Sound, acoustics Slides based on: Rossing, The science of sound, 1990. Acoustics 1 1 Introduction Acoustics 2! The word acoustics refers to the science of sound and is a subcategory of physics! Room acoustics (confusingly, sometimes just acoustics) studies sound propagation indoors (esp. concert halls) Contents: 1. Introduction 2. Vibrating systems 3. Waves 4. Resonance 5. Room acoustics! The goal in this lecture is to learn the principles, not so much the equations 2 Vibrating systems! Common to vibrating systems Motion repeats in each regular time interval (the period) Some force restores the system toward equilibrium 2.1 Simple harmonic motion! Spring-mass system F = K x Acoustics 3 force (spring constant x stretch) acts as the restoring force In equilibrium, earth gravity mg = F! In Simple harmonic motion, the restoring force is directly proportional to the distance from equilibrium In that case frequency f does not depend on amplitude Mass + spring: Kx = ma mx!!, K!! x + x = 0, x = Acos(ω t + ϕ) ω = m K m 2.2 Energy and damping (1) Acoustics 4! In a vibrating system, kinetic energy and potential energy are alternating! In the mass-spring case Kinetic energy E K = ½mv 2 Potential energy E P = Kx 2! Figure: top: displacement vs. time, bottom: speed vs. time At times t 1 and t 3 potential energy is at its maximum, At times t 2 and t 4 kinetic energy is at its maximum x
Acoustics 5 Acoustics 6 Energy and damping (2) 2.3 Simple vibrating systems (1)! In all realistic vibrating systems, there are energy losses due to friction etc.! Unless energy is brought to the system from outside, the amplitude of the vibration will decay (see figure) Typically a certain fraction of the mechanical energy is lost during each vibration. In that case, the amplitude envelope is exponentially decaying (see figure) Amplitude envelope! These are simple harmonic vibrating systems (in addition to the mass-spring system) 1. Pendulum (small angle) Mass attached to a string Gravity as the restoring force 2. Spring of air A piston of mass m moves freely in a cylinder, of area A and length l Acoustics 7 Acoustics 8 Simple vibrating systems (2) 2.4 Systems with two masses 3. Helmholz resonator Air in the neck acts as the mass Air in the cavity acts as the spring Examples of Helmholz resonators Blowing air across and ampty bottle Bass reflex tube in loudspeakers (figure) Sound hole in the guitar! In the above examples, one coordinate sufficed to describe the motion " only one degree of freedom! In the following, we consider systems with two or three degrees of freedom Then there are also more than one mode of vibration Typically each mode has a different frequency of vibration! Figure: system with two masses and three springs Two modes: masses moving (a) in the same direction (b) in opposite directions Modes are independent of each other and have different mode frequencies In realistic cases the movement is usually a combination of modes
Systems with two masses (2) Acoustics 9 2.5 Systems with many modes of vibration Acoustics 10! Figure: above-descrived two-mass system has two transverse vibration modes in addition to the longitudinal ones Vibration is perpendicular to the spring Transverse vibration: for example membrane of a drum Longitudinal vibration: for example air column in a wind instrument! Figure: in the general case a system with N masses has N longitudinal and N transverse vibration modes 2N modes, but only N frequencies, since corresponding longitudinal and transverse modes have the same frequency! More masses " wavelike shape emerges Vibrating string can be considered as a mass-spring system where N is very large Acoustics 11 Acoustics 12 2.6 Vibrations in musical instruments (1) Vibrations in musical instruments (2) 1. Vibrating string Can be viewed as a mass-spring system: string mass and elasticity Many vibration modes that are typically nearly exact integer multiples of a fundamental frequency " Harmonic modes (see bottom row of the figure on the previous slide) 2. Vibrating membrane Can be thought of as a two-dimensional string, tension of the membrane acts as a restoring force (membrane attached to a rim) figure: four vibration modes are illustrated 3. Vibrating bar For example marimba, xylophone, glockenspiel Stiffness of the bar provides a restoring force Vibration modes are not harmonic, but the frequencies in glockenspiel for example are 1 : 2.76 : 5.40 : 8.93 :... (harmonic would be 1:2:3:...) 4. Vibrating plate As in a vibrating bar, the stiffness of the plate itself acts as a restoring force (different from a stretched membrane of a drum)
Vibrations in musical instruments (3) Acoustics 13 2.7 Vibration spectra Acoustics 14 5. Air-filled pipes Vibrating air column For example organ pipe, trumpet Comparable to a vibrating string Many vibration modes! When a vibrating system is excited, it usually starts to vibrate in several modes at the same time Each mode has its specific frequency and amplitude " Spectrum of the vibration! Figure: spectrum of a plucked string " Mode frequencies of an instrument (for example) can be studied by recording its sound and then looking at its Fourier transform 3 Waves Acoustics 15 3.1 Progressing waves Acoustics 16! Waves transport energy and information through a medium so that the medium itself is not transported! In the case of sound, the medium is usually air! Sounds may reflect, refract, or diffract! Speed of sound in air 340 m/s (20 ºC) cf. speed of light 3 10 8 m/s! For a progressing wave v = fλ where v is velocity, f is frequency, and λ is wavelength figure: wave in a rope
3.2 Properties of waves Acoustics 17 Standing waves on a string Acoustics 18! Figure: reflection! Figure: linear superposition Waves may travel through each other without changing their properties! Standing wave Found e.g. on a stretched string Wave traveling in the string reflects from both ends so that the sum of waves traveling in opposite directions appears not to move Nodes and antinodes can be observed on the string! Standing wave is formed when a wave travels in opposite directions on a string, reflecting at both ends! Figure: resonance frequencies of a vibrating string Wavelength of the lowest resonance λ = 2 x string length " Fundamental mode frequency f 1 = v / 2L Higher modes: f n = n v / 2L = n f 1 where propagation speed v = T µ T is string tension and µ is mass per unit length (speed and frequency get lower by reducing tension or increasing string mass) 3.3 Sound waves Acoustics 19 3.4 Propagation in two or three dimensions Acoustics 20! Sound waves are longitudinal waves that travel in gas, liquid, or solid material Speed of sound is lowest in gas Hearing works also underwater, although due to the changed speed of sound, the direction of arrival of sounds is unclear! Figure: reflection of a sound pulse in a pipe (a) Sent positive pressure pulse (b) Reflection at open end (negated) (c) Reflection at closed end (d) Absorption (no reflection)! Usually sound waves propagate in two or three dimensions! Sources with different geometries radiate different kinds of patterns Point source radiates spherical waves (left figure) Line source radiates cylindrical waves (right figure) Large flat source radiates plane waves Real-life sources can only approximate these geometries
3.5 Doppler-effect Acoustics 21 3.6 Reflection Acoustics 22! Normally the frequency of the waves arriving to an observer is the same as the frequency of vibration at the sound source! The situation changes if either the source of the observer is in motion Observer meets the waves more frequently when moving towards the wavefront (" frequency increases) When moving apart from each other, observed frequency decreases = Doppler-effect! Reflection of sound waves can be experienced by clapping hands at some distance from a large wall! Figure: reflected waves appear to come from an imaginary source behind the reflecting surface Think of a mirror 3.7 Refraction Acoustics 23 3.8 Diffraction Acoustics 24! Refraction occurs when the speed of waves changes Direction of the waves changes! Figure: propagation speed changes abruptly as wave passes from one medium to another! Speed can also change gradually Figure: wind does not blow the sound back (speed of wind is small compared to sound), but because higher wind speed at higher altitude tends to refract the sound to the sky! Sound waves tend to bend around an obstacle! Figures: left: sound bends behind a wall (see arrows) right: sound waves traveling through a narrow opening appear as a new point source
4 Resonance Acoustics 25 4.2 Standing waves on a string Acoustics 26! Idea of resonance illustrated by a child in a swing: giving the swing a small push at a suitable frequency, its amplitude gradually increases 4.1 Mass-spring vibrator resonance figure: mass-spring systems attached to a crank Natural vibration frequency of the mass-spring system is f 0 Crank is revolved at frequency f, which is slowly varied " Vibration amplitude A changes and reaches its maximum A max when f = f 0! Curve: amplitude A as a function of frequency f! Standing wave is formed when a wave travels in opposite directions on a string, reflecting at both ends! Figure: resonance frequencies of a vibrating string Wavelength of the lowest resonance λ = 2 x string length " Fundamental mode frequency f 1 = v / 2L Higher modes: f n = n v / 2L = n f 1 as mentioned in 3.2 above 4.3 Partials, harmonics, overtones Acoustics 27 Partials, harmonics, overtones (2) Acoustics 28! Terminology for discussing vibration spectra, for example the spectra of musical instruments: Partial : any mode of a vibrating system (any component of sound) Harmonic : if partials are (nearly) integer multiples of the fundamental (as in a vibrating string for example), the partials are called harmonics (fundamental is the first harmonic) Harmonic sound : sound where partials are nearly integer multiples of the fundamental amplitude Fundamental mode (frequency = fundamental frequency) harmonic overtones (in a harmonic sound, frequencies are integer multiples of the fundamental) frequency! Relative strengths of the partials largely determine the timbre (tone colour) of the sound Temporal evolution of the partial amplitudes is another important factor of timbre (and there are also others like tonal vs. noisy quality etc.)
4.4 Open and closed pipes Acoustics 29 4.5 Sympathetic vibration Acoustics 30! Reflection of a positive sound pulse at the ends of a pipe Reflected as negative at open end and as positive at closed end! Human vocal tract can be modeled as an acoustic tube When varying the shape of the vocal tract, resonance frequencies (formants) move " phonemes! Left: vibration modes in a pipe open at both ends f n = n f 1, n = 1,2,3,...! Right: vibration modes in a pipe with one end closed f n = n f 1, n = 1, 3, 5,... " only odd harmonics!! The amount of sound radiated by a source is proportional to the amount of air it displaces as it moves Thin vibrating string displaces very little air and therefore radiates only a small amount of sound Membrane of a drum or the element of a loudspeaker displaces more air! Radiation can be increased by attaching the vibrating system to a wood plate or sounding box Vibrating system makes the plate move Due to its large area, sympathetic vibrations of the plate increase the amount of radiated sound, even though its resonance frequency would not be exactly right! In musical instruments, two or more vibrators often work together Piano strings and soundboard, guitar strings and body Clarinet reed and air column! String instruments are based on the sympathetic vibrations of a sounding box or soundboard Resonance frequencies of the sounding box largely determine the timbre of the instrument Acoustics 31 5 Room acoustics 5.1 Sound propagation outdoors and indoors! Free field Source is small enough to be considered a point source Source is outdoors and far away from reflecting objects " Sound waves propagate from the source in shape of sphreres and sound pressure is proportional to 1/r [Pa] (r: distance from source) " Average sound intensity I 1/r 2 [W/m 2 ] Indoors, free field can be found only in an unechoic room! Indoors sound waves encounter walls and other obstacles Figures: obstacles reflect and absorp sound in ways that determine the acoustic properties of the room 5.2 Direct, early, and reverberant sound Acoustics 32! In auditorium, direct sound reaches listeners in 20-200 ms Depends on the distance from the source to the listener! Soon after, the same sound reaches the listener from reflecting surfaces (walls, ceiling) These are called early reflections Time difference to direct sound usually < 50 ms! Last group of reflections is called reverberant sound Sound reflected several times from various surfaces Weaker, many reflections, close to each other in time When the source is turned off, reverberation decays in an approximately exponential manner (db level is a straight line)
Acoustics 33 Direct, early, and reverberant sound (2) 5.4 Reverberation time Acoustics 34! Figure: Room impulse response h(t) direct sound early reflections reverberation! Reverberation time is among the most familiar characteristics of auditoriums Reverberation reinforces direct sound Too much reverberation results in a loss of clarity Suitable reverberation time depends on the auditorium size and purpose (speech: short, organ music: long)! Figure: studying reverberation Switch on a steady source for time T Record the sound at another location in the auditorium " Level of the reverberant sound first increases in steps when early reflections arrive Reverberation time Acoustics 35! Reverberation time is usually denoted by T 60 Time where the sound level decreases by 60 db from its maximum Figure: sound level decreases approximately exponentially, thus db level decay is a straight line as a function of time p [lin.] L p [db] R(t) 10 log 10 (R(t)) Curve R(t) (see the figure) describing the sound level decrease can be also obtained from room impulse response h(t) by = 2 R ( t) h ( t) t