WAVE THEORY, SOUND WAVES & DOPPLER'S EFFECTS

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1 ITRODUCTIO OF WVS What is wae motion? J-Physics When a particle moes through space, it carries K with itself. Whereer the particle goes, the energy goes with it. (One way of transport energy from one place to another place) There is another way (wae motion) to transport energy from one part of space to other without any bulk motion of material together with it. Sound is transmitted in air in this manner. x. WV THORY, SOUD WVS & DOPPLR'S FFCTS You (Kota) want to communicate your friend (Delhi) Write a letter Use telephone x. st option inoles the concept of particle & the second choice inoles the concept of wae. When you say "amaste" to your friend no material particle is ejected from your lips to fall on your friends ear. Basically you create some disturbance in the part of the air close to your lips. nergy is transferred to these air particles either by pushing them ahead or pulling them back. The density of the air in this part temporarily increases or decreases. These disturbed particles exert force on the next layer of air, transferring the disturbance to that layer. In this way, the disturbance proceeds in air and finally the air near the ear of the listener gets disturbed. ote :- In the aboe example air itself does not moe. wae is a disturbance that propagates in space, transports energy and momentum from one point to another without the transport of matter. Few examples of waes : The ripples on a pond (water waes), the sound we hear, isible light, radio and TV signals etc. CL SSIFICTIO OF WVS Wae classification ccording to ecessity of medium Propagation of energy Dimension Vibration of particle \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65 (i) lastic or mechanical wae (ii) lectro magnetic wae (.M. wae) or non-mech. (i) Progressie (ii) Sationary (i) One dimensional (ii) Two dimensional (iii) Three dimentional (i) Transerse (ii) Longitudinal. Based on medium necessity :- wae may or may not require a medium for its propagation. The waes which do not require medium for their propagation are called non-mechanical, e.g. light, heat (infrared), radio waes etc. On the other hand the waes which require medium for their propagation are called mechanical waes. In the propagation of mechanical waes elasticity and density of the medium play an important role therefore mechanical waes are also known as elastic waes. xample : Sound waes in water, seismic waes in earth's crust.. Based on energy propagation :- Waes can be diided into two parts on the basis of energy propagation (i) Progressie wae (ii) Stationary waes. The progressie wae propagates with constant elocity in a medium. In stationary waes particles of the medium ibrate with different amplitude but energy does not propagate.

2 J-Physics 3. Based on direction of propagation :- Waes can be one, two or three dimensional according to the number of dimensions in which they propagate energy. Waes moing along strings are one-dimensional. Surface waes or ripples on water are two dimensional, while sound or light waes from a point source are three dimensional. 4. Based on the motion of particles of medium : Waes are of two types on the basis of motion of particles of the medium. (i) Longitudinal waes (ii) Transerse waes Direction of Disturbance Direction of Propagation Transerse wae Direction of Direction of Disturbance Propagation Longitudinal wae In the transerse wae the direction associated with the disturbance (i.e. motion of particles of the medium) is at right angle to the direction of propagation of wae while in the longitudinal wae the direction of disturbance is along the direction of propagation. TR SVRS WV MOTIO Mechanical transerse waes produce in such type of medium which hae shearing property, so they are known as shear wae or S-wae ote :- Shearing is the property of a body by which it changes its shape on application of force. Mechanical transerse waes are generated only in solids & surface of liquid. In this indiidual particles of the medium execute SHM about their mean position in direction r to the direction of propagation of wae motion. Particle Crest Trough Crest ormal leel Trough crest is a portion of the medium, which is raised temporarily aboe the normal position of rest of particles of the medium, when a transerse wae passes. trough is a portion of the medium, which is depressed temporarily below the normal position of rest of particles of the medium, when a transerse wae passes. LOGITUDIL WV MOTIO In this type of waes, oscillatory motion of the medium particles produces regions of compression (high pressure) and rarefaction (low pressure) which propagated in space with time (see figure). R C R C R C Particle ote : The regions of high particle density are called compressions and regions of low particle density are called rarefactions. The propagation of sound waes in air is isualized as the propagation of pressure or density fluctuations. The pressure fluctuations are of the order of Pa, whereas atmospheric pressure is 0 5 Pa. Wae \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65

3 Mecha ni cal Wae s i n Different Media J-Physics mechanical wae will be transerse or longitudinal depends on the nature of medium and mode of excitation. In strings mechanical waes are always transerse when string is under a tension. In gases and liquids mechanical waes are always longitudinal e.g. sound waes in air or water. This is because fluids cannot sustain shear. In solids, mechanical waes (may be sound) can be either transerse or longitudinal depending on the mode of excitation. The speed of the two waes in the same solid are different. (Longitudinal waes traels faster than transerse waes). e.g., if we struck a rod at an angle as shown in fig. () the waes in the rod will be transerse while if the rod is struck at the side as shown in fig. (B) or is rubbed with a cloth the waes in the rod will be longitudinal. In case of ibrating tuning fork waes in the prongs are transerse while in the stem are longitudinal. C C C C R C R C T Transerse-wae () T Longitudinal-wae (B) Further more in case of seismic waes produced by arthquakes both S (shear) and P (pressure) waes are produced simultaneously which trael through the rock in the crust at different speeds [ S 5 km/s while P 9 km/s] S waes are transerse while P waes longitudinal. Some waes in nature are neither transerse nor longitudinal but a combination of the two. These waes are called 'ripple' and waes on the surface of a liquid are of this type. In these waes particles of the medium ibrate up and down and back and forth simultaneously describing ellipses in a ertical plane. Ripple B \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65 CHR CTRISTICS OF WV MOTIO Some of the important characteristics of wae motion are as follows : In a wae motion, the disturbance traels through the medium due to repeated periodic oscillations of the particles of the medium about their mean positions. The energy is transferred from place to another without any actual transfer of the particles of the medium. ach particle receies disturbance a little later than its preceding particle i.e., there is a regular phase difference between one particle and the next. The elocity with which a wae traels is different from the elocity of the particles with which they ibrate about their mean positions. The wae elocity remains constant in a gien medium while the particle elocity changes continuously during its ibration about the mean position. It is maximum at the mean position and zero at the extreme position. For the propagation of a mechanical wae, the medium must possess the properties of inertia, elasticity and minimum friction amongst its particles. SOM IMPORTT TRMS COCTD WITH WV MOTIO Waelength () [length of one wae] Distance traelled by the wae during the time, any one particle of the medium completes one ibration about its mean position. We may also define waelength as the distance between any two nearest particles of the medium, ibrating in the same phase. Frequency (n) :umber of ibrations (umber of complete waelengths) complete by a particle in one second. 3

4 J-Physics Time period (T) : Time taken by wae to trael a distance equal to one waelength. mplitude () : Maximum displacement of ibrating particle from its equilibrium position. ngular frequency () : It is defined as n Phase : Phase is a quantity which contains all information related to any ibrating particle in a wae. For equation y sin (t kx); (t kx) phase. ngular wae number (k) : It is defined as k Wae number ( ) : It is defined as k number of waes in a unit length of the wae pattern. Particle elocity, wae elocity and particle's acceleration : In plane progressie harmonic wae particles of the medium oscillate simple harmonically about their mean position. Therefore, all the formulae what we hae read in SHM apply to the particles here also. For example, maximum particle elocity is ± at mean position and it is zero at extreme positions etc. Similarly maximum particle acceleration is ± at extreme positions and zero at mean position. Howeer the wae elocity is different from the particle elocity. This depends on certain characteristics of the medium. Unlike the particle elocity which oscillates simple harmonically (between + and ) the wae elocity is constant for gien characteristics of the medium. Particle elocity in wae motion : The indiidual particles which make up the medium do not trael through the medium with the waes. They simply oscillate about their equilibrium positions. The instantaneous elocity of an oscillating particle of the medium, through which a wae is traelling, is known as "Particle elocity". y Particle elocity () y t wae ( ) P x Wae elocity : The elocity with which the disturbance, or planes of equal phase (wae front), trael through the medium is called wae (or phase) elocity. Relation between particle elocity and wae elocity : Wae equation :- y sin (t - kx), Particle elocity y t 4 cos (t - kx). y Wae elocity P, T k x - k cos (t kx) - k cos (t - kx) - p ote : y x represent the slope of the string (wae) at the point x. y t y y x t Particle elocity at a gien position and time is equal to negatie of the product of wae elocity with slope of the wae at that point at that instant. Di fferential equation of harmoni c progressie waes : t y y y sin (t kx) k sin (t kx) x x y t P P \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65

5 Particle elocity ( p ) and acceleration (a p ) in a sinusoidal wae : J-Physics The acceleration of the particle is the second particle is the second partial deriatie of y (x, t) with respect to t, y(x, t) a P sin(kx t) y(x, t) t i.e., the acceleration of the particle equals times its displacement, which is the result we obtained for SHM. Thus, a P (displacement) Relation between Phase difference, Path difference & Time difference y, T, Phase () 0 Wae length () 0 Time-period (T) 0 4 T 4 T T Path difference T T 4 F I HG K J Phase difference T T 3T 4 x am ple progressie wae of frequency 500 Hz is traelling with a elocity of 360 m/s. How far apart are two points 60 o out of phase. We know that for a wae f So 360 f m Phase difference 60 o (/80) x 60 (/3) rad, so path difference x 0. 7 () x 3 0. m \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65 TH GR L QUTIO OF WV MOTIO Some physical quantity (say y) is made to oscillate at one place and these oscillations of y propagate to other places. The y may be, (i) (ii) (iii) displacement of particles from their mean position in case of transerse wae in a rope or longitudinal sound wae in a gas. pressure difference (dp) or density difference (d) in case of sound wae or electric and magnetic fields in case of electromagnetic waes. The oscillations of y may or may not be simple harmonic in nature. Consider one-dimensional wae traelling along x-axis. In this case y is a function of x and t. i.e. y f(x, t) But only those function of x & t, represent a wae motion which satisfy the differential equation. 5 y y The general solution of this equation is of the form y (x, t) f (ax ± bt)...(ii) Thus, any function of x and t and which satisfies equation (i) or which can be written as equation (ii) represents a wae. The only condition is that it should be finite eerywhere and at all times, Further, if these conditions are satisfied, then speed of wae () is gien by t coefficient of t b coefficient of x a x...(i)

6 J-Physics x am ple Which of the following functions represent a traelling wae? (a) (x t) (b) n(x + t) (c) e (x t) (d) x t lthough all the four functions are written in the form f (ax + bt), only (c) among the four functions is finite eerywhere at all times. Hence only (c) represents a traelling wae. quat ion of a Plane Progre ssie Wae If, on the propagation of wae in a medium, the particles of the medium perform simple harmonic motion then the wae is called a 'simple harmonic progressie wae'. Suppose, a simple harmonic progressie wae is propagating in a medium along the positie direction of the x-axis (from left to right). In fig. (a) are shown the equilibrium positions of the particles,, (a) 3 4 a y 5 9 x x Direction of wae (b) When the wae propagates, these particles oscillate about their equilibrium positions. In Fig. (b) are shown the instantaneous positions of these particles at a particular instant. The cure joining these positions represents the wae. Let the time be counted from the instant when the particle situated at the origin starts oscillating. If y be the displacement of this particle after t seconds, then y a sin t...(i) where a is the amplitude of oscillation and n, where n is the frequency. s the wae reaches the particles beyond the particle, the particles start oscillating. If the speed of the wae be, then it will reach particle 6, distant x from the particle, in x/ sec. Therefore, the particle 6 will start oscillating x/ sec after the particle. It means that the displacement of the particle 6 at a time t will be the same as that of the particle at a time x/ sec earlier i.e. at time t (x/). The displacement of particle at time t (x/) can be the particle 6, distant x from the origin (particle ), at time t is gien by x y a sin t But n, y a sin (t kx) k...(ii) x am ple y a sin t x T lso k...(iii) y a sin t x T This is the equation of a simple harmonic wae traelling along +x direction. If the wae is traelling along the x direction then inside the brackets in the aboe equations, instead of minus sign there will be plus sign. For t x example, equation (i) will be of the following form : y a sin T. If be the phase difference between the aboe wae traelling along the +x direction and an other wae, then the equation of that wae will be y a sin t x T The equation of a wae is, y(x, t) 0.05 sin (0x 40t) m 4 Find : (a) The waelength, the frequency and the wae elocity (b) The particle elocity and acceleration at x0.5 m and t0.05 s. 6...(i) \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65

7 J-Physics (a) The equation may be rewritten as, y(x, t) 0.05 sin 5x 0t m 4 Comparing this with equation of plane progressie harmonic wae, y(x, t) sin(kx t ) we hae, wae number k 5rad / m 0.4m The angular frequency is, f 0 rad /s f 0Hz The wae elocity is, f 4ms in x direction k y 5 (b) The particle elocity and acceleration are, p (0 )(0.05) cos t 4.m/s y 5 a p (0 ) (0.05) sin t 4 40 m/s ITSIT Y OF WV The amount of energy flowing per unit area and per unit time is called the intensity of wae. It is represented by I. Its units are J/m s or watt/metre. I f i.e. I and I. If P is the power of an isotropic point source, then intensity at a distance r is gien by, I P 4 r or I t (for a point source) r If P is the power of a line source, then intensity at a distance r is gien by, P I or I (for a line source) s, I r r \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65 Therefore, SUPRPOSITIO PRICIPL (for a point source) and r 7 r (for a line source) Two or more waes can propagate in the same medium without affecting the motion of one another. If seeral waes propagate in a medium simultaneously, then the resultant displacement of any particle of the medium at any instant is equal to the ector sum of the displacements produced by indiidual wae. The phenomenon of intermixing of two or more waes to produce a new wae is called Superposition of waes. Therefore according to superposition principle. The resultant displacement of a particle at any point of the medium, at any instant of time is the ector sum of the displacements caused to the particle by the indiidual waes. If y, y, y 3,... are the displacement of particle at a particular time due to indiidual waes, then the resultant displacement is gien by y y y y 3... Principle of superposition holds for all types of waes, i.e., mechanical as well as electromagnetic waes. But this principle is not applicable to the waes of ery large amplitude. Due to superposition of waes the following phenomenon can be seen Interference : Superposition of two waes haing equal frequency and nearly equal amplitude. B e a t s : Superposition of two waes of nearly equal frequency in same direction.

8 J-Physics Stationary waes : Superposition of equal wae from opposite direction. Li ssajous' fi gure : Superposition of perpendicular waes. Superposition of waes Interference Beats Stationary waes Lissaju's figures Longitudinal pplication Transerse pplication Organ pipe Resonance tube Stretched string Sonometer ITRFRC OF WVS : When two waes of equal frequency and nearly equal amplitude traelling in same direction haing same state of polarisation in medium superimpose, then intensity is different at different points. t some points intensity is large, whereas at other points it is nearly zero. Consider two waes y sin (t - kx) and y sin (t kx + ) By principle of superposition y y + y sin (t kx + ) where cos and tan s intensity I so I I I II cos Constructie i nterference (maxi mum i ntensity) : Phase difference n or path difference n where n 0,,, 3,... I I max + and Imax I I II Destructi e interference (mi ni mum i ntensity) : Phase difference (n+), or path difference (n ) I I min and Imin I I II sin cos where n 0,,, 3,... GOLD KY POITS I Maximum and minimum intensities in any interference wae form. Max I erage intensity of interference wae form %& < I >or I a 8 Min max min if and I I I then I max 4I, I min 0 and I V I Degree of interference Pattern (f) : Degree of hearing (Sound Wae) or Degree of isibility (Light Wae) f I I max max I I min min I I I I I + I I I In condition of perfect interference degree of interference pattern is maximum f max or 00% Condition of maximum contrast in interference wae form a a and I I then I max 4I and I min 0 For perfect destructie interference we hae a maximum contrast in interference wae form. 00 \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65

9 J-Physics VLOCIT Y OF TR SVRS WV Mass of per unit length m r Velocity of transerse wae in any wire d, m r d, where d Density of matter T m or T r d If m is constant then, T it is called tension law. T d r R W T m,d density If tension is constant then m it is called law of mass If T is constant & take wire of different radius for same material then r it is called law of radius If T is constant & take wire of same radius for different material. Then RFLCTIO FROM RIGID D law of density d When the pulse reaches the right end which is clamped at the wall, the element at the right end exerts a force on the clamp and the clamp exerts equal and opposite force on the element. The element at the right end is thus acted upon by the force from the clamp. s this end remains fixed, the two forces are opposite to each other. The force from the left part of the string transmits the forward wae pulse and hence, the force exerted by the clamp sends a return pulse on the string whose shape is similar to a return pulse but is inerted. The original pulse tries to pull the element at the fixed end up and the return pulse sent by the clamp tries to pull it down, so the resultant displacement is zero. Thus, the wae is reflected from the fixed end and the reflected wae is inerted with respect to the original wae. The shape of the string at any time, while the pulse is being reflected, can be found by adding an inerted image pulse to the incident pulse. quation of wae propagating in +e x-axis Incident wae y a sin (t kx) Reflected wae y a sin (t + kx + ) or y a sin (t + kx) \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65 RFLCTIO FROM FR D The right end of the string is attached to a light frictionless ring which can freely moe on a ertical rod. wae pulse is sent on the string from left. When the wae reaches the right end, the element at this end is acted on by the force from the left to go up. Howeer, there is no corresponding restoring force from the right as the rod does not exert a ertical force on the ring. s a result, the right end is displaced in upward direction more than the height of the pulse i.e., it oershoots the normal maximum displacement. The lack of restoring force from right can be equialent described in the following way. n extra force acts from right which sends a wae from right to left with its shape identical to the original one. The element at the end is acted upon by both the incident and the reflected wae and the displacements add. Thus, a wae is reflected by the free end without inersion. Incident wae y a sin (t kx) STTIORY WVS 9 Reflected wae y a sin (t + kx) * Definition : The wae propagating in such a medium will be reflected at the boundary and produce a wae of the same kind traelling in the opposite direction. The superposition of the two waes will gie rise to a stationary wae. Formation of stationary wae is possible only and only in bounded medium.

10 J-Physics LY TICL MTHOD FOR STTIORY WVS From rigid end : We know equation for progressie wae in positie x-direction y a sin (t kx) fter reflection from rigid end By principle of super position. y a sin (t + kx + ) a sin (t + kx) This is equation of stationary wae reflected from rigid end y y + y a sin (t kx) a sin (t + kx) a sin kx cos t mplitude a sin kx Velocity of particle pa dy dt a sin kx sin t Strain dy ak cos kx cos t dx stress lasticity strain dp dy dx Change in pressure dp dy dx ode x 0,,... 0, Vpa 0, strain max. Change in pressure max 3 ntinode x,... max, -Vpa max. strain 0 Change in pressure From free end : we know equation for progressie wae in positie x-direction y a sin (t kx) fter reflection from free end y a sin (t + kx) By Principle of superposition y y + y a sin (t kx) + a sin (t + kx) a sin t cos kx mplitude a cos kx, Velocity of particle Pa dy dt a cos t cos kx S t r a i n dy dx ak sin t sin kx Change in pressure dp dy dx ntinode : x 0,,... Max, Vpa dy dt max. Strain 0, dp ode : x,, , V pa dy dt 0, strain max, dp max PROPRTIS OF STTIORY WVS The stationary waes are formed due to the superposition of two identical simple harmonic waes traelling in opposite direction with the same speed. Important characteristi cs of stationary waes are: (i) (ii) (iii) (i) () (i) (ii) (iii) (ix) Stationary waes are produced in the bounded medium and the boundaries of bounded medium may be rigid or free. In stationary waes nodes and antinodes are formed alternately. odes are the points which are always in rest haing maximum strain. ntinodes are the points where the particles ibrate with maximum amplitude haing minimum strain. ll the particles except at the nodes ibrate simple harmonically with the same period. The distance between any two successie nodes or antinodes is /. The amplitude of ibration gradually increases from zero to maximum alue from node to antinode. ll the particles in one particular segment ibrate in the same phase, but the particle of two adjacent segments differ in phase by 80 ll points of the medium pass through their mean position simultaneously twice in each period. Velocity of the particles while crossing mean position aries from maximum at antinodes to zero at nodes. In a stationary wae the medium is splited into segments and each segment is ibrating up and down as a whole. 0 \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65

11 J-Physics (x) (xi) In longitudinal stationary waes, condensation (compression) and refraction do not trael forward as in progressie waes but they appear and disappear alternately at the same place. These waes do not transfer energy in the medium. Transmission of energy is not possible in a stationary wae. TR SMISSIO OF WVS We may hae a situation in which the boundary is intermediate between these two extreme cases, that is, one in which the boundary is neither rigid nor free. In this case, part of the incident energy is transmitted and part is reflected. For instance, suppose a light string is attached to a heaier string as in (figure). When a pulse traelling on the light reaches the knot, same part of it is reflected and inerted and same part of it is transmitted to the heaier string. Incident pulse Incident pulse Transmitted pulse (a) (a) Reflected pulse Transmitted pulse Reflected pulse (b) (b) s one would expect, the reflected pulse has a smaller amplitude than the incident pulse, since part of the incident energy is transferred to the pulse in the heaier string. The inersion in the reflected wae is similar to the behaiour of a pulse meeting a rigid boundary, when it is totally reflected. When a pulse traelling on a heay string strikes the boundary of a lighter string, as in (figure), again part is reflected and part is transmitted. Howeer, in this case the reflected pulse is not inerted. In either case, the relatie height of the reflected and transmitted pulses depend on the relatie densities of the two string. In the preious section, we found that the speed of a wae on a string increases as the density of the string decreases. That is, a pulse traels more slowly on a heay string than on a light string, if both are under the same tension. The following general rules apply to reflected waes. When a wae pulse traels from medium to medium B and > B (that is, when B is denser than ), the pulse will be inerted upon reflection. When a wae pulse traels from medium to medium B and < B ( is denser than B), it will not be inerted upon reflection. GOLD KY POITS Phenomenon of reflection and transmission of waes obeys the laws of reflection and refraction. The frequency of these wae remains constant i.e. does not change. i r t From rarer to denser medium y i a i sin (t k x) y r a i sin (t + k x) y t a t sin (t k x) From denser to rarer medium y i a i sin (t k x) y r a i sin (t + k x) y t a t sin (t k x) \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65 STTIORY WV R OF T WO TYPS : (i) Transerse st. wae (stretched string) (ii) Longitudinal st. wae (organ pipes) ( i ) Tra nser se Stat ionar y wae ( Fixed at Bot h ends) Fundamental Harmonic Second Harmonic Third Harmonic f f 3 f 3 3

12 J-Physics p th harmonic p f p p Law of length : For a gien string, under a gien tension, the fundamental frequency of ibration is inersely proportional to the length of the string, i.e, n (T and m are constant) Law of tension : The fundamental frequency of ibration of stretched string is directly proportional to the square root of the tension in the string, proided that length and mass per unit length of the string are kept constant. n T ( and m are constant) Law of mass : The fundamental frequency of ibration of a stretched string is inersely proportional to the square root of its mass per unit length proided that length of the string and tension in the string are kept constant, i.e., n ( and T are constant) m Melde's experiment : In Melde's experiment, one end of a flexible piece of thread is tied to the end of a tuning fork. The other end passed oer a smooth pulley carries a pan which can be loaded. There are two arrangements to ibrate the tied fork with thread. Tra nser se ar ra ngement : Case. In a ibrating string of fixed length, the product of number of loops and square root of tension are constant or p T constant. T Mg M Case. When the tuning fork is set ibrating as shown in fig. then the prong ibrates at right angles to the thread. s a result the thread is set into motion. The frequency of ibration of the thread (string) is equal to the frequency of the tuning fork. If length and tension are properly adjusted then, standing waes are formed in the string. (This happens when frequency of one of the normal modes of the string matched with the frequency of the tuning fork). Then, if p loops are formed in the thread, then the frequency of the tuning fork is gien by n p T m Case 3. If the tuning fork is turned through a right angle, so that the prong ibrates along the length of the thread, then the string performs only a half oscillation for each complete ibrations of the prong. This is because the thread only makes node at the midpoint when the prong moes towards the pulley i.e. only once in a ibration. Longitudi nal arrangement : The thread performs sustained oscillations when the natural frequency of the gien length of the thread under tension is half that of the fork. T Mg Thus if p loops are formed in the thread, then the frequency of the tuning fork is n p M T m \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65

13 SOOMTR : J-Physics Sonometer consists of a hollow rectangular box of light wood. One end of the experimental wire is fastened to one end of the box. The wire passes oer a frictionless pulley P at the other end of the box. The wire is stretched by a tension T. B B P The box seres the purpose of increasing the loudness of the sound produced by the ibrating wire. If the length the wire between the two bridges is, then the frequency of ibration is n To test the tension of a tuning fork and string, a small paper rider is placed on the string. When a ibrating tuning fork is placed on the box, and if the length between the bridges is properly adjusted, then when the two frequencies are exactly equal, the string quickly picks up the ibrations of the fork and the rider is thrown off the wire. l T m COMPRISO OF PROGRSSIV D STTIORY WVS Progressi e waes Stationary waes. These waes traels in a medium These waes do not trael and remain confined with definite elocity. between two boundaries in the medium.. These waes transmit energy in the medium. These waes do not transmit energy in the medium. 3. The phase of ibration aries The phase of all the particles in between two nodes continuously from particle to particle. is always same. But particles of two djacent nodes differ in phase by o particle of medium is Particles at nodes are permanently at rest. Permanently at rest. 5. ll particles of the medium ibrate The amplitude of ibration changes from particle and amplitude of ibration is same. to particle. The amplitude is zero for all at nodes and maximum at antinodes. 6. ll the particles do not attain the ll the particles attain the maximum maximum displacement position simultaneously. displacement \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65 SPD OF LOGITUDIL (SOUD) WVS ewton Formula medium (Use for eery medium) Where lasticity coefficient of medium & Density of medium For solid medium solid For liquid Medium liquid Y Where Y Young's modulas B Where B, where B olume elasticity coefficient of liquid 3

14 J-Physics For gas medi um The formula for elocity of sound in air was first obtained by ewton. He assumed that sound propagates through air and temperature remains constant. (i.e. the process is isothermal) so Isothermal lasticity P air t TP for air P.0 x 0 5 /m and.3 kg/m 3 so air (P / 79 m/s Howeer, the experimental alue of sound in air is 33 m/s which is much higher than gien by ewton's formula. Laplace Correcti on In order to remoe the discrepancy between theoretical and experimental alues of elocity of sound, Laplace modified ewton's formula assuming that propagation of sound in air is adiabatic process, i.e. diabatic lasticity p so that P i.e..4 x m/s [as air.4] Which is in good agreement with the experimental alue (33 m/s). This in turn establishes that sound propagates adiabatically through gases. The elocity of sound in air at TP is 33 m/s which is much lesser than that of light and radio waes ( 3 x 0 8 m/s). This implies that (a) (b) (c) If we set our watch by the sound of a distant siren it will be slow. If we record the time in a race by hearing sound from starting point it will be lesser than actual. In a cloud lightening, though light and sound are produced simultaneously but as c >, light proceeds thunder. n in case of gases s P PV mass M as mass olume V or RT s M [as PV RT] or s RT M w as mass M where M w Molecular weight M w M w nd from kinetic-theory of gases rms (3RT / M w ) So s rms 3 FFCT OF VRIOUS QUTITIS ( ) ffect of temperature For a gas & M W is constant By applying Binomial theorem. (i) For any gas medium t 0 ( ) ffect of Relati e Humi di ty T t 546 T T 4 t t 0 t 73 (ii) For air : t t m/sec. ( 0 33 m/sec. ) With increase in humidity, density decreases so in the light of P / ) We conclude that with rise in humidity elocity of sound increases. This is why sound traels faster in humid air (rainy season) than in dry air (summer) at same temperature. Due to this in rainy season the sound of factories siren and whistle of train can be heard more than summer. \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65

15 J-Physics ( 3 ) ffect of Pressure s elocity of sound P RT M So pressure has no effect on elocity of sound in a gas as long as temperature remain constant. This is why in going up in the atmosphere, though both pressure and density decreases, elocity of sound remains constant as long as temperature remains constant. Further more it has also been established that all other factors such as amplitude, frequency, phase, loudness pitch, quality etc. has partially no effect on elocity of sound. Velocity of sound in air is measured by resonance tube or Hebb's method while in gases by Quinke's tube. Kundt's tube is used to determine elocity of sound in any medium solid, liquid or gas. ( 4 ) ffect of Motion of ir If air is blowing then the speed of sound changes. If the actual speed of sound is and the speed of air is w, then the speed of sound in the direction in which air is blowing will be (+ w), and in the opposite direction it will be ( w). ( 5 ) ffect of Frequency There is no effect of frequency on the speed of sound. Sound waes of different frequencies trael with the same speed in air although their waelength in air are different. If the speed of sound were dependent on the frequency, then we could not hae enjoyed orchestra. x am ple piezo electric quartz plate of thickness m is ibrating in resonant conditions. Calculate its fundamental frequency if for quartz Y /m and.65 x 0 3 kg/m 3 We known that for longitudinal waes in solids Further more for fundamental mode of plate (/) L Y 0, So So x 5 x m But as f, i.e., f (/) so f [5.5 x 0 3 /0 ] 5.5 x 0 5 Hz 550 khz m/s x am ple Determine the change in olume of 6 liters of alcohol if the pressure is decreased from 00 cm of Hg to 75 cm. [elocity of sound in alcohol is 80 m/s, density of alcohol 0.8 gm/cc, density of Hg 3.6 gm/cc and g 9.8 m/s ] For propagation of sound in liquid B / i.e., B \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65 x am ple But by definition B V P V So V P V V( P), i.e. V Here P H g H g (75 00) dynes/cm So V cc ( a ) Speed of sound in air is 33 m/s at TP. What will the speed of sound in hydrogen at TP if the density of hydrogen at TP is (/6) that of air. ( b ) Calculate the ratio of the speed of sound in neon to that in water apour at any temperature. [Molecular weight of neon.0 0 kg/mol and for water apours.8 0 kg/mol] 5

16 J-Physics The elocity of sound in air is gien by P RT M ( a ) In terms of density and pressure H air P H H air P air air H [as P air P H ] H air ( b ) In terms of temperature and molecular weight air H 33 e W 6 M e e 38 m/s M W [as T T W ] ow as neon is mono atomic ( 5/3) while water apours poly atomic ( 4/3) so e W b g b g 5 / / W VIBR TIO I ORG PIPS When two longitudinal waes of same frequency and amplitude trael in a medium in opposite directions then by superposition, standing waes are produced. These waes are produced in air columns in cylindrical tube of uniform diameter. These sound producing tubes are called organ pipes.. Vibration of air column in closed organ pipe : The tube which is closed at one end and open at the other end is called close organ pipe. On blowing air at the open end, a wae traels towards closed end from where it is reflected towards open end. s the wae reaches open end, it is reflected again. So two longitudinal waes trael in opposite directions to superpose and produce stationary waes. t the closed end there is a node since particles does not hae freedom to ibrate whereas at open end there is an antinode because particles hae greatest freedom to ibrate. Hence on blowing air at the open end, the column ibrates forming antinode at free end and node at closed end. If is length of pipe and be the waelength and be the elocity of sound in organ pipe then Fundamental 4 I st Oertone II nd Oertone 3 4 (i) (ii) (iii) Case (i) Case (ii) Case (iii) 4 n n 3 4 n When closed organ pipe ibrate in m th oertone then (m ) 4 So 4 n (m ) (m ) 4 Hence frequency of oertones is gien by n : n : n 3... : 3 : 5... fundamental frequency. First oertone / III rd Harmonic Second oertone / V th Harmonic \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65

17 . Vibration of air columns in open organ pipe : The tube which is open at both ends is called an open organ pipe. On blowing air at the open end, a wae trael towards the other end after reflection from open end waes trael in opposite direction to superpose and produce stationary wae. ow the pipe is open at both ends by which an antinode is formed at open end. Hence on blowing air at the open end antinodes are formed at each end and nodes in the middle. If is length of the pipe and be the waelength and is elocity of sound in organ pipe. Case (i) n Fundamental Fundamental frequency. J-Physics I st Oertone II nd Oertone (i) (ii) (iii) 3 3 Case (ii) n First oertone frequency. Case (iii) 3 3 n 3 3 Second oertone frequency. Hence frequency of oertones are gien by the relation n : n : n 3... : : 3... When open organ pipe ibrate in m th oertone then (m ) so 4 GOLD KY POITS 4 m n (m ) rod clamped at one end or a string fixed at one end is similar to ibration of closed end organ pipe. rod clamped in the middle is similar to the ibration of open end organ pipe. If an open pipe is half submerged in water, it becomes a closed organ pipe of length half that of open pipe i.e. frequency remains same. Due to finite motion of air molecular in organ pipes reflection takes place not exactly at open end but some what aboe it so in an organ pipe antinode is not formed exactly at free end but aboe it at a distance e 0.6r (called end correction or Rayleigh's correction) with r being the radius of pipe. So for closed organ pipe L L + 0.6r while for open L L + 0.6r (as both ends are open) so that f C 4(L 0.6r) while f 0 (L.r) \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65 This is why for a gien and L narrower the pipe higher will the frequency or pitch and shriller will be the sound. For an organ pipe (closed or open) if constant. f (/L) So with decrease in length of ibrating air column, i.e., waelength ( L), frequency or pitch will increase and ice ersa. This is why the pitch increases gradually as an empty essel fills slowly. For an organ pipe if f constant. or L, f 7 constant i.e. the frequency of an organ pipe will remain unchanged if the ratio of speed of sound in to its wae length remains constant. s for a gien length of organ pipe constant. f So ( a ) With rise in temperature as elocity will increase ( T the pitch will increase. (Change in length with temperature is not considered unless stated) ( b ) With change in gas in the pipe as will change and so f will change ( / M ) ( c ) With increase in moisture as will increase and so the pitch will also.

18 J-Physics x am ple x am ple For a certain organ pipe, three successie resonant frequencies are obsered at 45, 595 and 765 Hz respectiely. Taking the speed for sound in air to be 340 m/s (a) xplain whether the pipe is closed at one end or open at both ends (b) determine the fundamental frequency and length of the pipe. ( a ) The gien frequencies are in the ratio 45 : 595 : 765, i.e., 5 : 7 : 9 nd as clearly these are odd integers so the gien pipe is closed pipe. ( b ) From part (b) it is clear that the frequency of 5th harmonic (which is third oertone) is 45 Hz so 45 5fc f C 85 Hz Further as f C 4L, L 4fC m B is a cylinder of length m fitted with a thin flexible diaphram C at middle and two other thin flexible diaphram and B at the ends. The portions C and BC contain hydrogen and oxygen gases respectiely. The diaphragms and B are set into ibrations of the same frequency. What is the minimum frequency of these ibrations for which diaphram C is a node? Under the condition of the experiment the elocity of sound in hydrogen is 00 m/s and oxygen 300 m/s. s diaphragm C is a node, and B will be antinode (as in a organ pipe either both ends are antinode or one end node and the other antinode), i.e., each part will behae as closed end organ pipe so that f H H 4L H Hz nd f 0 4L Hz C B s the two fundamental frequencies are different, the system will ibrate with a H O n common frequency f such that f n H f H n 0 f 0 i.e. n H 0 f f 0 H i.e., the third harmonic of hydrogen and th harmonic of oxygen or 6th harmonic of hydrogen and nd harmonic of oxygen will hae same frequency. So the minimum common frequency f or Hz PPR TUS FOR DTRMIIG SPD OF SOUD. Qui nck's Tube : It consists of two U shaped metal tubes. Sound waes with the help of tuning fork are produced at which trael through B & C and comes out at D where a sensitie flame is present. ow the two waes coming through different path interfere and flame flares up. But if they are not in phase, destructie interference occurs and flame remains undisturbed. 8 T B C T Suppose destructie interference occurs at D for some position of C. If now the tube C is moed so that interference condition is disturbed and again by moing a distance x, destructie interference occurs so that x. Similarly if the distance moed between successie constructie and destructie interference is x then x ow by haing alue of x, speed of sound is gien by n. Kundt's tube : It is the used to determine speed of sound in different gases. It consists of a glass tube in which a small quantity of lycopodium powder is spread. The tube is rotated so that powder starts slipping. ow rod CD is rubbed at end D so that stationary waes form. The disc C ibrates so that air column also ibrates with the frequency of the rod. The piston P is adjusted so that frequency of air column become same as that of rod. D \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65

19 J-Physics So resonance occurs and column is thrown into stationary waes. The powder sets into oscillations at antinodes while heaps of powder are formed at nodes. clamped at P P the middle rod R lycopodium power Let n is the frequency of ibration of the rod then, this is also the frequency of sound wae in the air column in the tube. For rod : rod For air : rod air air Where air is the distance between two heaps of powder in the tube (i.e. distance between two nodes). If air and air rod rod are elocity of sound waes in the air and rod respectiely, then n Therefore, air air air rod rod rod Thus knowledge of rod determines air. air rod ( i i ) RSOC TUB Construction : The resonance tube is a tube T (figure) made of brass or glass, about meter long and 5 cm in diameter and fixed on a ertical stand. Its lower end is connected to a water reseroir B by means of a flexible rubber tube. The rubber tube carries a pinch-cock P. The leel of water in T can be raised or lowered by water adjusting the height of the reseroir B and controlling the flow of water from B to T or from T to B by means of the pinch-cock P. Thus the length of the air column in T can be changed. The position of the water leel in T can be read by means of a side tube C and a scale S. \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65 Determination of the speed of sound in air by resonance tube First of all the water reseroir B is raised until the water leel in the tube T rises almost to the top of the tube. Then the pinch cock P is tightened and the reseroir B is lowered. The water leel in T stays at the top. ow a tuning fork is sounded and held oer the mouth of tube.the pinch cock P is opened slowly so that the water leel in T falls and the length of the air column increases. t a particular length of air column in T, a loud sound is heard. This is the first state of resonance. In this position the following phenomenon takes place inside the tube. (i) For first resonance /4 (ii) For second resonance 3/4 / ( ) If the frequency of the fork be n and the temperature of the air-column be t o C, then the speed of sound at t o C is gien by t n n ( ) The speed of sound wae at 0 o C 0 ( t 0.6 t) m/s. 9

20 J-Physics nd Correction : In the resonance tube, the antinode is not formed exactly at the open but slightly outside at a distance x. Hence the length of the air -column in the first and second states of resonance are (l + x) and ( + x) then (i) For first resonance + x /4...(i) (ii) For second resonance + x 3/4...(ii)) Subtract quation (ii) from quation (i) / ( ) Put the alue of in quation (i) + x ( ) 4 B T S + x 3 x When two sound waes of same amplitude traelling in same direction with slightly different frequency superimpose, then intensity aries periodically with time. This effect is called Beats. Suppose two waes of frequencies f and f (<f ) are meeting at some point in space. The corresponding periods are T and T (>T ). If the two waes are in phase at t0, they will again be in phase when the first wae has gone through exactly one more cycle than the second. This will happen at a time tt, the period of the beat. Let n be the number of cycles of the first wae in time T, then the number of cycles of the second wae in the same time is (n ). Hence, T nt (n ) T y y y y & y T T liminating n we hae T T T f f T T The reciprocal of the beat period is the beat frequency Wae s Inter ference On The Base s Of Beat s : 0 f f f T Conditions % Two equal frequency waes trael in same direction. Mathematical analysis If displacement of first wae y a sin t (, a) I a Displacement of second wae y a sin t (, a) By superposition y y + y quation of resulting wae y a {sin t + sin t} \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65

21 J-Physics y a ( ) ( ) sin t cos t ( ) a cos t sin t ( ) sin 't mplitude a Cos t a cos t ( ) Frequency ' For max Intensity ( ± a ): If cos ( ) t ± cos ( ) t cos n, n 0,,,... ( )t n t n 0,,, 3... For Minimum Intensity ( 0) : cos ( ) t 0 cos ( ) t cos (n + ) n 0,,... n ( ) t (n + ) t ( ), 3, 5... GOLD KY POITS When we added wax on tuning fork then the frequency of fork decreases. When we file the tuning fork then the frequency of fork increases. x am ple tuning fork haing n 300 Hz produces 5 beats/s with another tuning fork. If impurity (wax) is added on the arm of known tuning fork, the number of beats decreases then calculate the frequency of unknown tuning fork. The frequency of unknown tuning fork should be Hz or 305 Hz. When wax is added, if it would be 305 Hz, beats would hae increases but with 95 Hz beats is decreases so frequency of unknown tuning fork is 95 Hz. \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65 x am ple tuning fork haing n 58 Hz, produce 3 beats/s with another. s we file the arm of unknown, beats become 7 then calculate the frequency of unknown. The frequency of unknown tuning fork should be 58 ± 3 55 Hz or 6 Hz. fter filling the number of beats 7 so frequency of unknown tuning fork should be 58 ± 7 65 Hz or 5 Hz. s both aboe frequency can be obtain by filing so frequency of unknown 55/6 Hz. SPCIL POITS. Di splacement and pressure waes sound wae (i.e. longitudinal mechanical wae) can be described either in terms of the longitudinal displacement suffered by the particles of the medium (called displacement-wae) or in terms of the excess pressure generated due to compression and rarefaction (called pressure-wae). Consider a sound wae traelling in the x-direction in a medium. Suppose at time t, the particle at the undisturbed position x suffers a displacement y in the x-direction. The displacement wae then will be described by y sin (t kx)...(i)

22 J-Physics ow consider the element of medium which is confined with in x and x+x in the undisturbed state. If S is the cross-section, the olume element in undisturbed state will be V S x. s the wae passes the ends at x and x +x are displaced by amount y and y + y so that increase in olume of the element will be V S y. This in turn implies that olume strain for the element under consideration So corresponding stress, i.e. excess pressure V S y y... (ii) V S x x V P P P B as B V V V V V or y P B [From equation (ii))... (iii) x ote: For a harmonic -progressie - wae from dy dx i.e. pressure in a sound wae is equal to the product of elasticity of gas with the ratio of particle speed to wae speed. But from equation (i) So i.e. y k cos( t kx) x P kb cos (t kx) P P 0 cos (t kx)...(i) with P 0 Bk...(i) From equation dy, P B B dx pa (i) and (i) it is clear that sound wae may be considered as either a displacement wae y sin (t - kx) or a pressure wae P P 0 cos(t kx). s p pa p o s o T/ T The pressure wae is 90 0 out of phase with respect to displacement wae, i.e, displacement will be maximum when pressure is minimum and ice-ersa. This is shown in figure The amplitude of pressure wae: P 0 Bk k [as B /, /k]...() s sound-sensors (e.g. ear or mic) detects pressure changes, description of sound as pressure-wae is preferred oer displacement wae. x x. Ultrasoni c, Infrasonic and udible (sonic) Sound : Sound waes can be classified in three groups according to their range of frequencies. Infrasoni c Wae s Longitudinal waes haing frequencies below 0 Hz are called infrasonic waes. They cannot be heard by human beings. They are produced during earthquakes. Infrasonic waes can be heard by snakes. udi ble Wae s Longitudinal waes haing frequencies lying between 0-0,000 Hz are called audible waes. Ultrasoni c Wae s Longitudinal waes haing frequencies aboe 0,000 Hz are called ultrasonic waes. They are produced and heard by bats. They hae a large energy content. pplicat ions of Ultrasonic Wae s Ultrasonic waes hae a large range of application. Some of them are as follows: (i) (ii) (iii) (i) () The fine internal cracks in metal can be detected by ultrasonic waes. Ultrasonic waes can be used for determining the depth of the sea, lakes etc. Ultrasonic waes can be used to detect submarines, icebergs etc. Ultrasonic waes can be used to clean clothes, fine machinery parts etc. Ultrasonic waes can be used to kill smaller animals like rates, fish and frogs etc. \\node6\_od6 ()\Data\04\Kota\J-danced\SMP\Phy\Unit o-6\wae Motion\ng\Theory.p65