Chapter 16 Sound and Hearing by C.-R. Hu

Transcription

1 1. What is sound? Chapter 16 Sound and Hearing by C.-R. Hu Sound is a longitudinal wave carried by a gas liquid or solid. When a sound wave passes through a point in space the molecule at that point will execute a SHM in the direction of the wave. If its frequency lies in the audible range (between about 20 Hz and Hz but varies somewhat between individuals) it can be heard by a human ear. Sound with frequency above that range is called ultrasound. The frequency in the audible range is also called pitch. The intensity of an audible sound is also called loudness except that the latter has a different unit (decibel or db) which is more natural for describing how a human ear responds to the intensity of a sound. A decibel is one tenth of a bel. When the intensity of a sound is increased by a factor of 10 the human ear feels that the loudness or sound intensity level is only doubled. We say that the sound level has increased by 1 bel which is the same as 10 db. Thus the sound level β is defined as I β (in db) = 10 log 10 I where I 0 = W/m is the lowest intensity of sound that an average human ear can hear. Note that if I = I 0 we would get β = 0 ; if I = 10 I0 we would get β = 10db ; and if I = 100 I0 we would get β = 20db etc. Note that if you are comparing two different sound intensity levels β 1 and β 2 then their difference can be written as: I β = β β = lo g10 I 2 which no longer depends on I 0! This is because l og A log B= log( A/ B ). A sound wave is also a pressure wave. This is why a very loud sound can break your ear drum. In a solid pressure change is proportional to fractional volume change via the relation:

2 p = B( V / V ) where B is called the bulk modulus. (The minus sign is needed because when volume increases pressure actually decreases which means that if V is positive then p should be negative.) Next we must relate the volume change to the displacement ξ (x t) of the molecules. Consider a rectangular volume occupied by a fixed number of molecules. If all molecules move by the same displacement the volume containing the molecules moves but it won't change its size. The volume will increase if the molecules at the right face of the box move more than the molecules at the left face of the box (assuming that the movement is positive which means to the right). If the left face of the box is at x and the right face of the box is at x + dx then what we just said is that ξ (x + x t) must be larger than ξ (x t) in order for the volume to increase. For the box to always contain the same collection of molecules then its left face must move by ξ (x t) and its right face must move by ξ (x + x t). (Remember that sound wave is a longitudinal wave. The displacement is in the direction of the wave.) Thus after the displacements of the molecules the width of the box will change to: x + [ξ (x + x t) ξ (x t)] º x + [ ξ (x t)/ x] t x where the partial derivative [ ξ (xt)/ x] t just means taking derivative with respective to x at a fixed time t and will henceforth be simply written as ξ / x. Thus if A is the area of the box perpendicular to the wave direction x then the original volume of the box is V = A x and the volume after molecular displacement is A[ x + ( ξ / x) x ] = V 0 + ( ξ / x) V 0. Thus we have V = ( ξ / x) V 0 and therefore 0 p= B( ξ / x) = ( BAk) sin( kx ω t+ φ) assuming that ξ ( xt ) = Acos( kx ω t+ φ) for a sound wave moving in the +x direction. This equation tells us that the amplitude of the pressure wave is equal to (B A k) also known as the pressure amplitude. Also it is important to realize that the pressure wave nodes are located at the antinodes of the amplitude wave and the pressure wave antinodes are located at the nodes of the amplitude wave. This statement can be understood as follows: At some node of the amplitude wave the neighboring atoms both in the front and in the back of it are moving toward that point. So P is maximally positive at that point. At some other node of the amplitude wave the neighboring atoms both in the front and in the back of it are moving away from that point. So P is maximally negative at that point. Both cases are anti-nodes of the pressure wave. This is explained by the following figure":

3 This is a plot of the displacement variation: At this node of the displacement variation pressure is the lowest since a particle on its right is moving to the right and a particle on its left is moving to the left. At this node of the displacement variation pressure is the highest since a particle on its right is moving to the left and a particle on its left is moving to the right. Note that both of these two displacement nodes are pressure antinodes! The left one corresponds to a maximally negative p and the right one correspond s to a maximally positive p. To put it in another way we can say that the pressure wave is ahead of the displacement wave by the phase difference π/2. 2. The speed of sound In a fluid (i.e. a liquid or a gas) the speed of sound is given by υ = B / ρ (for a fluid) where B is the bulk modulus of the fluid and ρ is the mass density of the fluid. To derive this formula for the speed of sound we again need to apply Newton s law. Consider a cylindrical volume of the fluid of cross sectional area A and lying between x and x + x. The net force acting on this volume is equal to pxt ( ) A px ( + xa ) = [ pxt ( )/ x] xa (because the pressure on the left surface of the volume is pushing the volume forward but the pressure on the right surface of the volume is pushing the volume backward). According to Newton s law it must be equal to (total mass of the volume) (acceleration of the volume) = ρ ξ 2 2 ( A x)[ ( x t) / t ].

4 Then using pxt ( ) = B[ ξ ( xt )/ x] to evaluate pxt ( )/ x we obtain: 2 2 ξ( xt ) ρ ξ( xt ) = x B t 2 υ which is the wave equation except that where 1/ should show up we obtained ρ / B. Thus we have also obtained the formula for the speed of sound in a fluid. If the fluid is a gas one has B= γ p0 where p 0 is the equilibrium pressure of the gas and obeys the ideal gas law p0v = nrt where V is the volume of the gas n is the number of moles of the gas T is the absolute temperature of the gas and R = (15) J / mol K is called the ideal gas constant. Then we find γ RT υ = M where M = ρv / n is the molar mass (or mass per mole) of the gas. You can tell that this formula is for a gas because it has used the ideal gas law to get T to show up. In a solid or liquid the temperature dependence of υ is much weaker. In this equation γ Cp / CV is the ratio of (specific heat at constant pressure) and (specific heat at constant volume). (Details in Chap. 19). It is equal to 5/3 = for a monatomic gas and 7/5 = 1.40 for a diatomic molecular gas etc. Air can be treated as a diatomic gas since it is mostly N 2 and O 2. If the sound wave is propagating in a solid the formula changes to: where Y is the Young s modulus of the solid. 3. Sound intensity (for a gas) υ = Y / ρ (for a solid)

5 Sound intensity is defined to be the time-averaged power delivered by a sound to a (real or imaginary) wall perpendicular to the sound wave per unit area of the wall. Power (i.e. rate of energy transfer) is equal to force times velocity. Force per unit area is just pressure. Therefore sound intensity is equal to the pressure of the sound wave times the velocity of the sound wave averaged over a period of the wave. That is: I = p( x t) υ( xt. ) t But for ξ ( xt ) = Acos( kx ω t+ φ) we have already found that pxt ( ) = BAksin( kx ω t+ φ) and υ( xt ) = Aω sin( kx ωt+ φ). Therefore we obtain: I = BωkA = ρbω A where we have used the fact that ω= υk with υ = B / 2 the sound is in a liquid) and that the time average of sin ( kx ω t+ φ) ρ (assuming that gives a factor (1/2). Since the pressure wave has an amplitude p max = BAk we can also write the intensity as 1 ω 1 υ 1 1 I = p = p = p 2 Bk 2 B 2 ρb max max max. These intensity formulas are convenient for sound waves in liquids only for which the values of B are essentially fixed and are tabulated. For gases the values of B are pressure dependent and are not tabulated so a different formula is more convenient: I = pmax ρυ which is obtained by using υ = B / ρ to elliminate B in favor of ρ. In a solid bar one must still replace the bulk modulus B by the Young s modulus Y. 4. Standing sound waves and normal modes

6 Standing waves can also be established in a sound wave. The physics is the same as that in a string except that the two end points of a string must be nodes but the end of a fluid (air or liquid) column can be a node (if closed) or an anti-node (if open). Standing sound waves can also be established in a solid bar. If the end of a bar is free it is an anti-node but if it is clamped there it is a node. These nodes and anti-nodes refer to the displacement wave. It should be reminded that the nodes of the displacement wave is also the anti-nodes of the pressure wave and vice versa. The different standing waves of a fluid column or solid bar is again called its normal modes. Again you can speak of first second third... harmonics and fundamental first overtone second overtone etc. 5. Organ pipes and wind instruments (or vibrating air columns) An air column can have two closed ends or two open ends or one closed end and one open end. A closed end must be a node of the standing displacement wave where the air molecules do not move at all. An open end must be an anti-node where the air molecules execute SHM of the largest amplitude. These conditions let you figure out the relation between the length of the air column L and the wavelength λ. Both ends closed: The relation between L and λ is the same as that of a string with two fixed ends. The only difference is that on a string the vibration is transverse and in a air column the vibration is longitudinal. Thus we still have λ n = 2 L / n and fn = nv/ 2 L. This situation does not appear in music instruments because if both ends are closed then no sound can come out! Both ends open: This situation is called an open pipe. Change nodes to antinodes and antinodes to nodes and you get exactly the same situation as the case of both ends closed. Thus for this case we also have λ = 2 L / n and f = nv/ 2 L. n One end open and one end closed: This situation is called a stopped pipe. This is the only new situation to need a new analysis. For the fundamental mode or first harmonic one must put ¼ of a wavelength in the air column so that the left end is an antinode and the right (closed) end is a node: n

7 So for this mode we have λ 1 = 4L and f1 = v/ 4 L. For the next higher mode one must put (¼ + ½) of a wavelength in the air column so that the left end is still an antinode and the right end is still a node but now one has one more node inside the air column [at (1/3)L from the left end and (2/3)L from the right end] : (Notice that the internal node can not be at the center of the air column but must be at 1/3 of the way from the open end!) Comparing with the previous graph one sees that the quarter wavelength is 1/3 smaller in this case than in the previous case. Therefore the wavelength must also be 1/3 smaller for this mode than for the fundamental mode. The frequency is then a factor of 3 higher. That is this mode is the third harmonic and there is no second harmonics in this air column. More generally we have λ = 4L n and fn (2n 1) (2n 1) f1 2n 1 = v 4L = where n = (You may also say: λ n = 4 L/ n fn = nυ /4L= nf 1 if only you put n = ) For example in the above boxed formula for n = 2 one has λ 2 = 4 L /3 which is just the third harmonic we have just discussed and for n = 3 we have λ 3 = 4 L /5 corresponding to the fifth harmonic with (1/ /2) of a wavelength in the air column so that the left end is still an antinode and the right end is still a node but now one has two more nodes inside the air column [at (1/5)L and (3/5)L from the left end]: We see that the second and fourth harmonics are not available in a stopped pipe (i.e. a pipe with one end closed) nor is any other even harmonic. 6. Quality of sound and noise

8 Most sound is a mixture of frequencies of different proportions in amplitudes with a dominant one (i.e. the one frequency with the largest amplitude) to give you its pitch. Those minority frequency components that are related to the dominant frequency component as harmonics of a single fundamental gives the quality of the sound. (Note that the dominant frequency is not necessarily the fundamental frequency.) Those other frequency components in the mixture not related to the dominant frequency component this way add noise to the sound and make it noisy. The more noisy a sound is the less pleasant it is to our ears. Sounds of different qualities can be all pleasant to our ears as long as they are not very noisy. A noisy sound may or may not have a pitch depending on whether it has a dominant component with a definite frequency or not. A somewhat noisy sound with a definite pitch can still be used in music such as the sound of a drum. The third aspect of a sound is its loudness which is related to its intensity. It is already covered in section 1. Most noises are mixtures of practically all audible frequencies in different proportions. It would then be called a white noise. If analyzed of their frequency components one will not get a discrete set of lines in an amplitude versus frequency plot (which is called a discrete spectrum) but will give a continuous spectrum instead. 7. Resonance Resonance of sound waves can happen just like other waves. Large amount of energy can be transferred to a resonating body from the source of the sound wave. it can cause a glass to shatter. The final steady state amplitude of the vibration in the resonating body depends on the Q value of the resonating body as explained before. The key is to match the frequency of the sound source with one of the normal mode frequencies of the resonating body. The amplitude of the vibration of the resonating body will then build up until it reaches a steady-state value which will be very large if the matching condition is met. 8. Interference of sound waves; Beats Such concepts as reflection refraction focusing interference and diffraction etc. are properties of all waves so they surely also apply to sound waves. This is why good music halls use only rough absorbing walls otherwise the sound coming directly from the source and the sound reflected by a wall can interfere constructively or destructively to produce particularly loud or weak spots inside the hall!

9 Beat Interference between two sound waves of slightly different frequencies (and wavelengths). The result is a sound wave at the average frequency with an amplitude which varies sinusoidally in time at half of the difference frequency. This is because sin( A+ B) + sin ( A B) = 2 sin Aco s B. A + B = 2π f t and A B = 2π f2t we obtain Putting in 1 f f f + f sin (2 π f1t) + sin (2 π f2t) = [2 cos (2 π t)]sin (2 π t) where the factor inside the square bracket which varies in time slowly can be viewed as a time-varying amplitude of a wave at the frequency of the last factor. One can multiply both sides by A to get the interference of two sinusoidal waves. The factor inside the square bracket on the right hand side would simply gain a factor A. When such a sound is heard its intensity which is proportional to the square of the amplitude will vary in time according to: f f It I t () = 4 0 cos (2 π ) where I 0 is the intensity of each wave assumed equal. Note that it varies in time between 4 I 0 and 0 with an average equal to 2 I 0. The average intensity is still that of the sum of the two incoming sound waves only now it comes as strong-weak-strong-weak... kind of variation in time at the difference frequency. This phenomenon is called beat. The beat frequency is the difference frequency f1 f2 [not ( f1 f2) /2 in spite of the fact that the above formula seems to say so. This is because the frequency of the function cos (2π f t) is f but the frequency of the function cos 2 (2π f t) is 2f twice as large! Plot the two curves and you will see why. The second function has a period half as long and a frequency twice as big!] 9. Doppler effect The sound frequency will appear to be higher if either the source is moving toward the listener or if the listener is moving toward the source or if both are moving toward each other. On the other hand the sound frequency will appear to be lower if either the source is moving away from the listener or if

10 the listener is moving away from the source or if both are moving away from each other. If one is moving away and the other is moving toward then it will depend on which one is moving faster: As long as the net result is that they are moving toward each other the frequency will appear to be higher and as long as the net result is that they are moving away from each other the frequency will appear to be lower. This is the Doppler effect. The formula is: f L - = v v v -v L S f S where v L is the velocity of the listener v S is the velocity of the source and v is the velocity of sound. The signs of these quantities are all defined in relation to the direction of the sound wave from the source to the listener. v L and v S are positive if they are moving in that direction and negative if they are moving in the opposite direction. Do not claim that you can look at the situation from the point of view of the source and find the source not moving and the listener moving at the difference velocity v - v L S or look at the situation from the point of view of the listener and find the listener not moving and the source moving at the difference velocity vs- vl. If you do that you must also change v to v- vs or v- v L respectively because the sound velocity is originally measured with respect to the sound-wave carrier (such as air or a solid bar). Indeed if the carrier is moving with a velocity v C (as when there is a wind in the air) you should change v to v+ vc since the carrier velocity v C (positive if in the same direction as the sound velocity) would then be helping the sound to move faster. When you move with the source (or listener) to look at the situation the carrier would be moving with velocity -vs (or -vl ) relative to you hence v+ v would become v- v (or v- v ). C It is easy to get the signs straight: Just make sure that they change the fre- quency in the right direction. If both v L and v S are zero you will simply get f = f Then consider each velocity at a time and see whether they L S. S L

11 change the frequency in the right direction. Finally get their combined effect by having both the numerator factor and the denominator factor. Why is this formula correct? Because a wave coming toward the listener (or receiver) can be viewed as a regular sequence of baseballs being thrown toward the listener. If the listener moves toward the source (the thrower) he can catch more balls (waves) in the same time period which means higher frequency. If he moves away from the source he will catch less balls lower frequency. On the other hand if the source (thrower) is moving toward the listener the spacing between the consecutive balls (waves) will be reduced shorter wavelength and if the source is moving away from the listener the distance between consecutive balls (waves) will be increased longer wavelength. With the speed of sound fixed shorter (longer) wave length also implies higher (lower) frequency. This is why the factor v- is in the denominator whereas the factor v- is in the numerator. (Each can be viewed as already divided by v as a correction factor.) Once you have obtained the new (apparent) frequency you can always obtain the new (apparent) wavelength by using the relation λnew = υ/ f new. v C Use v+ instead of v if the carrier of the wave is moving with velocity v C in the direction of v. Light is also a wave so it can also exhibit the Doppler effect. But the formula is different due to the relativistic effect (discovered by Einstein): f = c -v c + v v L R f S where f R is the apparent frequency to the receiver c is the speed of light and v vl-v S is the relative velocity of the receiver with respect to the source. It is positive if the listener is moving away from the source i.e. if the relative velocity is along the light wave direction. Note that we no longer care a- bout the separate velocities of the source and receiver but only their relative velocity. This is because light (an electromagnetic wave) propagates in vacuum with no carrier so different frames moving at constant speed with respect to each other are completely equivalent so one should get the same result by going to the frame moving with the source or the receiver. (The speed of light is always c in all of these frames! There can be no carrier velocity in this case.) v S (for light waves)

12 Thus if a star is moving away from the earth all the characteristic frequencies of light emitted by the atoms in that star will be shifted lower. This is called the red shift. It can be easily observed with a telescope and a spectrometer. From such measurements one can tell that the universe is expanding with all stars moving away from each other: The farther is the star the faster it is moving away from us. Tracing this expansion backward we can find out the age of the universe when the universe was just a point. This is the big bang theory of the universe: There was a big bang in the beginning about 13.7 billion years ago when the universe started the expansion from just a point. It is not just a theory if you think a theory is just a hypothesis. There are now a good number of concrete and quantitative evidences for it. A funny situation happens if the source of a sound wave is moving faster than the sound wave itself. The source will reach the listener before the wave it emitted reaches the listener. For sound this is no problem. But if this happens to light then you can move faster than the light you emitted and then turn around to see yourself at another place. Einstein found this situation not acceptable. He therefore postulated that nothing can move faster than the speed of light. This postulate let him discover the special theory of relativity from which he deduced the famous formula E = mc Shock waves and the Sonic boom Mach number (of a rapidly moving object) the speed of the object v = the speed of sound v When the Mach number of an object exceeds unity it is moving faster than the speed of sound (in the medium in which the object is moving). The speed is said to be supersonic. Otherwise it is subsonic. A supersonic object will generate a shock wave. If the object is very small it can be represented by a point. The shock wave is then a cone-shaped wave front of sound opening up in the backward direction from the direction of motion with the tip of the cone at the present location O of the small object. (A large object can be viewed as being made of many small objects located at slightly different locations each generating its own sonic cone.) The half angle of the sonic cone α obeys: S. υ S

13 v sin α =. v S A α O To see why consider a point A which is the position of the object a time earlier. The sound generated by the object at A at that time would have created a spherical wave front of radius AP= v t by now (the dotted circle). But AO = v t. This immediately gives the above formula for α S since sin α = AP/ AO. The wave front of the shock wave is the whole cone of half angle α since t can be any time period from 0 up to a large value (not too large since sound generated too long ago would have been absorbed by air molecules already). It should be clear that there is no sound at all outside of the cone and there is only weak sound deep inside the cone. The main shock wave is at the surface of the cone and just inside of it. So you will hear a loud but short sound pulse when a supersonic plane flies by and the shock-wave cone sweeps by you. This is called sonic boom. 11. Applications: sonar ultrasound and ultrasound imaging Sonar is a device to generate a sound pulse underwater or at earth surface and then to detect its reflection from an underwater or under-earth object after some time delay. The length of this time delay tells you how far the object is from you since the sound pulse has to travel to the object and then back to the detector. Since the sound is sent in a controlled direction you also know (roughly) in what direction the object is located. Usually the sound frequency is above the audible range and is about 20 to 100 khz and is referred to as ultrasound. Ultrasound is also used to image tumors inside a human body. High frequency sound is used for its short wavelength to reduce diffraction so that a good image can be obtained. Sound velocity inside a human body is about 1540 m/s (close to that inside water). In air the sound velocity depends on temperature T and obeys the formula: v sound in air ( T ) = 331m/s m/s T (in C ). This is because the air density changes with temperature. Note that if one combines the principle of sonar with the principle of Doppler effect one can also detect the speed of an object moving toward or away P t

14 from you. For this purpose one simply measures the frequency of the reflected wave and compare it with the frequency of the sound that is sent out by the sonar device. The moving object acts as a moving mirror since it can reflect the sound wave. If the sonar device is not moving then the reflected sonar wave can be regarded as being emitted by an image of the sonar device created by this mirror at equal distance behind the mirror as the sonar device is in front of the mirror. If the sonar device is not moving and this mirror is moving at speed v m toward the sonar device then this image is moving at 2v m toward the sonar device. Since the reflected wave can be regarded as emitted by this image of the sonar device this source is moving at the speed 2v m toward the sonar device. The frequency of the reflected wave is therefore f reflected sound v = v-2v In this way one can detect the location and speed of a submarine or a school of fish under water some distance away. But we are only talking about the speed of motion toward or away from the listener or receiver of the sound. Any motion perpendicular to the line connecting the source of the sonar wave and the reflector of the sonar wave can not be detected this way. If the sonar device is also moving with some speed then the analysis is a bit more complicated and is left as an exercise. Note that the receiver (detector) of the reflected wave is usually moving at the same speed as the sonar device because they are usually on the same boat or ship. So this speed actually enters in both the source speed and the listener s speed. m f Sonar.