Lecture #8-6 Waes and Sound 1. Mechanical Waes We hae already considered simple harmonic motion, which is an example of periodic motion in time. The position of the body is changing with time as a sinusoidal function. The total mechanical energy of the system is consered (in the absence of damping), while potential and kinetic energies of the system are changing as the square of the sinusoidal function. Een though the body was moing periodically, this process does not inole any motion of energy in space. All the energy is concentrated in the system and it is not traeling anywhere except changing from one form of mechanical energy to another form of mechanical energy inside of oscillator. Today we are going to consider a different periodic phenomenon, known as waes. This physical phenomenon gies us an example of how motion of energy (or information) can be conducted through space without actual motion of matter. Let us consider the following example. If you want to communicate with your friend who lies far away, you can either send him/her a letter by regular mail or you can use e-mail to do the same thing. Nowadays many people prefer the later method, because it is much faster and more reliable. The first method will inole the actual motion of matter (the letter) to your friend and this is the type of motion we hae already studied. The second way, howeer, does not inole such translational motion of particles or matter. Instead you send an electromagnetic wae which only delieries energy and information, but not any physical object. You will learn more about electromagnetic waes in Phys 1420. Now we are going to concentrate on so-called mechanical waes. The most common example of such a wae is the sound wae. We all know that we can pass information by means of sound, at the same time we do not actually hae to pass some material objects from person to person. Particle and wae are two essential concepts in physics. The former is the tiny concentration of matter capable of transmitting energy and located at certain point in space, the later is a broad distribution of energy filling the entire space without exact point of location. Almost all the areas of physics hae to do with these two concepts. Moreoer, you will see later that sometimes, it is not een possible to distinguish between
particle and wae, as in the case of elementary particles which can demonstrate both types of properties. The main types of waes in nature are 1) Mechanical waes. This is the most familiar for eeryday life type of waes, because we are actually using them almost eery day in the form of sound. We can see them as waes on the water surface and een sometimes can feel them as seismic waes. All these waes are goerned by Newton's laws and they can only exist in material medium, such as gas, liquid or solid. 2) Electromagnetic waes. These waes are also ery common, but maybe less familiar. They include light waes of all types, radio waes, microwaes, radar waes and others. These waes do not require any medium to exist. The uniersal property of these waes, that they all hae the same speed in acuum c=299792458 m/s. 3) Matter waes. These waes are associated with electrons, protons and other fundamental particles. They are subject for the modern quantum mechanics and field theory. The only type of waes we shall study in Phys 1410 is the first type of waes: mechanical waes. The simplest example of the mechanical wae is the wae sent along the stretched taut string. This becomes possible because the string is under tension. So, if the string is pulled up and down at one end, it begins to pull up and down the adjacent section of the string and then the next section and so on. As a result the pulse will trael along the string with some elocity. If a person who pulls the string will continue to do that on periodic basis the continuous wae will trael along this string. If the external force causing this motion has the form of the simple harmonic motion, then this wae will hae the shape of sinusoidal function and will be called harmonic wae. In this example we hae made an assumption that there is no friction in this string, so this wae will not die out while traeling. And the string is long enough, so we do not hae to take into account the other wae traeling in the opposite direction as a result of rebound from the opposite end of the string. If we consider the motion of some element of this string we will see that it moes up and down in the direction perpendicular to the direction of propagation of the wae. This is why we call that type of wae transerse wae.
Transerse wae is not the only type of mechanical wae. As an alternatie we can consider a wae produced by a piston in a long air-filled pipe. In a same way as the person was pulling on the string, he/she can moe the piston rightward and leftward. This motion of the piston can be a simple harmonic motion. At first it will cause the increase of air pressure right next to the piston then this pressure increase will moe to the next section of the pipe and so on. When the piston moes back, the pressure will decrease, this decrease will also trael along the pipe. As a result we hae a sound wae in this pipe. Howeer, this wae is different from the one we considered before, because now different elements of air in the pipe are moing in the same direction as the direction of wae s trael. We shall call such a wae to be longitudinal wae. Both transerse wae and longitudinal wae are said to be traeling waes, because they both trael from one point in space to another point in space. It is the wae that moes but not the material. In both cases elements of the string or elements of the air in the pipe do not trael anywhere. They just oscillate near the same positions. Waes are traeling at certain speed. In solid substances, waes of both types: transerse waes and longitudinal waes can exist. Moreoer, those waes hae different speeds. This fact is often used to determine the unknown distance to some object. The longitudinal wae with speed 1 usually traels faster than transerse wae with speed 2. Let d be the unknown distance to the object, which emits both types of waes. It will take time t 1 d 1 for the first wae to trael from the object to the obserer and time t 2 d 2 for the second wae. The obserer can measure the time difference t t2 t1, which passes between detection of two waes, so d t 2 1 t d d, 1 2 1 2. This is how one can find the unknown distance.
Let us consider the transerse wae in the string. This is an example of the plane wae, which propagates in the direction perpendicular to the plane surface. Let us call this direction to be x-direction. Since it is transerse wae, the elements of the string will moe in the direction perpendicular to the x-direction. We will call that direction to be y- direction. This means that positions of the elements of the string will change with time and it will be different for different alues of x. To describe this wae we will need to know this function y b g for the displacement of the elements in the string. We shall y x, t only consider sinusoidal waes, behaing according to, sin y x t A kx t. (8.6.1) In this equation A is called the wae amplitude, k is called the wae number and is wae angular frequency. The wae described by equation 8.6.1 is moing in positie x direction. Let us clarify the physical significance of all these constants in equation 8.6.1. A stands for the wae amplitude, because it is the maximum alue, which y ariable can achiee (the sin-function can not be larger than one). The phase of the wae is the argument of the sin-function kx t. If we fix x in this argument, we can watch how the position of the string changes at certain location in space. According to equation 8.6.1 it will oscillate in a simple harmonic motion. If we fix time t, we can see the snapshot of this wae in space. It will look like a sin-function, which changes along x direction. It is periodic function which will reproduce its shape along axis x. The smallest distance between repetitions of the wae shape along axis x is called a waelength. Let us find how this quantity is related to other parameters in equation 8.6.1. According to definition of the waelength, at certain moment in time,, 2, y x t y x t Asin kx t Asin k x t, kx t k x t 2 k. This means k 2. (8.6.2)
So the wae-number of the wae equal 2 diided by the waelength. The SI unit for the wae-number is radian per meter (rad/m). Now let us see how the position of the string's element changes with time at certain location x in space. This element moes up and down in simple harmonic motion. The motion, as we know, is the example of the periodic motion. What is the period of this motion and how it is related to the other parameters in equation 8.6.1. Since the position of string's element has to be the same after the time interal equal to the period, we hae,,, 2 y x t y x t T Asin kx t Asin kx t T, kx t kx t T 2 T. This means that 2 T. (8.6.3) The wae angular frequency is equal to 2 diided by the period of wae. The SI unit for the angular frequency is radian per second (rad/s). We can also define the linear frequency of wae as f 1 T 2. (8.6.4) It is the number of oscillations made per unit of time, usually measured in Hertz. We hae already mentioned that wae moes in space with certain elocity. Een though this moement is not actually related to the motion of matter from one point in space to another, it has to do with motion of the wae pattern. This wae pattern moes for distance x during time t, so we can talk about wae s speed x t. This speed is also known as a phase speed, because the wae's phase remains constant, as the wae moes in positie x direction, which is kx t const. So, we hae x k 0, t x. t k
Then the wae s speed is f. (8.6.5) k T The last equation shows that the wae moes for the distance of one waelength during the time interal equal to the wae's period. So far we were considering wae moing in the positie direction of axis x. We can also introduce a wae, which moes in the negatie direction by replacing the sign in equation 8.6.1, which is y Asinkx t. (8.6.6) Exercise: Show that this wae is indeed moing in negatie direction (has a negatie speed). Now let us consider a special example of wae traelling in the stretched string and see how the speed of this wae is related to the properties of the medium of the string. Since the elements of the string will oscillate, when the wae passes the medium, those elements should hae both mass and elasticity to make these oscillations possible. So the speed of the wae can be calculated in terms of the string's elements mass and tension in the string. At this point we can use the technique we learned during ery first lectures, known as dimensional analysis. Since we are looking for speed, it has the dimension of length died by time l t, Talking about a small element of the string, we can introduce its mass by means of the linear mass density, which has the dimension of mass per unit of length m l. Finally to describe elasticity of the string, which takes place due to the stretching under tension, I shall introduce force of tension F, which has dimension of force So I hae ml t. F 2
F q, p l ml m 2 t t l 1 pq, p q, 1 2 p, 0 pq, the solution of this set of equations is 1 1 p, q, 2 2 F. The wae s speed is proportional to the square root from tension oer mass density. But we can not find coefficient of proportionality just based on dimensional analysis, so we need to use Newton's second law to find it. It turns out that this coefficient is indeed equal to 1. F. (8.6.7) The speed of the wae along a stretched ideal string depends only on the tension and linear density of the string but not on the frequency of the wae. 2. Sound The most common example of mechanical wae is a sound wae. The sound wae from the point like source in air is longitudinal spherical wae. On the other hand sound waes can exist in a ariety of media such as fluids, solids and/or gases. Sound waes in water can be used by submarines to detect positions of the unknown objects. Sound waes in solid medium, such as ground, are used to study the seismic actiity and other properties of the Earth's crust. The simplest type of these waes is the sound wae in air. We are using them eery day, the only reason why you can hear my lecture today is due to the existence of the sound waes. When we discussed kinematics, we introduced the idea of the point-like or particlelike object. The size of this object is much smaller compared to the distance of its trael. Using the same idea, we can introduce the point-like source of sound. For instance, I am almost point like source of sound compared to the size of this auditorium. This means
that sound waes (of my oice) are traeling in all directions from me, so eerybody can hear. This is different compared to situation of the string, where transerse wae was going in one direction of x-axis only. In the case of the string a waefront, the surface where oscillations of matter hae the same phase is a planar surface. In the case of the sound wae propagating in all directions along so called rays in the air, the waefronts are spherical. Those waefronts are perpendicular to the rays, showing the direction of the wae s propagation. In the case of the longitudinal wae the oscillations of air are also taking place in the direction of those rays perpendicular to waefronts, while for transerse wae they take place in the direction of the waefront perpendicular to the rays. If, howeer, we consider a spherical wae at ery large distance from the source, the waefront's curature is almost negligible and we can treat it as a planar wae. We saw that the speed of the transerse wae in a medium depends on both inertial properties of the medium and elastic properties. In the case of the string the inertial property associated with its kinetic energy was the linear mass density and elastic property associated with its potential energy was tension in the string F. This general idea remains alid for any type of wae, but the particular choice of ariables aries. Let us see how one can find the speed of sound in fluid. Since this wae is propagating in three dimensions in space, we hae to replace the linear density by the olume density (mass of the fluid per unit of olume). The elastic properties of the fluid are related to periodic compressions and expansions of the small olume elements of this fluid. The socalled bulk modules p B V V (8.6.8) is responsible for these changes. In this equation p is change of pressure (force acting per unit of area) and V V is fractional change of olume. B is always positie quantity, because for any stable substance the increase of pressure causes decrease of olume. B has the same dimension as pressure, which is dimension of force per unit of area. Following the dimensional analysis one can arrie to the following equation for speed of sound B, (8.6.9)
which is actually correct equation, een with correct coefficient of proportionality. Exercise: Proe equation 8.6.9 based on dimensional analysis. Equation 8.6.9 can also be proed based on the Newton's laws. The speed of sound, as we can see from equation 8.6.9 depends on substance, since different substances hae different densities as well as on temperature, since density also depends on temperature. In the air under the normal atmospheric pressure and temperature, the speed of sound is 343m s 770 mi h. Table 16.1 in the book shows alues of speed of sound for some other substances at different conditions. Let us consider traeling sound wae in the tube filled by air. This wae appears due to the motion of the piston at the end of the tube from the left to the right and backward in the sinusoidal pattern. This motion will cause the air elements at the end closer to the piston change their density according to the same sinusoidal law as a result of compression and then those changes of density will trael along the tube as a sound wae with speed, which we just calculated. Each element of air with thickness x in the tube will oscillate according to the same sinusoidal law as the piston does. In the longitudinal wae, those oscillations take place in the same direction x as this wae propagates. To aoid confusions I will use letter s to show this displacement. Let this wae hae a form of the cos-function, so, cos s x t A kx t, (8.6.10) where A is the amplitude, which is the maximum displacement of the air element to the either side from equilibrium position. All other characteristics of the wae, such as k,, f,, T are defined in a same way as they were in the case of transerse wae. The change of pressure along the wae takes place in the opposite phase compared to the change of displacement, but also as a harmonic function. Equation 8.6.10 inoles angular frequency of the sound wae 2 f. It is the frequency of sound, which makes a great impression, when we hear sound. The human ear can hear sound in between 20 Hz on the low frequency end to 20,000 Hz on the highfrequency end. The sound with frequency aboe this rage is referred as ultrasonic, lower than this range as infrasonic.
Another important characteristic of sound is its intensity, which actually shows how strong the sound is at particular point in space. For instance, you cannot sleep at night if somebody plays music loudly. This is because intensity of this sound is too high. Intensity of sound I at certain surface is the aerage power per unit area transferred by this wae trough or onto the surface I P. (8.6.11) A If we consider a point-like source of sound, which emits sound waes isotropically (with equal intensity in all directions) and if the sound wae's energy is consered, so it is not dissipated in space, then we can write that Ps I. (8.6.12) 4 2 r Here P s is the power of the source and r is the distance from the source. As you can see, the sound intensity gets smaller with distance, not een because of the reflections and scattering from different objects, but because the same energy spreads oer larger and larger areas of the spheres 4r 2 surrounding this point-like source. When talking about sound we usually mean sound range, which can be heard by ordinary human ear. This rage is enormous. Human ear can distinguish between sounds, which intensity aries up to 10 12 times. This is why it is not ery conenient to use intensities on such a huge scale. Instead the logarithmic scale is often used to deal with this enormous range. Let us introduce a new quantity, known as intensity leel or sound leel, which is I b10dbglog I. (8.6.13) 0 Here db is abbreiation for decibel, the unit of sound leel, named after Alexander Graham Bell. I 0 10 12 2 W m stands for the standard reference intensity, chosen because it is near the low limit of the human range of hearing. This standard reference leel corresponds to zero decibels. So if the sound intensity increases 10 times, the sound leel increases only by 10 decibels. Intensity of sound is related to all the characteristics of sound wae, such as its amplitude, elocity and frequency. It is proportional to the square of amplitude of sound as well as to the square of its frequency.
I will only be talking about Doppler Effect (the rest of the lecture starting from here) if we hae time left. Another interesting question is how sound changes if the source of sound and the obserer who receies this sound are in the state of the relatie motion. You can see this effect in eeryday life. It is quite often, when a firefighter's car with siren on passes you on the street. You might notice that the sound of this siren is ery different before and after the car passes you. It changes frequencies depending if the car is moing towards you or from you. This effect is known as the Doppler Effect, since it was first considered by the Austrian physicist Johann Christian Doppler in 1842. Een though this effect works for any type of waes, here we only consider it for sound waes in the air. The air itself will be considered as a reference frame. So we can measure elocity of the source of sound S as well as of the detector D relatie to the air at which sound is traeling. For simplicity, I will assume that S and D are moing along the straight line with speeds less than speed of sound in air. In this case the emitted frequency of sound f is related to the detected frequency f as f f D S, (8.6.14) where is speed of sound, D is the detector's speed relatie to the air and S is speed of the sound's source relatie to the air. When the motion of detector or source is towards each other, the sign of its speed must gie an upward shift in frequency. When the motion of the detector or source is away from each other, the sign of speed must gie downward shift in frequency. In the example I mentioned, when the firefighters car is moing towards you, you hear the higher frequency compared to one which is actually emitted, when it is moing away from you, you hear lower frequency. Let us consider two special cases 1) Detector moes relatie to the air and the source is stationary. The detector moes towards a stationary source, which emits a spherical sound wae (in all directions) with speed, waelength and frequency f. If the detector is stationary, the waefront will moe during time t for distance t. In this case the number of waefronts detected by the detector for this time is t, so the detected frequency is f t as it should be for the real wae. If, howeer, the detector is moing towards t
the source, their relatie elocity is and so the detected frequency is going to be f b g t t f D f D D D D. If detector moes away from the source then the relatie elocity is going to be D and we hae f f D. So, the equation 8.6.14 is proen for the moing detector and stationary source. 2) Source moes relatie to the air and detector is stationary. The source moes towards D with speed S and emits wae with frequency f. It takes time T 1 f between the emissions of two waefronts. During this time S moes for distance T S and waefront for distance T. This means that the waelength detected by D, which is distance between two waefronts will be T S T, so f T T f f f S S S. If the source moes in the opposite direction, we need to change the sign of its elocity. So the equation 8.6.14 is also proen for the moing source. The general Doppler equation is just combination of those two cases. This equation only works for the speed of the source less than speed of sound. Indeed the detected frequency becomes infinite, if the source moes with speed of sound, which means it keeps pace with its own spherical waefronts. In the case of the supersonic speed the source will moe faster than its own sound wae and it will create a cone of waefronts known as the Mach cone. A shock wae exists along the surface of this cone, producing a burst of sound known as the sonic boom.