 This is the second lecture of the four lecture series on acoustics. In the last lecture which was the first one, we considered sound generation and propagation various types of wave motion in particular harmonic waves. So, it was pointed out that the elasticity of the medium is the very important aspect. Sound waves are mechanical waves. The particles of the medium vibrate and therefore elastic properties determine how the wave will propagate in the medium. We also considered an equation, a very general equation for wave motion for propagation along the x axis. This equation was a partial differential equation in x and t. We considered its general solution which were in the form of the solution of the combination x minus vt or x plus vt giving rise to waves moving along the positive direction of x axis or the negative direction of x axis. No other form of the solution is admissible for this equation. Then we considered waves of different types. The transverse waves where the particle motion is perpendicular to the direction of propagation. It is in the transverse plane. It can have any direction, but always remaining in the transverse plane. A typical example was pointed out as the vibrations of a stretched string which are a string for example and the longitudinal waves where the particle motion is along the direction of propagation. This is a very common thing because the sound waves in air are longitudinal. Then we considered in particular harmonic waves because the particles vibrate. The most of the time they vibrate simple harmonically because the force restoring force due to elasticity comes out to be proportional to the displacement. We considered various properties of the harmonic waves, their main characteristics. Now in the present lectures this one we shall consider principle of superposition, formation of beads and stationary waves. We shall consider phenomena of reflection, refraction and diffraction of sound. Principle of superposition. Let us now see how the resultant of two or several waves is evaluated. Since the equation of wave motion is linear and is homogeneous. Therefore, the displacement psi and its derivatives occur always in the form of first degree. This psi 1 and psi 2 are any two solutions of the wave equation. Any combination like A1 psi 1 plus A2 psi 2 is also a solution where A1 and A2 are arbitrary constants. From this we conclude that we may superpose any number of individual solutions to form new functions which are also solutions in themselves. Therefore, in general it may be stated that when two or more wave trains are superposed the resultant displacement at any point is equal to the vector sum of the individual displacements there. This is known as the principle of superposition. Thus psi will be equal to psi 1 plus psi 2. If psi 1 and psi 2 are two displacements and psi is the corresponding resultant displacement see this principle has wide applicability. Let us consider superposition of two harmonic waves moving in the same direction. Let psi 1 equal to A cos of omega 1 t minus k 1 x plus alpha 1 for the first wave travelling along the positive direction of x axis and the other one psi 2 equal to A times cos of omega 2 t minus k 2 x plus alpha 2. This is the second one. So, these two are there alpha 1 and alpha 2 are arbitrary initial phases. Now according to the principle of superposition the resultant displacement is given by psi is equal to psi 1 plus psi 2 just vector sum this gives A times cos of omega 1 t minus k 1 x plus alpha 1 plus A times cos of omega 2 t minus k 2 x plus alpha 2 this gives 2 A times cos of omega 2 minus omega 1 by 2 times t minus k 2 minus k 1 by 2 times x plus alpha 2 minus alpha 1 by 2 multiplied by cos of omega 2 plus omega 1 by 2 times t minus k 2 plus k 1 by 2 times x plus alpha 2 plus alpha 1 by 2. This is a general form of the expression for the two waves of equal amplitude both having amplitude A progressing in the same direction both per progressing along the direction of positive x axis superposing on each other. We shall make its use in the following. Let us consider the phenomena of beats. This phenomena occurs when two wave trains of nearly equal frequencies omega 1 is almost equal to omega 2 k 1 is almost equal to k 2 when such waves they overlap again using principle of superposition psi is equal to psi 1 plus psi 2. This is the same expression as I had a little while ago but now consider this equation with initial phases at set to 0 just for simplicity. If you put alpha 1 and alpha 2 equal to 0 we get what psi which is equal to psi 1 plus psi 2 2 A times cos of omega 2 minus omega 1 by 2 times t minus k 2 minus k 1 by 2 times x multiplied by cos of omega t plus omega 1 by 2 times t minus k 2 plus k 1 by 2 times x. This equation represents a wave motion determined by the factor cos of omega 2 plus omega 1 by 2 times t minus k 2 plus k 1 by 2 times x. Remember the frequencies are nearly equal omega 1 is almost equal to omega 2 k 1 is almost equal to k 2. Essentially this factor is cos of omega t minus k x with the amplitude usually the amplitude is constant but here in this case the amplitude is 2 A times cos of omega 2 minus omega 1 by 2 times t minus k 2 minus k 1 by 2 times x. If we put omega 2 equal to omega 1 and we put k 2 equal to k 1 this will be cos of 0 which is 1 and the amplitude will be just equal to a constant value 2 A. But here we find this is the expression which varies with t and also x. The nature of this wave motion can be easily understood by analyzing the amplitude term as a function of t as a function of time at some fixed point let us take x is equal to 0 just for simplicity does not affect any physics. The amplitude is then 2 A times cos of omega 2 minus omega 1 by 2 times t this oscillates with time in the maximum value 2 A and the minimum value 0. The amplitude is maximum when the force factor is equal to plus minus 1 this means at times t equal to 0 or 2 pi upon omega 2 minus omega 1 or 4 pi upon omega 2 minus omega 1 or 6 pi upon omega 2 minus omega 1 or in terms of frequency at times t equal to 0 1 upon nu 2 minus nu 1 2 upon nu 2 minus nu 1 3 upon nu 2 minus nu 1 where nu 1 is omega 1 by 2 pi nu 2 is omega 2 by 2 pi. Similarly the amplitude is minimum when the force factor is 0 this means the argument is like pi by 2 or 3 pi by 2 or pi by 2 that is at t equal to pi upon omega 2 minus omega 1 or 3 pi upon omega 2 minus omega 1 or 5 pi upon omega 2 minus omega 1 like this or in terms of frequency at times t equal to 1 upon twice of omega 2 minus omega 1 or 3 upon twice the omega 2 minus omega 1 or 5 upon twice the omega 2 minus omega 1 these types. You will find that there is a minimum amplitude between any two consecutive maxima. You see the time interval between any two consecutive maxima or minima is 1 upon omega 2 minus omega 1 with an interval of this a maxima repeats or a minima repeats. Hence the frequency of appearance of maxima or minima of this amplitude is nu 2 minus nu 1 this phenomena the maxima and minima constitute beats and therefore, we say that the number of beats per second is nu 2 minus nu 1 just the difference of the two frequencies. Here this figure this figure is an envelope of this transient amplitude modulation resulting from the superposition of the waves a and b of slightly different frequencies. The frequency of the waves is in the audible range one can hear it there will be vexing and waning of sound which is detectable by the. So, this is a very interesting phenomena whenever we have two sources of slightly different frequencies and they are sounded together. For example, if the two tuning forks say a frequencies 500 vibrations per second and 502 vibrations per second the difference of two if they are sounded together we expect two beats per second this means in a second the sound will be maximum at two instance and minimum at two instance is very interesting and easy is a simple experiment can be done in anti lab. Let us consider stationary waves now these occur when two identical plane harmonic waves moving in a positive direction remember earlier we consider superposition of two waves in the same direction. Now, we are considering two plane harmonic waves moving in a positive direction incident and reflected waves and they now they overlap. These two waves moving respectively towards right and left are psi 1 equal to a cos of omega t minus kx moving towards right that is positive direction of x axis and the other one psi 2 a cos of omega t plus kx plus alpha moving towards the other direction negative direction of x axis then using the principle of superposition as before the total displacement psi is given by psi 1 plus psi 2 remember these displacements by themselves can be along the direction of x axis if these waves are longitudinal or they can be in the transverse plane if these waves are transverse all right. So, psi is equal to psi 1 plus psi 2 it means a cos of omega t minus kx plus a cos of omega t k plus kx plus alpha which means 2 a cos of kx plus alpha by 2 multiplied by cos of omega t plus alpha by 2 this equation corresponds to what is called a stationary wave. Since there is no result in progressive motion there is no energy transfer to the right or left that is why these waves are called stationary waves as against progressive waves with progress in some direction. Now, the amplitude of the stationary wave is 2 a cos of kx plus alpha by 2 which varies from point to point remember again the amplitude of a progressive wave is constant does not change does not vary from point to point, but here for a stationary wave we find the wave is a stationary, but we find that the amplitude of such a wave is not same for all values of x the amplitude is 0 at places where this factor is 0 that is where kx plus alpha by 2 is equal to an odd multiple of pi by 2 taking n as integer value 1 2 3 which means for x equal to 2 n minus 1 pi by 2 minus alpha by 2 times lambda by 2 pi which gives 2 n minus 1 minus alpha by pi times lambda by 4 does these values are values of x 1 minus alpha by pi times pi by 4 3 minus alpha by pi times pi by 4 5 minus alpha by pi pi by 4 like this the successive points we see at which the amplitude of the displacement is 0 or lambda by 2 distance apart these points are known as displacement nodes the particles at these points remain permanently at rest they just do not move it is very interesting phenomena the waves are there in the region as a result of superposition of a direct and the reflected wave and we find these are the points which remain permanently at rest the displacement is 0 for all times you have a maximum amplitude where the cos factor is equal to plus minus 1 that is kx plus alpha by 2 is equal to n pi this pi 2 pi 3 pi and this gives x equal to n pi minus alpha by 2 times lambda by 2 pi which gives 2 n minus alpha by pi times lambda by 4 for x this gives the values 2 minus alpha by pi times lambda by 4 4 minus alpha by pi lambda by 4 or 6 minus alpha by pi lambda by 4 like these these points having maximum amplitude or again lambda by 2 distance apart these are known as displacement anti nodes these are the points where the amplitude is maximum which is equal to 2a remember a is the amplitude in the individual waves and at anti nodes the amplitude is 2a and 2 consecutive anti nodes as I said are separated by lambda by 2 a node is separated by a distance lambda by 4 from its nearest anti node between 2 anti nodes there is a node similarly between 2 nodes there is an anti node now there is another interesting feature consider again the stationary wave equation gave an above psi equal to 2 a cos of kx plus alpha by 2 times cos of omega t plus alpha by 2 we had this earlier the interesting thing is at time t given by this expression omega t plus alpha by 2 is equal to an odd multiple of pi by 2 the time dependent cos factor is 0 and therefore, psi is 0 for all x all through for all values of all the particles are passing through the mean position simultaneously for all values of x psi is 0 what about that velocities at this incident we find that those between first and second nodes I am leaving sec between second and third first and second nodes third and fourth node fifth and the sixth nodes etc those in the alternate segments have their velocity in one direction similarly those between second and third nodes fourth and fifth nodes sixth and seventh nodes they have all their velocities in the opposite direction it appears something like this the region is just gets divided into segments by the nodes all the particles in any one segment between any 2 consecutive nodes are in the same phase all of them pass through their mean positions at the same time and have their velocities in the same direction they pass through their mean positions at the same time and in the same direction all of them are in the same phase then those in the adjacent segment are in the opposite phase this means if the particles in one segment are going towards right all those in the adjacent segment are going towards left at the same time ok now we consider another interesting thing an experimental setup could stew experiment this experiment early in the beginning it was devised to measure velocity of sound in different materials but we are using it for a different purpose it provides a very simple setup to visually demonstrate formation of nodes and anti nodes in the stationary waves the setup consists of a horizontal glass tube about 50 meter long and a few centimeters in diameter at one end of it an adjustable piston is fitted the other end is closed by a loosely fitted cardboard cap B within the tube it is firmly attached to the metal rod BC the rod is clamped in the middle at the point D the tube itself is clamped on horizontal heavy table now before performing the experiment the tube is thoroughly dried and then a small amount of lycopodium powder is scattered in the gap AB of the tube the part DC of the rod is now rubbed with the resin cloth rubbed along the length by doing so the rod is set up in longitudinal stationary vibrations with node in the middle at the point D which is clamped and anti nodes at the two free ends at B and C the disc B now vibrates forward and backward you see the rod is a longitudinal motion so the disc B vibrates forward and backward due to which the air column inside the tube also vibrates the same frequency the frequency of the rod now the position of the piston a is adjusted in such a way that the air column in the tube resonates resonates means the natural frequency of the air column in the tube is same now as the frequency with which the rod is vibrating the natural frequency of the air column can be adjusted by adjusting the position of the piston a and this is the experiment so the position of the piston a is now adjusted in such a way that the natural frequency of the air column becomes equal to the frequency of the rod and the resonate and the air column sounds loudly to the node produced by the rod this is indicated by the violent motion of the lycopodium powder which is there in the tube at various places along the tube now we are there are stationary waves in the tube these waves are formed by the superposition of direct and reflected waves they are reflected by the piston a you have incident and direct wave of the same frequency one traveling in one direction the other traveling in the opposite direction and stationary waves are formed nodes and anti nodes are formed the powder gets gathered in a small heaps at the nodes as there is no motion there and gets displaced from the anti nodes as shown in the figure it is a very clear simple demonstration of the formation of nodes and anti nodes alternately and also the property that there is no motion at the nodes now we come to the study of this phenomena of reflection a reflection and the fraction of sound waves when a sound wave is incident upon a surface a portion of its energy is absorbed by the surface and the remainder bounces back have become reflected from the surface a perfectly hard surface will reflect back all of the energy a perfectly hard surface really does not exist the idea is if this the harder surfaces more will be the reflection coefficient this figure shows the incidence of a series of plane wave fronts on the reflecting surface a a prime the arrows normal to the wave fronts or rays which represent the direction of propagation or drawn to represent the incidence and the consequent reflection of the wave front the angle of incidence theta i is equal to the angle of reflection theta r note that is stationary wave patterns will occur from these reflections let us consider the sound field resulting from the reflection consider plane harmonic waves the intersection of these waves along the normals to the reflecting surface constitutes a projection of the incident and reflected waves from the concept of a motion the distance between crests I mean between consecutive crests or consecutive compressions or consecutive rare factions along the normal is like projected wavelength lambda prime which is related to the wavelength lambda of the incident wave as follows lambda prime is equal to lambda divided by cos of theta i which is equal to lambda sec of theta i in obeying the laws of reflection the reflected wave also produces a traveling wave the projected wavelength also equal to lambda prime hence where occurs along any normal line the superposition of two waves traveling in opposite directions with wavelength lambda prime from the concept of a stationary waves it can be inferred the straight away that nodes and anti nodes occur along the normal line and the spacing between them needs only to be modified by the factor sec theta i for the spatial case of theta i equal to 0 which means the normal incidence the nodal spacing reduces to the standard value lambda by 2 between any two nodes or any two anti nodes as the angle of incidence increases the spacing between the nodes likewise increases and in the limit theta i equal to pi by 2 there is no reflected wave and thus the stationary wave field vanishes the phenomena of sound wave reflection finds many applications see the time it takes for a sound wave pulse to travel from a transducer at sea level to the ocean bottom and for the eco to travel back gives a measure of depth of the water further comparison of the spatial characteristics of the reflected wave with those of the original generated waves provides an ample measure the geological composition of the ocean bottom for example the occurrence of silt or rock or sand or coral and so on reflected sound is also used in an analogous way by geologists to gauge the depth and composition of a stratified layers in the earth crust to locate the occurrence of oil natural gas or mineral deposits let us now consider refraction this phenomena is more familiar in optics than in acoustics here the direction of the advancing wave front is bent away from the state line of travel refraction occurs as a result of the difference in the propagation velocity as the wave travels for one medium to a different medium in the optical situation refraction occurs rather suddenly is the wavelengths are very small when the light waves cross the sharp interface between the atmosphere outside and say glass at the surface of a lens because light travels with slower speed in glass then what it does in air at audible frequencies of sound waves the wavelengths are so long that refracting apparatus would have to be extremely large in order to render observable acoustic refractions this picture very similar to what one have in optics the propagation is from medium 1 to medium 2 velocity in medium is v 1 then velocity in the second medium is v 2 the refracted ray moves away from the normal or towards the normal is really determined whether the velocity v 2 is more or less relatively basically the basic structure is essentially the same the basic law of refraction sin theta incident divided by v 1 is equal to sin theta refraction divided by v 2 theta incidence is the angle of incidence theta area refract is the angle of refraction v 1 is the speed of sound in medium 1 v 2 the speed of sound in medium 2 the above relation is analogous to this nels law or light refraction see the analysis of acoustic refraction does not easily figure prominently we most of the time we really do not bother very much about it in a caustic studies but we cannot overlook the fact that zones of severe temperature difference and thereby severe velocity difference do occur in the atmosphere and oceans when sound travels from zone to zone often across regions of severe temperature gradients the direction of propagation changes majorly to an extent which cannot be ignored for example the surface of the earth heats up more rapidly then the atmosphere on a sunny day the temperature of the earth close to the ground rises correspondingly now as the speed of sound is higher in the warmer lower surface sound waves traveling horizontally are refracted upwards similarly on a clear night the earth crust cools more quickly and a layer of cooler air forms and bends the sound waves downwards towards the surface towards the earth surface does noise from industrial plant for example would be refracted downwards at night and which seem louder to a homeowner residing near the plant then during the day when the upward refraction occurs this is quite often the case let us now consider the fraction of sound this figure shows sound waves incident on a partial barrier some of the sound is reflected back some continues onwards unimpeded and some of the sound bends are diffracts over the top the barrier costs and acoustical shadow which is not defined sharply another example of the fraction is bending of sound around the building corner usually can hear voices on the other side of a wall that is approximately 3 meters or so high it is a wavelength dependent effect the sound at lower frequencies larger wavelengths tend to deflect over partial barriers more easily than the sound at higher frequencies moreover the sharpness and extent of the sound zone behind the barrier depends on the relative positions of the source and the receiver the closer the source is to the barrier the longer is the shadow zone on the other side of the barrier and that is greater is the sound reduction that is all we need to know about sound diffraction so we have come to the end of this lecture.