Transverse Wave - Only in solids (having rigidity), in liquids possible only on the surface. Longitudinal Wave

Wave is when one particle passes its motion to its neighbour. The Elasticity and Inertia of the medium play important role in the propagation of wave. The elasticity brings the particle momentarily at rest in their extreme positions, while inertia carries them across their equilibrium positions. The phase difference between two neighbouring particles must depend on their interaction (nature of bonding) Transverse Wave - Only in solids (having rigidity), in liquids possible only on the surface. Longitudinal Wave Wave transmits only their motion and not the particles themselves. Wave transmits energy and momentum. A) Writing a wave equation y = a sin t At present y = a sin (t x/v) [for Harmonic wave]. In general, y = f (vt x) = a sin ν (t x/v) = a sin v/ (t x/v) = a sin / (vt x) = a sin / ( T t x) = a sin (t/t x/ ) You need to mug up only this > y = A sin (ωt kx) Here y can be a) Displacement (in that case it is called a Displacement Wave, b) pressure Difference (pressure wave Ex. Sound Wave), c) E- Field or B-Field (EM Wave) The 4 equations y = A sin(ωt kx) y = A sin(ωt + kx) y = -A sin(ωt kx) y = -A sin(ωt + kx) Wave Motion by Dr. Rajeev Tyagi. 981044396, rajeev_tyagi@rediffmail.com 1

Few Terms: a) Amplitude, A b) Time period, T c) Frequency, f or ν; ω = ν d) Phase: In y = f (vt ± x), phase is vt ± x. For any point, this quantity is constant. Putting vt ± x = constt, phase vel. or wave velocity is obtained by dx/dt = v e) Wavelength, λ; propagation constant, k = /λ f) Particle velocity, dy/dt = Aω cos(ωt kx) = A ν cos(ωt kx), g) maximum particle velocity = Aω h) Wave speed V w = νλ = ω/k = In general dx/dt i) particle acceleration, maximum particle acceleration [IIT84] A transverse wave is given by y = y o sin (ft x/λ). The max. particle velocity is equal to four times the wave velocity if (λ = y o /) Q. Given E y = 3 sin (4z t). Find direction of a) propagation, b) oscillation of E and of B, c) values of ω, k, d) speed of wave [IIT screening 004] A source of sound of frequency 600 Hz is placed inside water. Speed of sound in water is 1500 m/s and in air is 300 m/s. The frequency of sound recorded by on observer who is standing in air is (600 Hz, as frequency does not change from one medium to another) [IIT005, M] A harmonically moving transverse wave on a string has max. particle velocity and acceleration of 3 m/s and 90 m/s respectively. Velocity of the wave is 0 m/s. Find the waveform. [Ans. y = (0.1) sin [(30 t + 1.5x + )] Relation between V p, V w, slope etc. y = A sin / (vt x) = A sin(ωt kx) Particle velocity, dy/dt = Aw cos(ωt kx) = A f cos(ωt kx) Slope, dy/dx = -Ak cos(ωt kx) = - A/ cos(ωt kx) dy/dx represents strain or compression. When dy/dx is positive a rarefaction takes place, when dy/dx is negative a compression takes place. So, particle velocity dy/dt = -V (dy/dx) = wave velocity x slope of displacement curve Differentiating twice we get t x V Wave Motion by Dr. Rajeev Tyagi. 981044396, rajeev_tyagi@rediffmail.com

Other Forms: The general form of a wave travelling towards right is y = f (vt x) The general form of a wave travelling towards left is y = f (vt + x) So, general form of a progressive wave is y = f (vt ± x). It has to be x + vt, x vt, vt + x, vt x. so (x vt) and (x vt) represent wave but x v t does not. Also it has to be finite everywhere at all times. If it is in form of vt x ot vt + x then its direction of propagation can be straightway made out. Its differential forms are (means that if a function satisfies these conditions, then it represents a wave): y y V t x V t x Q. Which of the following represents one-dimensional wave equations? a) y = 4 sin x cos vt, b) y = 9 sin x cos vt, c) y = x v t, d) y = x 5t a) = - V y, = - y. Hence it represents a wave motion. t x b) = - V y, = - 4y. Hence it does NOT represents a wave motion. t x c) = - V, =. Hence it does NOT represents a wave motion. t x d) = - 5, =. Hence it represents a wave motion with v = 5/. t x 0.8 [IIT99] y(x, t) = represents a non-periodic travelling pulse. What is the (4x 5 t) 5 velocity of the pulse and its direction of motion? What is the maximum displacement (or amplitude) in this moving pulse? [1.5 m/s, -ve X-axis as it is of form x + vt, 0.16 m]bring it in form of x + vt by taking 4 common. Q. If at t=0 a travelling pulse is described as y = 6 / (x + 3), what will be the amplitude and wave function representing the pulse at time t, if the pulse is propagating along positive x axis with speed 4 m/s? [6/3 = m, y = 6 / [(x - 4t) + 3] Q. If y = 3 / (x + 3t) determine the wave velocity and its direction of propagation. [1.5 m/s, towards ve x-axis as it is of form x + vt] Q. y = ln (x + vt) and y = 1 / (x + vt) do not represent wave as they are not finite for all values of x and t. Wave Motion by Dr. Rajeev Tyagi. 981044396, rajeev_tyagi@rediffmail.com 3

B) Speed of mechanical waves for transverse waves (on a String), v = (T/μ) [Derivation on pg. 307, HCV1] Diagram of sonometer experiment T might have to be found (ex acc sys in sono expt) T might be diff at diff pt (vertical string with mass) Instead of Mg, it might be P pulling it for longitudinal waves v = (E/ρ). For waves passing through solids and liquids, E = B, for 1-D (waves in a rod) E = Y For gases v = P (Newton s formula) P Laplace s correction, v = [IIT96] The extension in a string obeying Hooke s law is x. The speed of the transverse wave in the stretched string is v. if the extension in the string is increased to 1.5x, the speed of the transverse wave will be (1. v) Factors affecting speed of the wave RT Effect of temperature using PV = nrt, V = M Effect of Pressure If T is constant, then velocity is independent of Pressure Effect of Humidity The presence of moisture in air decreases the density of air. So velocity of sound in moist air is more than V in dry air. Effect of frequency No effect [IIT 1984, 6M] A uniform rope of length 1 m and mass 6 kg hangs vertically from a rigid support. A block of mass kg is attached to the free end of the rope. A transverse pulse of wavelength 0.06 m is produced at the lower end of the rope. What is the wavelength of the pulse when it reaches the top of the rope? [Ans. 0.1 m] Energy, Intensity, Power transmitted along the string by a sine wave: Covered later For a point source I 1/r For a cylindrical source I 1/r Wave Motion by Dr. Rajeev Tyagi. 981044396, rajeev_tyagi@rediffmail.com 4

C) Interference of Waves First we discuss, superposition of two waves, moving in the same direction, differing in phase by. y 1 = A 1 sin (ωt kx) y = A sin [(ωt kx) + ] The result is that the interference of two waves gives rise to another wave of Amplitude R = A A A A cos 1 1 R is max. when φ = 0, π, 4 π In general, φ =n π where n = 0, 1,, 3 & the maximum value of R is, R max = A 1 + A R is min. when φ = π, 3 π, 5 π In general, φ =(n-1) π where n = 1,, 3 & the minimum value of R is, R min = A 1 A Q. If A 1 and A are given, find the ratio R max / R min Q. If R max / R min is given, find ratio A 1 /A As Intensity I = ka, so the same questions can be asked in terms of I 1 and I also. Wave Motion by Dr. Rajeev Tyagi. 981044396, rajeev_tyagi@rediffmail.com 5

Reflection and Transmision [NOTE wave gets reflected from BOTH denser as well as rarer medium. Proof Total Internal Reflection. Actually at the INTERFACE, both reflection and transmission occurs. How much of which one occurs depends!] Rarer Denser Incident wave Transmitted wave y = A i sin (t x/v 1 ) y = A t sin (t x/v ) Reflected wave y = -A r sin (t + x/v 1 ) INTERFACE Denser Rarer Incident wave, A i Transmitted wave, A t Reflected wave, A r NOTE Phase change of π occurs only when reflection occurs from a denser medium In case of a string Rarer medium might mean a thin string, and a denser medium mean a thick string. Also reflection from a very-very dense medium might be reflection from a fixed end (the wall to which the string is fixed acts as the denser medium). Reflection from a very-very rarer end might mean a free end. V V1 A r = Ai V V 1 A t = V V V Ai 1 In case of sound, a closed end of a pipe corresponds to fixed end of a string, an open end of pipe corresponds to free end of a string. Wave Motion by Dr. Rajeev Tyagi. 981044396, rajeev_tyagi@rediffmail.com 6

Let us now consider superposition of two waves moving in opposite directions, either reflected from a denser medium or from a rarer medium. Let s also assume that the amplitude of incident and reflected wave remain the same (A) y 1 = A sin( t + kx) y = - A sin( t - kx) Ex. Wave on a guitar string y = A sin(kx) cos( t) y 1 = A sin( t + kx) y = A sin( t - kx) Ex. Holding string in hand y = A cos(kx) sin( t) Here y represents the resulting displacement due to two waves, of a particle at x, at time t It is an SHM of amplitude A sin(kx) or of A cos(kx) [Amplitude depending on position x of the particle] Proof the motion of particle at sin(kx) = 1 is y = - A cos( t) Discussions for Standing waves Case 1 (Reflection from a free end) y = (A cos kx)(sin t) This is SHM [Single sine function in time] The term A cos kx represents the amplitude. 1. All points do not have the same amplitude. The value of amplitude ranges from 0 to A. In general, amplitude, A = A cos(kx). There are some points (some values of x ) for which cos kx is zero. For these points, y = 0, whatever be the value of t. (So these points never move. These are the points whose amplitudes are zero). These are called nodes. So the positions of nodes are obtained by putting cos kx = 0. 3. There are points (some values of x ) for which cos kx = 1. For these points amplitudes are maximum (equal to A). These are called anti-nodes. 4. There are times when sin t = 0. At these times, y = 0 for any value of x. So all points pass through the mean position together. 5. There are times when sin t = 1. The displacements of all particles at these times depends on their position and is equal to their amplitudes. So, all particles reach their extreme positions together. 6. All particles between two nodes, move in the same direction. After that (on the two sides of a node) the particles move in different directions. [IIT88] A wave y = a cos(kx wt) is superimposed with another wave to form a stationary wave such that x = 0 is a node. The equation for the other wave is Wave Motion by Dr. Rajeev Tyagi. 981044396, rajeev_tyagi@rediffmail.com 7

Now, let us learn how to draw standing waves. [Then we will calculate their frequencies] Rules for drawing standing waves Rule 1 A Fixed end has to be a Node. Rule Between two Nodes, there must be an anti-node, and vice-versa. Rule 3 A free end has to be an anti-node. Case 1 String fixed at both ends 1 st Harmonic OR Fundamental Note nd Harmonic OR 1 st Overtone Case String fixed at one end, other end is free 3 rd Harmonic OR 1 st overtone 5 th Harmonic ( nd & 4 th are missing) nd overtone Case 3 String free at both ends. 1 st Harmonic OR Fundamental Note nd Harmonic OR 1 st Overtone Wave Motion by Dr. Rajeev Tyagi. 981044396, rajeev_tyagi@rediffmail.com 8

Let us now calculate the frequencies of various modes drawn above for string. Result If both ends are free, OR if Both ends are fixed (i.e Both Ends are similar) Fundamental note, 0 = v/l And all Harmonics are present. ( nd harmonic means x 0, 3 rd harmonic mean 3 x 0 etc. Harmonic mean that multiple of fundamental frequency ex. 7 th harmonic is 7 x 0 As All harmonics are present, so the 1 st overtone is the nd harmonic, the nd overtone is the 3 rd harmonic etc. If one end is fixed and the other end is free Fundamental note, 0 = v/4l And only ODD Harmonics are present. (i.e. 3 rd, 5 th, 7 th etc.) As only ODD harmonics are present, so the 1 st overtone is the 3 rd harmonic, the nd overtone is the 5 th harmonic etc. [OVERTONE means the next possible standing wave] For a rod fixed in between, the ends will always be anti-nodes, and the fixed point in between will be a node. For sound waves, instead of displacement, we talk of pressure difference. The discussion is exactly the same except that nodes and anti-nodes interchange their positions. In case of Resonance with Sound waves in organ pipes, there are only two cases Both Ends open [Open Organ Pipe] or One End Closed [Closed Organ Pipe] A closed end is a Displacement Node [pressure anti-node] An open end is a Displacement Anti-Node [pressure node] End Correction in an organ pipe at the open end [Resonance Experiment] For each open end e = 0.6r. for a closed pipe, L L + e for an open pipe, L L + e BEATS (Superposition of two waves of slightly different frequencies, moving in the same direction) If there are two sources of frequencies n 1 and n, then number of Beats n = n 1 n It means that n times in a second that there will be waxing and waning of sound. It is also referred to as Beat Frequency. Filing a tuning fork: Frequency of tuning fork increases Loading a tuning fork: Frequency of tuning fork decreases Wave Motion by Dr. Rajeev Tyagi. 981044396, rajeev_tyagi@rediffmail.com 9

[IIT81] A cylindrical tube, open at both ends, has a fundamental frequency f in air. The tube is dipped vertically in water so that half of its length is in water. The fundamental frequency of the air column is now (f) [IIT86] A tube closed at one end and containing air produces the fundamental note of frequency 51 Hz. If the tube is opened at both ends the fundamental frequency that can be excited (in Hz) is (104 Hz) [IIT88] An organ pipe P 1 closed at one end vibrating in its first harmonic and another pipe P open at both ends vibrating in its third harmonic are in resonance with a given tuning fork. The ratio of length of P 1 and P is (1/6) [IIT95] An object of specific gravity is hung from a thin steel wire. The fundamental frequency for transverse standing waves in the wire is 300Hz. The object is immersed in water so that half of its volume is submerged. The new fundamental frequency in Hz is 1 {300[ ] 1/ } [IIT96] An open pipe is suddenly closed at one end with the result that the frequency of third harmonic of the closed pipe is found to be higher by 100 Hz than the fundamental frequency of the open pipe. The fundamental frequency of the open pipe is (00Hz) [IIT98] A string of length 0.4m and mass 10 - kg is tightly clamped at its ends. The tension in the string is 1.6 N. Identical wave pulses are produced at one end at equal intervals of time t. The minimum value of t which allows constructive interference between successive pulses is (0.10 s) [IIT000] Two vibrating strings of the same material but of lengths L and L have radii r and r respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental mode. The one of length L with frequency ν 1 and the other with frequency ν. The ratio of ν 1 / ν is (1) [IIT00] A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a mass of 9 kg is suspended from the wire. When this mass is replaced by mass M, the wire resonates with the same tuning fork forming three antinodes for the same position of bridges. The value of M is (5kg) [IIT006] A massless rod is suspended by two identical strings AB and CD of equal length. A block of mass m is suspended from point O such that BO is equal to x. Further it is observed that the frequency of 1 st harmonic (fundamental frequency) in AB is equal to nd harmonic frequency in CD. Then the length BO is Ans. L/5 A C B O D x L - x [IIT008] m Wave Motion by Dr. Rajeev Tyagi. 981044396, rajeev_tyagi@rediffmail.com 10

The pressure wave in detail: Let us consider two cross sections of area A at x and x+ x. If they travel different distances in time t then compression or rarefaction takes place. Let s say plane1 travels distance y, then dy/dx represents rate of change of displacement with distance. So the displacement of plane is y +(dy/dx) x. The change in volume between the planes can be shown to be (dy/dx) A x. Volume strain = change in volume / Initial volume = [(dy/dx) A x] / [A x] = dy/dx As the wave has created a volume strain in air between the planes, so there is variation of pressure too. From B = stress / strain = -p / (dy/dx) where p is excess pressure, change in pressure (or excess pressure), p = -B (dy/dx) dy/dx also represents the slope of the displacement wave. Hence in pressure wave (longitudinal wave) the slope of displacement curve dy/dx measures pressure change (compression or rarefaction) at that point. When dy/dx is negative, p is positive (compression); when dy/dx is positive, p is negative (rarefaction) Let us consider y = A sin / (v w t x) If p is the instantaneous change in pressure at point x, then p = -B(dy/dx) = (-B) [- / ] A cos / ( v w t x) = (-V Vw A w ) [- / ] A cos / ( v w t x) = cos / ( v w t x) = p o cos / ( v w t x), Vw A where p o = = ν A V w represents the pressure amplitude. The pressure wave is 90 o out of phase with displacement wave i.e. when displacement at a point is zero, the pressure change is maximum and vice-versa. By measuring dy/dx at different points x on a displacement curve, the corresponding values of p = -V (dy/dx) can be obtained. Then a curve may be plotted between p and x. This would be the pressure curve. Kinetic Energy per unit volume = (1/4) ω A Potential Energy per unit volume = (1/4) ω A Total Energy per unit volume (Energy Density) = (1/) ω A = ν A Energy Current or Wave Intensity : All progressive waves transmit energy. The energy flowing per second per unit area perpendicular to the direction of wave propagation is called Energy flux or Energy current or Wave Intensity. As the wave travels distance V w per second, so Wave Intensity, I = Energy density x wave velocity = ν A po V w = Vw [IIT001] The ends of a stretched wire of length L are fixed at x=0 and x=l. In one experiment the displacement of the wire is y 1 = A sin (πx/l)sin(wt) and the energy is E 1 and in the other is y = A sin (πx/l)sin(wt) and energy is E. Then (E = 4E 1 ) Wave Motion by Dr. Rajeev Tyagi. 981044396, rajeev_tyagi@rediffmail.com 11

Doppler Effect: Apparent Actual { VSound Vmedium } Vobserver [ ] { V V } V Sound medium Source Rules 1. Towards => Apparent frequency increases; Away => App. frequency decreases. Source comes in denominator, Observer comes in numerator 3. When moving perpendicular, take component of velocity along/away the line joining source and observer. Whenever you get component of velocity along or away, then Doppler effect will be there. 4. Dealing with reflection cases: Treat it as a case of reflection from mirror with wall being the mirror. Now the image will act as source of sound. 5. If reflection takes from a moving car etc then it is equivalent to : S (v=0) v S(v = v) S (v = v) Questions on Doppler s effect 1. [005, M] An observer standing on a railway crossing receives frequency of. khz and 1.8 khz when the train approaches and recedes from the observer. Find the velocity of the train. V sound = 330 m/s. [Ans. 30 m/s]. [IIT003, M] 3. [1981, 4M] A source of sound of frequency 56 Hz is moving rapidly towards a wall with a velocity of 5 m/s. How many beats per second will be heard by the observer on source itself if V sound = 330 m/s? [Ans. 7.87 Hz] 4. [1997, 5M] A band playing music at a frequency f is moving towards a wall at a speed V b. A motorist is following the band with a speed V m. If V is the speed of sound, obtain expression for the beat frequency heard by the motorist. Vbf ( V Vm) [Ans. ] ( V Vb ) 5. [1996, 3M] A whistle emitting sound of frequency 440 Hz is tied to a string of length 1.5 m and rotated with an angular velocity of 0 rad/s in the horizontal plane. Calculate the range of frequencies heard by an observer stationed at a large distance from the whistle? V sound = 330 m/s. [403.3 Hz to 484 Hz] 6. [1990, 7M] A source of sound is moving in a circle of r = 3 m with w = 10 rad/s. A sound detector located far away from the source is executing linear SHM along the line BD (see figure) with amplitude BC = CD = 6m. The frequency of oscillation of the detector is 5/ per sec. The source is at A when the detector is at B. If source frequency is 340 Hz, find the maximum and minimum frequencies recorded by the detector. V sound = 340 m/s. [Ans. 438.7 Hz, 57.3 Hz] r = 3 m 6 m 6 m A B C D Wave Motion by Dr. Rajeev Tyagi. 981044396, rajeev_tyagi@rediffmail.com 1

7. [001, 10M] A boat is traveling in a river with a speed of 10 m/s along the stream flowing with a speed of m/s. From this boat a sound transmitter is lowered into the river through a rigid support. The wavelength of sound emitted from the transmitter inside water is 14.45 mm. Assume attenuation to be negligible. a) What will be the frequency detected by a receiver kept inside the river downstream? [Ans. 1.0069x10 5 Hz] b) The transmitter and the receiver are now pulled up into air. The air is blowing with a speed of 5 m/s in the direction opposite to stream. Determine the frequency of sound detected by the receiver? [Ans. 1.0304x10 5 Hz] (Temperature of air and water = 0 0 C; Density of water = 10 3 kg/m 3. B water =.088x10 9 Pa; Gas constant, R = 8.31 J/mol-K; Mean molecular mass of air = 8.8x10-3 kg/mol; C P / C V for air = 1.4) 8. [1988, 5M]A train is approaching a hill at a speed of 40 km/hr sounds a whistle of frequency 580 Hz when it is at a distance of 1 km from a hill. A wind with a speed of 40 km/hr is blowing in the direction of motion of the train. Find a) The frequency of the whistle as heard by an observer on the hill, b) the distance from the hill at which the echo from the hill is heard by the driver and its frequency. V sound = 100 km/hr. [Ans. a) 599.33 Hz, b) 0.935 km, 61.43 Hz] 9. [1986, 8M] Two tuning forks with natural frequencies of 340 Hz each move relative to a stationary observer. One fork moves away from the observer, while the other moves towards him at the same speed. The observer hears beats of frequency 3 Hz. Find the speed of the tuning fork. [Ans. 1.5 m/s] 10. [1983, 6M] A sonometer wire under tension of 64 N vibrating in its fundamental mode is in resonance with a vibrating tuning fork. The vibrating portion of the sonometer wire has a length of 10 cm and a mass of 1 g. The vibrating tuning fork is now moved away from the vibrating wire with a constant speed and an observer standing near the sonometer hears one beat per second. Calculate the speed with which the tuning fork is moved if the speed of sound = 300 m/s. [Ans. 0.075 m/s] Wave Motion by Dr. Rajeev Tyagi. 981044396, rajeev_tyagi@rediffmail.com 13