30 minutes Sailplane Waveform Preparation Before a filter could be estimated from the sailplane data, the waveforms needed to be preprocessed. This involved several steps. The first step was to remove the pressure variation due to changes in altitude of the sailplane. The next step

59 48 Figure 4.10: Picture of the ground microphone used in determining filter functions. Note the 2 ft x 2 ft (0.6 m x 0.6 m) ground board. 18 was to apply a low frequency correction to extend the low frequency range of the measurement microphone. Then a Doppler correction was made to the waveform to account for the motion of the sailplane. The last step was to truncate the waveform. This process is outlined below Altitude Pressure Correction This subsection discusses the removal of the slow background pressure changes in the sailplane waveforms. These slow background changes result from fluctuations in sailplane altitude away from a constant descent rate. In order to remove them, the sonic boom was temporarily removed from the waveform and then the remaining signal was fit to a 5th order polynomial to simulate the slow fluctuations. The sonic boom was removed by plotting the waveform and manually picking points before and after the boom. A linear fit between the the two points was then temporarily inserted into the waveform. The result from the curve fit was then removed from the original waveform (after the boom had been reinserted). Figure [4.11] shows an example of the sailplane waveform before and after removing the background pressure fluctuations due changes in altitude. A small DC offset has been removed. Once the background pressure fluctuations were removed, the next step was to extend the low frequency performance of the measurement.

60 Before correction After correction 0.4 Pressure (psf) Time (s) Figure 4.11: Background pressure variations removed from the first sailplane measurement made on June 22, Note the small DC offset which has been removed Low Frequency Microphone Correction The microphones used to collect the data in the current study were B&K 1/2 inch pressure microphones, model These microphones are capable of measuring acoustic signals down to 1 Hz. Sonic booms contain energy below 1 Hz. B&K sells a low frequency coupler which improves low frequency performance of their B&K 4193 microphone down to.1 Hz but at the expense of dynamic range. (Such couplers were not used in the present study.) The low frequency coupler moves the pole associated with the microphone preamplifier from.57 Hz or.51 Hz down to.1 Hz, effectively extending the low frequency range of the microphone, but as a tradeoff the microphone loses dynamic range. Marston and Gabrielson have developed a numerical technique to mimic the effect of the low frequency coupler, without the loss of dynamic range. 41 Their technique applies a filter which numerically represents the electrical components contained

61 50 within the low frequency coupler and their interaction with the microphone and preamplifier. This low frequency correction filter can be applied to sonic boom waveforms after the measurements are made, providing sonic boom measurements that have extended low frequency content, as though the low frequency coupler were used to make the measurement, without the loss in dynamic range. Their method applies two filters using values representing the tolerance range of the components in the coupler. The resulting two new waveforms represent the range of what could have been measured, had the low frequency coupler been used. Figure [4.12] shows the sonic boom measurement both with and without the low frequency correction, based on the technique of Marston and Gabrielson. The original waveform is in black. The other two waveforms were computed based on the tolerances of the low frequency coupler. The two corrected waveforms were averaged and the averaged result was then used in the next step of the waveform preprocessing, the Doppler correction No correction f0 = 0.57 f0 = 0.51 Pressure (psf) Time (s) Figure 4.12: Low frequency microphone correction applied to the flight 603 sailplane measurement made on June 22, 2006.

62 51 Figure [4.12] shows the advantage gained by applying the low frequency correction method of Marston and Gabrielson. The corrected waveforms have better low frequency response, characterized by the fact that the corrected waveforms cross from positive to negative pressure later in time than the waveform without the correction Doppler Correction Next, the sailplane waveform is modified using a Doppler correction into what would be measured by a stationary microphone at the location and altitude of the sailplane. This process is completed in the same manner as described above in section Waveform Truncation The last step in preparing the sailplane waveform is to truncate the boom. In order to truncate the boom, the point of inflection associated with the bow shock is determined. The point of inflection associated with the tail shock is then determined. The difference in time between these two points defines the boom duration. The start of the waveform is taken to be before the bow shock point of inflection. The amount of the waveform included prior to the point of inflection is equal to 1/20 of the boom duration. Figure [4.13] shows the sailplane waveform after all the processing for flight 603. The single black arrow shows the point of inflection of the bow shock. The double arrows show the boom duration, used in determining the amount of the waveform to be included prior to the bow shock point of inflection. Only the positive pressure portion of the sailplane waveform is used in estimating the filter. It should be noted that the ground boom has pre-boom noise of a longer duration than the sailplane boom. This difference ensures that the bow shock of the sailplane boom occurs before the bow shock of the ground boom. This difference is necessary so that the system is causal. If the ground boom occurred before the sailplane boom, (when comparing the waveforms after the preprocessing) the solver would not converge on a meaningful solution. Figure [4.14] shows an example of the first.025 s of the sailplane and the ground waveforms.

63 Pressure (psf) Time (s) Figure 4.13: Waveform truncation for the sailplane measurement from Dryden 2006, flight 603. The single black arrow shows the point of inflection associated with the bow shock. The double arrows show the boom duration. The dashed vertical lines indicate the extent of the waveform used to determine the filter Ground waveform preparation The ground waveform is conditioned in much the same way as the sailplane waveform. The only major differences are that the ground waveform has no altitude pressure correction or Doppler correction and that the ground waveform has more pre-boom noise than the sailplane waveform. Lastly, the ground waveform was downsampled from 25.6 khz to 24 khz, so that it matched the sampling frequency of the sailplane waveforms.

64 Sailplane Boom Ground Boom 0.8 Pressure (psf) Time (s) Figure 4.14: A closeup of the sailplane and ground waveforms measured for NASA Dryden 2006 flight 606. Note how the pressure rise of the sailplane waveform starts before the ground waveform Low Frequency Microphone Correction The first step in preparing the ground waveform for deconvolution is to apply the BK4193 correction filter. 41 This process is completed in the same manner as described above in section Waveform Truncation Lastly, the ground waveform is truncated. To do this, first the point of inflection associated with the bow shock is determined. Then the point of inflection of the tail shock is determined. The difference in time between these two points is the

65 Pressure (psf) Time (s) Figure 4.15: Waveform truncation for the ground measurement from Dryden 2006, flight 603. The single black arrow shows the point of inflection associated with the bow shock. The double arrows show the boom duration. The dashed vertical lines indicate the extent of the waveform used to determine the filter. boom duration. The start of the waveform is taken to be the point in time which is 1/15 of the boom duration earlier than the bow shock point of inflection. Note that this value was chosen arbitrarily. Figure [4.15] shows a diagram of the waveform. The ground waveform length is equal to 3 times the boom duration plus 1/15 of the ground boom duration for pre-boom noise Filter Estimation by Matrix Deconvolution Once the waveforms have been properly conditioned, they can be used in the matrix deconvolution method, as described in section 3.4. The Matlab code can be found in Appendix B.1. The filters that were generated from the Dryden 2006 data set

66 55 were based on flights 603, 606, 609, 610, and 612. These filters will be referred to as α, β, γ, δ, and ɛ Filter Agreement Once the filters were estimated, they were checked to see how well they were able to convert the input signal (the at-altitude measurement) into the output signal (the ground measurement). The filters were checked by convolving the filter with the input measurement, computing the PL value for the resulting waveform, and then comparing that PL value of the original ground waveform with the PL value of the ground estimate. Figure [4.16] shows the difference in PL between the estimated ground waveform (based on the filter and the input signal) and the PL value of the actual ground measurement. The significance of Figure [4.16] is that it shows that the filters were able to reproduce the ground waveform to within.2 db. The filters do a good job of recreating the ground waveforms, as differences in PL of.2 db would not be audible. Figure [4.17] shows the agreement between the ground measurement and the ground measurement estimate based on filter β for flight 606. The reconstruction is excellent, as indicated by a difference in PL of less than.16 db Applying the 2006 Filters Figure [4.18] shows the result when a CFD based low-boom waveform is convolved with the five filters obtained from the Dryden 2006 data set. Only 50 ms of filter coefficients are used in the convolution. 50 ms worth of filter coefficients were chosen because this corresponds to one estimate of the temporal window of the ear. 39 Additionally, the filter coefficients had decayed to nearly zero by that point. Earlier efforts used 25 ms of filter coefficients. The amount of filter coefficients used was increased to 50 ms because that value is closer to the dividing line between two signals being perceived as one continuous event or two separate events, as mentioned above. The turbulized low-boom waveforms show considerable variability. The large range of pressure values is due to the fact that the sailplane was not always in position to record the sonic boom that hit the ground microphone. Consequently,

67 PL, db α β γ δ ε Figure 4.16: Difference between the PL values of the ground measurements and the PL values of the ground estimates. Positive values indicate the filter estimate yielded a waveform which had a larger PL value than the actual measurement. the waveform measured at-altitude was rarely on the same ray path as the waveform measured on the ground. Note that sailplane positioning was much improved in the 2007 test. Figure [4.19] shows the Perceived Level (PL) values when the 2006 filters are convolved with 6 low-boom waveforms. The low-boom represented in the first column corresponds to the results shown in Figure [4.18]. The colors of the symbols in Figure [4.19] (corresponding to the different filters) match the colors of the different waveforms in Figure [4.18] (also corresponding to the different filters). The variability in the PL values shown in Figure [4.19] is indicative of what is often seen in field measurements of sonic booms. Some of the waveforms are peaked (leading to an increase in PL), but most of them are rounded (leading to a

68 glider waveform ground waveform filtered reconstruction Pressure (psf) Time (s) Figure 4.17: Filter agreement for Dryden 2006 filter β. decrease in PL). This limited set of filters is by no means showing the complete statistical variability possible from different realizations of the atmosphere. 4.3 NASA Dryden 2007 In 2007, a third supersonic flight experiment was conducted. A vacant house was outfitted with over 100 microphones and accelerometers. Additionally, a 10 m tall microphone tower approximately 130 m from the house held an array of ten microphones at 0, 1.2, 2, 3, 4, 5, 6, 7, 8, and 10 m above the ground. (See Figure [4.20]) Four microphones on 2 foot square ground boards were positioned 5 and 10 m east, and 5 and 10 m south of the tower. 36 The five ground microphones in the vicinity of the tower were used in determining filter functions. Sonic boom

69 Pressure (psf) Original Filter α Filter β Filter γ Filter δ Filter ε Time (s) Figure 4.18: Low-boom #1 convolved with 5 filters derived from the Dryden 2006 data set. Only.05 s of filter coefficients are used. The original waveform is the solid black line. measurements were taken over the course of several days. The results presented in this section are based solely on measurements made on July 17, Three sonic booms were considered for making filter functions: Flight 1086 Pass 3, Flight 1086 Pass 4, and Flight 1087 Pass 5. These three flights were considered because the ray paths associated with each boom intersected the ground near the tower. The ray paths were determined based on a ray tracing computer program, PCBoom4. 42 (PCBoom4 is a ray tracing program which is capable of estimating where a sonic boom will be heard on the ground based on atmospheric conditions and flight parameters.) Sailplane measurements of the sonic booms from Flight 1086 Pass 3 and 1087 Pass 5 were rounded so they didn t have short rise time. Consequently, they could not be used to derive filter functions. (Sailplane

70 59 PL, db Original Filter α Filter β Filter γ Filter δ Filter ε low boom number Figure 4.19: Stevens Mk VII Perceived Level (PL) values for 6 low-boom waveforms and the results when those wave forms are convolved with 5 filters filters derived from the Dryden 2006 data set. The 6 waveforms represent low-booms based on different operating conditions (altitudes, speeds, etc.) The black stars represent the PL values for the low-booms assuming a non-turbulent atmosphere. waveform rise times must be shorter than ground waveform risetimes for the matrix deconvolution process to work properly.) Flight 1086 Pass 4 was the only flight from which turbulence filter functions were obtained Meteorological Conditions Meteorological data taken during the Dryden 2007 experiment was limited to weather balloons which were periodically launched from Dryden Flight Research Center. A weather balloon was launched on July 17th, 2007 at 9:30 a.m. PST. Figure [4.21] shows temperature and relative humidity measured by the balloon as a function of altitude. Figure [4.22] shows the wind profile as a function of altitude. Unfortunately, weather conditions were not measured at the ground microphone

71 60 Figure 4.20: 10 m tall tower with microphones at 2 m to 10 m shown. The four guy-wires can be seen connected at the apex of the tower. 36 locations, so there was no way to correlate the weather data to the filters Sailplane Waveform Preparation The 2007 sailplane waveform preparation is nearly the same as the 2006 sailplane waveform preparation found in section The major difference between the waveform preprocessing done to the 2007 data set and the waveform preprocessing done to the 2006 data set had to do with the Doppler correction. In 2007 additional data was available, provided by PCBOOM, which improved the Doppler correction technique. Another small difference between the two data sets was that the altitude pressure correction technique done to the 2007 data set used a 3rd order polynomial, as compared to the 2006 data set which used a 5th order polynomial. The difference was simply due to the fact that the 5th order fit seemed to work better for the 2006 data set, whereas the 2007 data set was fine with a 3rd order fit.

72 Altitude (meters) Temperature Relative Humidity Temp ( C), R. H. () Figure 4.21: Temperature and Relative Humidity as a function of altitude at Dryden Flight Research Center. Measured on 17 July 2007 at 9:30 a.m. PST by a weather balloon Altitude Pressure Correction The first step in the process is to do a curve fit to the pressure waveform in order to remove the pressure variations that are due to slow changes in the altitude of the sailplane. This is done by manually removing the sonic boom from the waveform and then fitting the background waveform to a 3rd order polynomial. Note that the 2006 data set used a 5th order polynomial instead of a 3rd order polynomial Low Frequency Microphone Correction The next step in preparing the sailplane waveform for deconvolution is to apply the BK4193 correction filter. 41 This process is completed in the same manner as described above in section

73 Alt. (m) 62 Wind speed (m/s) North South West East 5 Figure 4.22: Wind speed profile, in m/s, at Dryden Flight Research Center on 17 July 2007 at 9:30 a.m. PST, as measured by a weather balloon. The black arrow points north Doppler correction In 2007 the sailplane was flying in the opposite direction as the propagation of the sonic boom. Consequently, the measurements were compressed in time as compared with a stationary observer. The compression amount is a function of the dot product of the sailplane velocity with the wavefront velocity. GPS data on the sailplane determined the true time the bow shock is measured, t bow, and the

74 63 stationary observer time, t so is given by ( t so = (t L23 t bow ) 1 v ) nw(v ns + w ns ) + v ew (v es + w es ) + v dw v ds vnw 2 + vew 2 + vdw 2 (4.4) where t L23 is the measured time on the sailplane; v ns, v es, and v ds are the GPS velocity components of the sailplane; w ns and w es are the north and east components of wind at the sailplane; and v nw, v ew, and v dw are the wavefront north, east, and down velocity components. 36 The wavefront velocity components are determined from the air temperature at the sailplane (from the weather balloon data) and the wavefront azimuth and elevation angles angles at the sailplane determined by PCBoom Waveform Truncation The last step in preparing the sailplane waveform is to truncate the boom. In order to truncate the boom, the point of inflection associated with the bow shock is determined. The point of inflection associated with the tail shock is then determined. The difference in time between these two points defines the boom duration. The start of the waveform is taken to be 1/20 of the boom duration earlier than the bow shock point of inflection. It should be noted that the ground boom has pre-boom noise of a longer duration than the sailplane boom. This difference ensures that the bow shock of the sailplane boom occurs before the bow shock of the ground boom. This difference is necessary so that the system is causal. If the ground boom occurred before the sailplane boom, the solver would not converge on a meaningful solution Ground waveform preparation The ground waveform is conditioned in much the same way as the sailplane waveform. The only major differences are that the ground waveform has no altitude pressure correction or Doppler correction and that the ground waveform has more pre-boom noise than the sailplane waveform. The ground waveform was downsampled from 25.6 khz to 24 khz, so that it matched the sampling frequency of the sailplane waveforms.

75 Low Frequency Microphone Correction The first step in preparing the ground waveform for deconvolution is to apply the BK4193 correction filter. 41 This process is completed in the same manner as above in section Waveform Truncation Next, the bow and tail shocks are determined, along with the boom duration, as described above. The waveform start point is determined by the point of inflection associated with the bow shock plus points before the point of inflection equal to 1/15 of the ground boom duration. The end of the boom is defined as the bow shock point of inflection plus 3 times the boom duration. It should be noted that the ground boom has more pre-boom noise than the sailplane boom. This difference ensures that the bow shock of the sailplane boom occurs before the bow shock of the ground boom. Again, this difference is necessary so that the system is causal Filter Estimation by Matrix Deconvolution Once the waveforms have been properly conditioned, they can be used in the matrix deconvolution method, as described in section 3.4. The Matlab code can be found in Appendix B.1. In 2007 only one sailplane measurement was used to generate filter functions for five ground measurements. In contrast, the 2006 data set had one sailplane measurement for each ground measurement. The 2007 filters were referred to as α, β, γ, δ, and ɛ Filter Agreement Once the filters were estimated, they were checked to see how well they were able to convert the input signal (the at-altitude measurement) into the output signal (the ground measurement). The filters were checked by convolving the filter with the input measurement, computing the PL value for the resulting waveform, and then comparing that PL value of the original ground waveform with the PL value of the ground estimate. Figure [4.23] shows the difference in PL between the estimated

76 65 ground waveform (based on the filter and the input signal) and the PL value of the actual ground measurement PL, db α β γ δ ε Figure 4.23: Difference between the PL values of the ground measurements and the PL values of the ground estimates. Positive values indicate the filter estimate yielded a waveform which had a larger PL value than the actual measurement and negative values indicate the filter estimate yielded a waveform which had a lower PL value than the actual measurement. The significance of Figure [4.23] is that it shows that the filters were able to reproduce the ground waveform to within.04 db. The filters do an excellent job of recreating the ground waveforms, as differences in PL of.04 would be imperceptible. Figure [4.24] shows the filter agreement between the ground measurement and reconstructed ground estimate. The reconstruction is excellent, as indicated by a difference in PL of less than -.01 db.

77 glider waveform ground waveform filtered reconstruction Pressure (psf) Time (s) Figure 4.24: Filter agreement for filter β from Dryden Applying 2007 Filters Figure [4.25] shows the result when a low-boom is convolved with the five filters obtained from the Dryden 2007 data set. Only.05 s worth of filter coefficients are used in the convolution. The source waveform was provided by industry. The turbulized low-boom waveforms show some degree of variability. The limited variability is likely due to the fact that the filters in the 2007 data set all came from the same sonic boom event. The ground microphones were all relatively closely spaced (less than 15 meters apart). 36 Figure [4.26] shows the Stevens Mk VII Perceived Level of noise (PL) values for 6 industry supplied low-boom waveforms and the results when those wave forms are convolved with 5 filters derived from the Dryden 2007 data set. The 6 waveforms represent low-booms based on different operating conditions (altitudes, speeds, etc.)

78 Pressure (psf) Original Filter α Filter β Filter γ Filter δ Filter ε Time (s) Figure 4.25: Low-boom #1 convolved with 5 filters derived from the Dryden 2007 data set. Only.05 s worth of filter coefficients are used. The original waveform is the solid black line. The black stars represent the PL values for the low-booms without the effects of turbulence. The turbulized waveforms show a spread of between 1 and 2 db PL. The overall trend is that the 2007 filters increase the PL values of the low-booms, but in some instances the PL value is reduced. In contrast, the trend for the SSBE and Dryden 2006 filters is a reduction in PL. 4.4 Filter estimation evolution The process associated with estimating filter functions evolved as new data sets and techniques became available. Previous sections in this chapter discuss in detail the data sets used in estimating filters. Initially, filters were estimated using the entire

79 68 Stevens Mk VII PL, db Original Filter α Filter β Filter γ Filter δ Filter ε low boom number Figure 4.26: Stevens Mk VII Perceived Level (PL) values for 6 low-boom waveforms and the results when those wave forms are convolved with 5 filters filters derived from the Dryden 2007 data set. The 6 waveforms represent low-booms based on different operating conditions (altitudes, speeds, etc.) The black stars represent the PL values for the low-booms without the effects of turbulence. sailplane waveform and the entire ground waveform (as was the case with the SSBE data set). When the Dryden 2006 data set became available, it was discovered that better filter agreement was attainable when using just the positive pressure portion of the sailplane waveform, instead of the entire sailplane waveform. The Dryden 2007 data set became available at the same time as the low frequency microphone correction technique. 41 This technique of improving low frequency response of the microphones was immediately put to use. It was also applied to the Dryden 2006 data set. The SSBE data set was not revisited because of the limited bandwidth associated with the sailplane measurements. It is worth noting that the key correction made to the sailplane waveforms was the Doppler correction, because it is time based. Without it, the solver would not converge on a solution for waveforms recorded when the sailplane was traveling

80 69 in the direction of sonic boom propagation. (The waveform would be stretched in time when compared to the ground waveform.) The other corrections made could all be categorized as low frequency corrections. Changes in the low frequency content of the waveforms will yield little changes in PL because the PL algorithm is most susceptible to high frequency variations. In the process of analyzing the Dryden 2007 data set, Mr. Edward A. Haering of NASA Dryden Flight Research Center, pointed out that filter agreement was best when the ground waveform had pre-boom noise of a longer duration than the sailplane waveform. It was realized that if that condition were not met, the system would not be causal and therefore the agreement between the ground measurement and the estimated ground measurement would not be as accurate. Consequently, both the 2006 and 2007 Dryden data sets were reprocessed accordingly. The SSBE data set was not reprocessed. One additional point is worth review, and this regards the comparisons between the ground estimates and the actual ground measurements. Comparing the PL plots in Figures [4.6] (SSBE), [4.16] (Dryden 2006), and [4.23] (Dryden 2007) one sees that the PL values greatly decreased as the filter estimation procedure improved over time. For the Dryden 2007 data one can be fairly confident that the sailplane and ground measurement were on the same ray path, but this cannot be claimed for the SSBE and Dryden 2006 data sets. This is probably why the PLs for the Dryden 2007 comparisons were the best. There may be other factors that could contribute to the improved performance of the 2007 data set. One factor that improved the performance of the 2007 data set is the fact that there were 5 closely spaced microphones on the ground to measure the sonic boom, yielding five filter functions. In 2006 there was only 1 microphone to measure each boom on the ground. The SSBE data set had multiple ground microphones but they were separated by large distances. The 2007 results are all based on the flight where the sailplane and the ground measurements were on the same ray path.

81 Chapter Summary This chapter has discussed the three experimental data sets used to estimate filter functions- the SSBE, NASA Dryden 2006, and NASA Dryden The circumstances of these data sets, along with the techniques used to prepare the waveforms for matrix deconvolution have been described. The filters obtained were then applied to low-boom waveforms and the PL results were presented. The evolution of the matrix deconvolution concept was also discussed. The next chapter focuses on the Nonlinear Progressive wave Equation, or NPE, which can be used to obtain filter functions by numerically propagating an N-wave through statistical realizations of atmospheric turbulence.

82 Chapter 5 The Nonlinear Progressive Wave Equation (NPE) The previous chapter described three data sets used to estimate filter functions which could be applied to low-boom waveforms to turbulize them. This chapter will focus on a numeric propagation algorithm which can generate waveforms which in turn can be used to create filter functions which represent the effects of atmospheric turbulence. Figure [5.1] gives an overview of this process. The numeric technique presented in this chapter propagates an N-wave through a statistical representation of turbulence. The initial condition of the propagation code can be used as the input to a transfer function (like the sailplane waveform in the previous chapter) and the result of the propagation code can be used as the output of a transfer function (like the ground waveform in the previous chapter). A filter function which converts the initial waveform into the final waveform, after the numerical propagation run, can be estimated using the technique described in Chapter 3. To this end, an existing propagation code, based on the nonlinear progressive wave equation (NPE) was modified to include a turbulent representation. This chapter will discuss the NPE, turbulence and how it was incorporated into the NPE, the filter functions obtained from the NPE waveforms, and the results obtained when NPE based filter functions were applied to low-boom waveforms.

83 72 Figure 5.1: A transfer function which approximates the input-output relationship between the NPE initial waveform and the waveforms after propagating through a statistical representation of turbulence. The transfer function is a linear approximation of all the processes affecting the waveform. The sonic boom is represented in the NPE as a 2D wavefront propagating through the PBL. 5.1 NPE Overview The NPE was originally developed by McDonald and Kuperman to investigate finite amplitude sound propagation underwater It was later modified to account for dissipation appropriate for sonic booms in air. 46 The NPE is mathematically equivalent to the frequency domain parabolic equation 47 for wave propagation, but it is formulated in the time domain, making it useful for visualizing the progression of a full 2D wave field with time. This benefit allows one to gain insight by visualizing the evolution of the wavefront as it propagates through turbulence. The NPE equation can be written ( p t + c 0 + β p ) p ρc 0 x + c x 0 2 p 2 p 2 y 2 dx δ eff x = 0 (5.1) 2

84 73 in which p represents acoustic pressure and the constants c 0, ρ, β, and δ eff are the ambient sound speed, density, coefficient of nonlinearity, and effective dissipation coefficient for air, respectively. The spatial variable, x, represents the primary direction of propagation. The third term is characteristic of the NPE formulation: it accounts for scattering of sound at small angles from x. The effective dissipation δ eff includes the thermoviscous dissipation as well as an effective increase in the dissipation to mimic the effects of molecular relaxation absorption by O 2 and N 2. Equation (5.1) is solved numerically in time over a 2D computational domain. A 2D array of grid points is used to represent a 2D pressure field. The grid of pressure points propagates in the primary propagation direction (in this case x) with the ambient speed of sound, in the frame of reference of the wavefront. This formulation allows the computational domain to be limited to the region of interest (the vicinity of the sonic boom), rather than requiring that the pressure be computed at all points within the entire space through which the sonic boom would propagate. The NPE marches in time, computing the wavefront as it propagates. The time step used in this research is 10 5 s. This value was the largest time step that still maintained code stability. Larger time steps would cause the propagation algorithm to become unstable. The computational grid can be described as a 2D matrix of acoustic pressure values divided into rows and columns. Successive grid columns correspond to grid points in the direction of propagation. Adding columns corresponds to adding to the computational domain in the x direction. The spatial separation between columns, x, is m. This value was chosen to match the propagation time step such that the computation grid would physically propagate one grid point at each time step. The rows in the computational grid correspond to points in the transverse or y direction. Additional rows correspond to a wider computational domain. The spatial separation between rows, y, is.1 m. It was chosen based on work done previously by Piacsek. 21 Figure [5.2] shows a diagram of one row of the computational grid. The NPE code used in the present work was originally written by Piacsek in the Fortran 77 programming language. 21 Computational resources limited Piacsek s work in several ways. He only propagated a step shock, rather than a full N-wave. He also represented turbulence as an initial condition where the 2D wavefront was

85 74 Figure 5.2: One row of the NPE, showing the N-wave placed inside the computational grid. Initially all rows of the NPE are identical, comprising a 2D wavefront. Rows and columns are counted starting from the top left corner of the computational domain. The bow shock is in the right half of the domain, corresponding to high column numbers. The tail shock is in the left half of the computational domain, corresponding to low column numbers. The computational grid and the N-wave both propagate to the right at the ambient speed of sound, which remains constant. curved, simulating the effect of the wavefront passing through a single turbulent eddy. Advances in computational resources allow the present work to represent the entire N-wave and to incorporate a statistical representation of turbulence. In addition to incorporating turbulence, the code was updated to the Fortran 90 standard. The code was also modified to run on parallel computers using the Message Passing Interface (MPI). 48 The code is considered embarrassingly parallel because the same code runs on multiple processors without any inter-processor communication as part of the NPE algorithm. (The processors do communicate on a limited basis as part of the MPI framework.) Each processor is assigned a unique random start seed, which allows each processor to generate a different instance of the atmosphere. The code was

86 75 parallelized in preparation for future work. The results presented later in this chapter were all obtained by running the code on a single processor of a parallel computer, but no parallelism was used NPE Initial and Boundary Conditions The initial waveform used within the NPE is an N-wave based on a Taylor shock profile. 49 The user is able to specify the distance between the bow and tail shocks. The initial distance between shocks used in the present work was 30 m. The boundary conditions on the NPE are unique. The pressure before and after the N-wave is set to zero. The density is taken to be constant. The numeric algorithm computes the pressure from the right side of the computational domain (high column numbers, in front of the N-wave) to the left side of the computational domain (low column numbers, behind the N-wave). The algorithm stops computing the pressure just before it gets to the left edge of the computational domain (the first column) so there is no explicit boundary condition. This non explicit boundary condition is a consequence of the integral in equation (5.1). The top and bottom boundaries (in the transverse direction) are periodic. The periodic boundary condition was chosen over a rigid condition because the turbulence representation combined with the rigid boundary condition created a non-physical standing wave across the bow shock. The periodic boundary condition required that the turbulence also be periodic. One of the consequences of having periodic turbulence is that the computational domain computes the pressure twice, thus reducing the useable portion of the NPE by half. Throughout the remainder of this work, unless stated explicitly, only the bottom half of the computational domain will be shown, as the top half is simply the mirror image of the bottom half. Figure [5.3] shows the computational domain with the boundary conditions and the propagation direction. The periodic turbulence caused non-physical spiking in the regions of the boundary and along the center line of the domain (where the mirror images of the turbulence meet.)

87 76 Figure 5.3: Boundary condition diagram showing the direction of propagation. Pressure in front of and behind the N-wave is zero. The top and bottom boundary conditions are periodic. One row of the pressure within the NPE shows the N-wave placed inside the computational grid. Initially all rows of the NPE are identical, making up a 2D wavefront. 5.2 Turbulence Turbulence within the atmosphere can be described as rapid variations in atmospheric temperature and particle velocity (wind). It is caused by the interaction of the atmosphere with the surface of the earth. It is also influenced by obstacles on the surface of the earth as well as convection caused by solar radiation. Turbulence causes the formation of eddies of many different length scales. Most of the energy is contained in the largest scale structures. The energy cascades from large scale eddies down to smaller scale eddies. Eventually, the energy is dissipated in the smallest eddies when it is converted into heat due to viscosity. The notion of energy cascading from large eddies to small eddies was originally introduced by Richardson 50 and later developed by Kolmogorov. 51

88 77 It is possible to describe turbulence statistically in terms of a turbulent spectrum. This spectrum can be divided into three parts: energy-containing subrange, inertial subrange, and dissipative subrange. Most of the energy is input into the turbulence in the energy-containing subrange. The length scales in the energy-containing subrange can range from tens of meters to several kilometers. The characteristics of the spectral densities for the energy-containing subrange depend on many variables: wind, surface roughness, altitude of the PBL, etc. Eddy sizes that are less than the height of the PBL, but greater than the eddy size associated with dissipation, are termed to be part of the inertial subrange. The inertial subrange experiences no energy input or dissipation. Rather, energy is transferred from large eddies to smaller eddies. The turbulence represented in the current work is limited to the energy-containing and inertial subranges. The dissipative subrange is where turbulent energy is converted to heat due to viscosity. The length associated with the dissipative subrange near the earth s surface is on the order of a millimeter Incorporating Turbulence into the NPE In a turbulent atmosphere, the index of refraction fluctuates about some average value. Fluctuations in the index of refraction are due to turbulent temperature and wind fluctuations η = T 2T 0 u t c 0, (5.2) where η is the index of refraction, T is the turbulent fluctuation in temperature, T 0 is the ambient temperature (taken to be 293 K in this work), u t is the turbulent fluctuation in wind, and c 0 is the ambient speed of sound (taken to be 343 m/s in this work). Multiplying equation (5.2) by the ambient speed of sound yields c = c 0T 2T 0 u t, (5.3) which represents variations in the speed of sound due to turbulent fluctuations and is a function of position. For a more detailed discussion on representing turbulence as fluctuations in the speed of sound, the reader is referred to the work by Salomons. 47 For simplicity, the work presented here is based solely on temperature fluctua-

89 78 tions. The implication of this simplification is that the effects of turbulence will not be as pronounced as if turbulent wind were also included. 52 Dropping the wind variations in (5.3) leads to the equation which will be added to the NPE to incorporate turbulence. It is c = c 0T 2T 0. (5.4) Turbulence is incorporated into the NPE by substituting c 0 + c for the ambient sound speed c 0 in equation (5.1). The modified NPE equation can be rewritten as ( p t + c 0 + c + β p ) p ρc 0 x + c x 0 2 p 2 p 2 y 2 dx δ eff = 0. (5.5) x2 The quantity c is a function of space and represents the sound speed perturbation field that arises from turbulent fluctuations of temperature; it is calculated using the modified von Kármán energy spectrum employed by Blanc-Benon, described in the next subsection, and is represented as an array of values with the same spatial representation as the pressure array. Figure [5.4] shows an example of c for one instance of the NPE. The periodic boundary condition mentioned above required that the turbulence also be periodic. The periodic boundary condition caused non-physical focusing of the wavefront at the top and bottom boundaries, which will be discussed later. The top boundary is located at y=200 m and the bottom boundary is located at y=0 m. The turbulence is mirrored top to bottom about the line y=100 m. Turbulence is computed initially for the entire domain. Then, as the NPE propagates, at the start of each time step, the array representing c is shifted one column in the negative x direction. The new values of turbulence are computed (in the right most column, corresponding to the part of the domain which is farthest ahead of the N-wave) and the NPE propagation algorithm computes the wavefront at the new time step Turbulence Spectrum The turbulence model used here is based on temperature fluctuations and follows the work by Blanc-Benon et al. 52 The effects of wind are not explicitly modeled.

90 79 Figure 5.4: Speed of sound fluctuations representing periodic turbulence for a 200 m wide computation domain. Note the periodic nature of the turbulence. The arrow along the bottom indicates the direction of N-wave propagation. Approximately 35 million grid points shown. Variations in the speed of sound c, are based on equation (5.3). T is defined as T (x) = N T (K i ) cos (K i x + φ i ) (5.6) i=1 where x is a position vector, K i is the ith Fourier component of the wave number vector, and φ i is the phase fluctuation for the ith Fourier mode. Both K and φ are independent random variables with uniform distributions to ensure statistical isotropy and homogeneity. The magnitude of K is logarithmically spaced between 1/L 0 and 1/l 0, where L 0 is the outer length scale, L 0 = 100 m, and l 0 is the inner length scale, l 0 =.005 m. The values of L 0 and l 0 were chosen based on discussions