The Helmholtz Resonance

Transcription

1 The Helmholtz Resonance A Brief and Not-Too-Technical Introduction to the History and Theory of the Lowest Sound-Producing Mode, and Some Practical Considerations for Instrument Designers R.M. Mottola Copyright (c) 2004 by R.M. Mottola Research in physics and acoustics of stringed musical instruments shows us the mechanism by which sound is produced by those instruments. The plates of the instruments and the air inside vibrate in various patterns, each pattern producing sound in a range around a certain frequency. Each of these patterns can be considered to be a resonator, each with its own characteristics. Some of these resonators exist as modes of vibration of different areas of the plates of an instrument, and some are modes of vibration of the air inside the instrument. One of the air resonators is composed of the mass of air inside the instrument and the mass of air within and around the sound hole. The natural frequency of this resonator is near the lowest note that an instrument can make. It is generally labeled the A0 resonance, the letter A standing for the word air and the numeral 0 indicating that this is the first in a series of air resonances. This resonance is also referred to as the so-called Helmholtz resonance. Understanding how this resonance works in stringed instruments is not difficult, particularly given a historical perspective. Complete understanding involves some math, but a practical understanding can be had without it. Therefore, I am putting off presenting the formulae in the main article and have included them in a companion sidebar. Hermann Ludwig Ferdinand von Helmholtz was born in 1821 and was educated as a medical doctor. He worked in this profession only reluctantly, spending his free time in the study of physics, chemistry and mathematics. Research projects included work in the areas of optics and Figure 1 - A schematic of a Helmholtz resonator

2 acoustics. His major work, as least as concerns our topic, was one undertaken in the mid nineteenth century on human perception of musical tone. In his acoustic research he needed the ability to isolate and identify tones of different frequencies from sound sources. To this end he proposed a series of acoustic filters, each one for a specific frequency. One of these filters could be held to the ear and when a complex sound was heard through it the components of that sound nearest in frequency to that of the filter would be heard the loudest, all other frequencies being diminished to some extent by the filter. The filters were simple and quite ingenious. Fig 1 is a schematic representation of one of these filters, generally referred to as a Helmholtz resonator. It consists of a rigid hollow sphere with two holes in it on opposite ends. A rigid hollow cylinder is attached to one of these holes. At the other end of the sphere a small conical earpiece is attached to the other hole. To use the resonator, the earpiece is inserted into the ear, the other ear plugged, and the open end of the cylinder pointed at the source of sound. Helmholtz proposed a series of such resonators, each tuned to a different frequency. The frequency of the resonator is determined by the amount of air inside the sphere and by the amount of air inside the cylindrical tube. Increase the volume of the spherical part and the frequency of the resonator goes down. Decrease it and the frequency goes up. Increase the volume of the cylindrical part by extending the tube s length and the frequency goes down. Decrease it and the frequency goes up. Figure 2 - The mass spring - a spring with a weight on top 2 To understand how von Helmholtz resonators work we ll first take a look at another resonating system. This one is beloved by physics teachers and is called a mass spring. It consists of a spring supporting a weight (mass) as in fig 2. Push the weight down so the spring compresses and then let it go and the weight will bob up and down rhythmically until friction causes it to stop. See fig 3. One interesting thing is that it will bob up and down at a fixed frequency every time it is impulsively set into motion. You can push it down soft or hard and this will affect the displacement (how high and how low the weight moves) but the frequency will remain constant. Another interesting thing is that the resonant frequency at which it will oscillate is completely defined by how massive the weight is and by how stiff the spring is. Increase the weight or make the spring less stiff and the mass spring will oscillate at a lower frequency. Decrease the weight or make the

3 Figure 3 - The mass spring will oscillate at a frequency determined by the weight and the stiffness of the spring spring stiffer and the frequency of oscillation will increase. Since both mass and the stiffness of the spring determine the frequency, it is possible to construct a mass spring with an extremely small weight - just adjust the stiffness of the spring to compensate. Other configurations of mass springs are possible. Consider a Slinky toy, with one end held in the hand and the other end let go to fall like a yo-yo. The bottom end of the Slinky will oscillate at a rate fixed by the stiffness of the spring and the mass of the oscillating part. In this mass spring the mass and the spring are made of the same material - spring. Figure 4 - In the Helmholtz resonator the air in the neck is the mass and the air in the spherical part behaves like a spring The Helmholtz resonator works just like a mass spring. In fact it is a mass spring. See Fig 4. Air is compressible and in the Helmholtz resonator the air in the sphere behaves like a spring, while the piston of air in the tube part is the mass. As mentioned, increasing the size of the sphere will result in a lower resonant frequency. This is because a greater volume of air makes for a less stiff spring. You can demonstrate this yourself with a bottle and a solid, tight-fitting stopper. Feel the effort it takes to push the stopper into the empty 3

4 bottle, then remove it, fill the bottle almost all the way with water, then insert the stopper again and compare the effort it takes to do it. It will be considerably more difficult to insert the stopper in the bottle containing just a small quantity of air. Increasing the length of the tube part of the Helmholtz resonator will also result in a lower resonant frequency, as the increased amount of air here has greater mass. The compression part of a sound wave at or near the natural frequency of the Helmholtz resonator pushes the air piston in the cylinder in, compressing the air spring in the sphere, which rebounds, pushing the air piston out again at or near the rarefaction part of the sound wave. But the bigger the difference in frequency between the sound wave and the natural frequency of the resonator, the more the resonator will impede transmission of the sound through the resonator. So Helmholtz resonators work as selective filters, allowing only waves close in frequency to that of the resonator to pass unimpeded. For a more familiar example of such filtering, consider what happens if you rest your finger ever so lightly on an electric guitar string tuned to pitch and then start singing near it. If you begin singing or humming at a frequency much below that of the string you will feel no vibration, but vibration in the string will increase in amplitude as the pitch you are singing nears that of the string. Maximum amplitude will be felt when you are singing at the pitch that the string is tuned to. As you raise the pitch higher, the amplitude of vibration of the string will become increasingly lower. The body of a stringed musical instrument will behave in part like a Helmholtz resonator. The body encloses a mass of air which serves as the spring, while the sound hole(s) encloses a mass which serves as the air piston. There is a different type of Helmholtz-like resonator that doesn t have a cylindrical neck, and it looks a little more like the body of a stringed instrument. This resonator is a rigid rectangular box with a round hole cut in it (fig 5). The sidebar contains a simplified formula for this box type resonator, and this can work pretty well for approximating the Helmholtz frequency of stringed instruments as long as the size of the hole is small compared to the dimensions of the face of the box. I haven t researched the history of this resonator model but it is very likely to have been created for the purpose of building tuned speaker cabinets. The deal here is the same as for the Figure 5 - A resonator consisting of a box with a hole in one surface 4

5 original Helmholtz resonator. Increasing the capacity of the box decreases the resonant frequency and vice versa. The Helmholtz resonance of an instrument supports the lowest notes the instrument can produce. When designing an instrument it is highly desirable to make the capacity of the box and area of the sound hole such that the air inside will resonate near that lowest note. Sometimes this is not possible. The Helmholtz resonance of the double bass is nowhere near that of the lowest note of the instrument. To get it that low the instrument would have to be so big that playing it (not to mention transporting it) would be difficult. This is because decreasing the Helmholtz frequency by one octave requires a quadrupling of the volume enclosed by the instrument, everything else being equal. For most instruments though, the formula for the box type resonator could be used to design an instrument with its Helmholtz resonance near that of the lowest note. In practice this doesn t quite work, as the following example will show. I have a small bodied steel string guitar. Using Simpson s Rule, the technique for calculating the area of an irregular surface outlined by Dave Raley in American Lutherie #70, I calculated the volume of the body to be cubic meters. Plugging that and the sound hole radius ( M) into the simplified formula for the natural frequency of the box type resonator yielded Hz. But the actual Helmholtz resonance of the instrument was at Hz, 16% below the predicted value. Why the difference? The primary reason is that the walls of the guitar are not rigid. This flexibility makes the air spring less stiff, which lowers the actual frequency. So, when designing an instrument the formula for the box can get you in the neighborhood, but the actual frequency will be somewhat lower. This is why the term so-called Helmholtz resonance was used in the introductory paragraph of this article, and why you ll often see the term in quotes in the musical instrument research literature. A true Helmholtz resonator uses a rigid enclosure. A typical experimental technique appearing in the research literature is to immobilize the walls of the instrument in sand. When this is done the measured resonant frequency ends up much closer to the predicted frequency. There is a method described in the research literature that yields estimates of the Helmholtz frequency with better accuracy than the formula for the box resonator. It makes use of first order perturbation theory, and it is quite complex. How can you measure the Helmholtz frequency of an instrument? Spectrographic analysis of a digital recording of a thump on the instrument top, with the microphone pointing at the sound hole will show the Helmholtz frequency as the lowest peak. Fig 6 shows the results of a thump on the small bodied steel string guitar. The vertical reference line on the left side of the graph is set to 117 Hz and the high energy band behind it stands out readily. Want to measure the frequency but don t want to throw too 5

6 much technology at the problem? Try blowing across the sound hole, like blowing across the top of a bottle. Remember the pitch you hear. Take it to the piano and identify the note. I have a little trouble getting the octave right when doing this sometimes. The tone has a very breathy quality. Approximate Helmholtz frequencies of typical instruments include 95 Hz for classical guitars, 120 Hz for steel string guitars, 284 Hz for violins,102 Hz for celli, and 60 Hz for double basses. Figure 6 - Fourier analysis showing Helmholtz frequency of small bodied steel string guitar at 117 Hz When designing an instrument, the size of the body (and thus its enclosed volume of air) and the size and depth of the sound hole can be used to tune the Helmholtz resonance, but as a practical matter doing this is fraught with complications. Changing the volume of the box is most problematic. A stringed instrument really is a quite complex system of coupled resonators. Changes made intending to affect one undoubtedly affect others, and not necessarily in desirable (or even predictable) ways. Changing the shape of the box alters other air resonances and couplings, and if those changes involve reshaping of the plates then plate resonances and couplings are altered as well. Side effects aside, increasing the volume of the box by a factor of four will reduce the Helmholtz frequency by one octave. Attempting to alter the Helmholtz frequency by changing the area of the sound hole is generally of limited practicality. In theory, increasing the radius of the hole should almost proportionally increase the frequency, so doubling the radius of the hole should result in a frequency increase of about one octave. But keep in mind that this makes for one large hole, and the formula only works if the hole size is small compared to the size of the top. In practice enlarging the size of the hole may result in an increase in the Helmholtz frequency in some instruments, but only up to a point. Make the hole too big and the instrument stops behaving like a Helmholtz resonator and starts behaving more like a box with one open side. Enlarging the sound hole of an instrument like a flat top guitar could have unintended consequences too, especially if the edges of the hole approach the top braces. This would tend to make the hole appear deeper, that is, more like a tube, and this could have the unintended consequence of decreasing the resonant frequency. 6

7 Enlarging the hole also puts the hole closer to the edges of the top and this also can have an effect on the frequency. Note that increasing the area of the hole increases the resonant frequency. You d think increasing the area would result in increased mass of the piston of air and thus decrease the frequency but it works just the opposite. This is because the bigger the radius of the hole or tube, the less the air in it behaves like a solid piston. There are compelling reasons not to alter the hole size at all. Plugging typical values for a guitar into a formula used to calculate the efficiency of a speaker cabinet port, Evan Davis found that the near universal 4 inch diameter guitar sound hole was optimum. The sound hole can be viewed as a very short tube, the length of which is just the thickness of the top. Attempts to lower the Helmholtz frequency by adding a longer tube inside the sound hole can work but the effect is usually smaller than may be desired. A four fold increase in the effective length of the sound hole would decrease the Helmholtz frequency by one octave. But the effective length of the hole is longer than the actual length. I lined the sound hole of the small steel string guitar with a 2.5 cm long tube, which lowered the measured frequency by only 7%. You d think increasing the depth of the hole from 2.5 mm (the thickness of the top) by a factor of ten would have a more dramatic effect. That it does not demonstrates something interesting, and that is that the hole does not work as just a very short tube. In the box type resonator particularly, the hole actually behaves like a much longer tube, the effective length of which is the actual length plus the radius of the hole times a constant, the classic value of which is 1.7. See fig 7. The 10 cm diameter sound hole in the little guitar behaves like a tube with an effective length of 8.75 cm. So adding the 2.5 cm long extension tube in the hole did not increase the length of the hole by a factor of ten, but only by about a third. Figure 7 - The effective length of the sound hole is a function of its radius, and is quite a bit longer than its actual length 7 Empirical evidence of the subtlety of effect of attempts to lower the Helmholtz frequency with the addition of sound hole tubes is provided by historical instruments that were originally fitted with such tubes only to have them removed to no obvious ill effect at some later date. Sound hole tubes are a nice way to experiment with altering the Helmholtz frequency though, as they can be added and removed with no permanent modification to the instrument. When performing such experiments do keep in mind that in addition to increasing the Helmholtz frequency,

8 the tube adds mass to the top plate, which could and probably would affect the vibrating qualities of the top. Moving the location of the sound hole can have an effect on the Helmholtz resonant frequency too, but it should be pointed out again that this type of alteration undoubtedly affects the structure of the top and thus its plate resonances. It would also affect other air resonances within the body. Therefore without thorough measurement and analysis following such a change it really would be impossible to attribute the effects of the change solely to the altered Helmholtz frequency. The issue of plate dynamics aside, moving the hole towards the upper edge of the top should lower the frequency, but the effect is generally small unless compounded by other factors like coincident changes to the effective depth of the hole. The effect will be larger in instruments that are inherently long and skinny, like dulcimers. I have performed no experiments to confirm the effects of moving the hole around, wanting to keep the small steel string guitar more or less intact. I m taking this from a nice article by W.D. Allen entitled Basics of Air Resonances which was published in American Lutherie #1 and appears in The Big Red Book of American Lutherie Vol. 1. The examples in the preceding discussion about sound holes are all about guitars with single round holes. This is just for convenience of explanation. All of this applies to instruments with multiple and/or complex shaped holes as well. In such instruments the areas of all of the holes are added together and behave as if there was a single hole with that area, at least as far as the Helmholtz formulae calculations go. As with single round holes, the placement of multiple holes can have decided effects on both top and other air resonances. The fact that attempts to modify the Helmholtz frequency by addition of tubes or repositioning of the sound hole result in such small changes could be exploited by the instrument designer and builder to some advantage. Because hole depth and placement matter little, the designer may find it desirable to focus attention on the design of the plate, allocating space for the hole where it is most convenient. Since the Helmholtz frequency formulae yield only approximate results, and since these results will deviate still further from the measured frequency of an instrument, a conservative approach to the design of an instrument for a specific Helmholtz frequency would start with an existing instrument and make small calculated changes, measuring frequency after each. That it may require the construction of a number of instruments before achieving the desired result demonstrates just how approximate the results of the formulae are for real stringed instruments. 8