PW W A V E S
PW CONCEPTS C C Equation of a Travelling Wave The equation of a wave traveling along the positive x-ax given by y = f(x vt) If the wave travelling along the negative x-ax, the wave funcion represented by y = f(x + vt) In general, the wave motion in one dimension given by y = f(x ± vt) Sine Harmonic Wave The wave function that represents a harmonic wave given by y = A sin[k(x ± vt)] or y = Asin(kx ± t) The negative sign used when the wave travels along the positive x-ax, and vice-versa. The term k called the wave number which defined as k, where called the wavelength. It the dtance between the two successive points with the same phase (for example, two crests). The term called the angular frequency (measured in rad/s) f where T the time period T and f the frequency. A the dplacement amplitude. Wave velocity given by f Particle Velocity : Particle velocity the rate at which particle s dplacement vary as a function of time, i.e., y A cos(kx t ). t Acceleration of the Particle : The acceleration of the particle obtained by differentiating above equation w.r.t. time y t Asin(kx t ).. A wave equation which gives the dplacement along y-direction given by y = 0 4 sin (60t + x) where x and y are in m and t in s. Th represents a wave travelling with velocity of 30 m/s in the negative x-direction of wavelength () m of frequency (30/) Hz all the above. A transverse sinusoidal wave of amplitude a, wavelength and frequency f traveling on a stretched string. The maximum speed of any point on the string v/0, where v the speed of propagation of the wave. If a = 0 3 m and v = 0 ms, then and f are given by = 0 m = 0 3 m f = 0 3 /() Hz both and are correct 3. A transverse wave described by the equation y = A sin (ft x/). The maximum particle velocity equal to n times the wave velocity then the relation between A and n = A n = A n = 3A n = 4A [Answers : () d () d (3) b]
PW 3 C3 Wave Speed on Stretched String : The speed of a wave on a stretched string set by properties of the string. The speed on a string with tension T and linear density µ T v. µ C4 Energy Transmitted by a Wave deav The average power transmitted by the wave Pav µ A v where v the wave velocity. The dt mass per unit length of a wire given by µ = a where the density and a the cross-sectional area. The P intensity of the wave given by I a av. A singing a note and at the same time B singing a note with exactly one-eighth the frequency of the note of A. The energies of the two sounds are equal. The amplitude of the note of B Same as that of A Twice that of A Four times that of A Eight times that of A [Answers : () d] A v C5 Velocity of Longitudinal Waves = C6 density of the medium in which wave will propagate. Velocity of Sound Wave E, where E the modulus of elasticity of the medium and the The speed of longitudinal waves in a fluid given by v B where B the bulk modulus defined. Velocity of sound in air given by v p where the adiabatic exponent of the gas. Using ideal gas equation p RT RT, v p M M Substituting the values of and M for air we obtain the approximate value of speed of sound in air at absolute temperature T, as v 0 T. The ratio (V s /V rms ) of the velocity of sound in a gas (V s ) and the root mean square velocity (V rms ) of its molecular at the same temperature : ( = ratio of the two specific heats of the gas) / 3 3 / 3. The ratio of the speed of sound in nitrogen gas to that in helium gas, at 300 K / 7) ( ( / 7) ( 3) / 5 ( 6) / 5 3. The temperature of air increased from 300 K to 30 K. The fractional change in velocity of sound /300 /600 /900 /00 4. The speed of sound in a gas at NTP 300 m/sec. If the pressure increased four times, without change in temperature, the velocity of sound will be 50 m/sec 300 m/sec 600 m/sec 00 m/sec [Answers : () a () b (3) b (4) b] 3
PW 4 C7 Beats : Adding Wave That Differ in Frequency Only : If two or more waves of slightly different frequencies f and f are superimposed, the intensity of the resulting wave have alternatively maxima and minima. The number of minima or maxima in one second called the beats frequency which given by f f. When two tuning forks A and B are sounded together x beat/s are heard. Frequency of A n. Now when one prong of fork B loaded with a little wax, the number of beat/s decreases. The frequency of fork B n + x n x n x n x. The speed of sound in a gas in which two waves of wavelengths 50 cm and 50.4 cm produce 6 beats per second 338 m/s 350 m/s 378 m/s 400 m/s [Answers : () a () c] C8 Standing Waves : Adding Waves That Differ In Direction Only : Equation of standing wave : y = a sin kx cos t, here a sin kx the amplitude of the wave. The above equation shows that the string executes simple harmonic motion such that every point on the string vibrates in same phase with same frequency but different amplitudes which depends on the position x of the point along the string. Th type of wave motion represented by equation called a standing wave because it appears to travel neither to the left nor to the right. There are positions along the string for which the amplitude of oscillation always zero (called nodes), and other positions where the amplitudes of oscillation always a (called antinodes). The dtance between two successive node. The dtance between two successive antinodes. Also, nodes and antinodes occur alternatively and equally spaced from each other.. For the stationary wave x y 4 sin cos( 96t), the dtance between a node and the next antinode 5 7.5 units 5 units.5 units 30 units. A standing wave having 3 nodes and antinodes formed between two atoms having a dtance. Å between them. The wavelength of the standing wave. Å 6.08 Å 3.63 Å.4 Å [Answers : () a () a] C9 Standing Wave Pattern on the String Standing waves on a string can be set up by reflection of traveling waves from the ends of the string. If an end fixed, it must be the position of a node. For a stretched string of length L with fixed ends, the resonant v v frequences are f n, for n =,, 3,... (where v = L The oscillation mode corresponding to n = called the fundamental mode or the first harmonic; the mode corresponding to n = the second harmonic; and so on.. Two vibrating strings of the same material but length L and L have radii r and r respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental modes, the one of length L with frequency v and the other with frequency v. The ratio v /v given by 4 8 T ). µ
PW 5. 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 suspended from the wire. When th mass replaced by a mass M, the wire resonates with the same tuning form forming three antinodes for the same positions of the bridges. The value of M 5 kg 5 kg.5 kg /5 kg 3. In order to double the frequency of the fundamental note emitted by a stretched string, the length reduced to 3/4 th of the original length and the tension changed. The factor by which the tension to be changed 3/8 /3 8/9 9/4 [Answers : () d () a (3) d] C0 Standing wave pattern in pipes Standing sound wave pattern can be set up in pipes. A pipe open at both ends will resonate at frequencies v nv f, n =,, 3,... L where v the speed of sound in the gas in the pipe. For a pipe closed at one end and open at the another, the v nv resonant frequencies are f, n =, 3, 5,... 4L. Velocity of sound in air 30 m/s. A pipe closed at one end has a length of m. Neglecting end correction, the air column in the pipe cannot resonate for sound of frequency 80 Hz 40 Hz 30 Hz 400 Hz. If the fundamental frequency of a pipe closed at one end 5 Hz, the fundamental frequency of a pipe of the same dimensions but open at both ends will be 04 Hz 5 Hz 56 Hz 8 Hz [Answers : () c () a] C Doppler Effect : The Doppler effect a change in the observed frequency of a wave when the source or the detector moves relative to the transmitting medium (such as air). For sound the observed frequency f given in terms of v v the source frequency f by f f v v O S, where v 0 the speed of the detector relative to the medium, v S that of the source, and v the speed of sound in the medium. The signs are chosen such that f tends to be greater for motion (of detector of source) toward and less for motion away. Wind Effect : The above formulae can be modified by taking the wind effects into account. The velocity of sound should be taken as : v + v w or v v w if the wind blowing in the same or opposite direction as source to observer.. A vehicle with a horn of frequency n moving with a velocity of 30 m/s in a direction perpendicular to the straight line joining the observer and the vehicle. The observer perceives the sound to have a frequency n + n. Then (if the sound velocity in air 300 m/s) n = 0 n n = 0 n = 0. n n = 0. n. A train moves towards a stationary observer with speed 34 m/s. The train sounds a whtle and its frequency regtered by the observer f. If the train s speed reduced to 7 m/s, the frequency regtered f. If the speed of sound 340 m/s then the ratio f /f 8/9 / 9/8 3. A siren placed at a railway platform emitting sound of frequency 5 khz. A passenger sitting in a moving train A records a frequency of 5.5 khz while the train approaches the siren. During h return journey in a different train B he records a frequency of 6.0 khz while approaching the same siren. The ratio of the velocity of train B to that of train A 4/5 5/6 /6 [Answers : () b () d (3) b]
PW 6 INITIAL STEP EXERCISE. Consider the following statements : Assertion (A) : The velocity of sound in air increases due to the presence of moture in it. Reason (R) : The presence of moture in air lowers the density of air. Of these statements both A and R are true and R the correct explanation of A both A and R are true but R not the correct explanation of A A true but R true A false but R true. How does the intensity (I) of a wave depend on the dtance (r) from a line source? I r I r I r I r ½ 3. The dplacement of particles in a string stretched in the x-direction represented by y. Among the following expressions for y, those describing wave motion are cos kx sin t k x t cos (kx + t) both and are correct 4. A wave represented by the equation y = a cos (kx t) superposed with another wave to form a stationary wave such that the point x = 0 a node. The equation for the other wave y = a sin(kx + t) y = a cos (kx + t) y = a cos (kx + t) y = a sin (kx t) 5. A boat at anchor rocked by waves whose crests are 00 m apart and whose velocity 5 m/sec. These waves strike the boat once every 500 sec 0.5 sec 500 sec 4 sec 6. A source producing sound of frequency 70 Hz approaching a stationary observer with a velocity 7 ms. The apparent change in the wavelength of sound heard by the observer (speed of sound in air = 340 m/s) 0. m 0. m 0.4 m 0.5 m 7. The wavelength of light observed on the earth from a moving star found to decrease by 0.05%. Relative to the earth the star moving away with a velocity of.5 0 5 m/s coming closer with a velocity of.5 0 5 m/s moving away with a velocity of.5 0 4 m/s coming closer with a velocity of.5 0 4 m/s 8. Standing waves can be produced on a string clamped at both the ends on a string clamped at one end and free at the other when incident wave gets reflected from a wall all are correct 9. A sinusoidal wave of frequency 500 Hz has a speed of 350 m/s. The dtance between two points that differ in phase by /3 rad. 5.7 cm 6.9 cm.7 cm.0 cm ANSWERS (INITIAL STEP EXERCISE). a. a 3. d 4. b 5. d 6. a 7. b 8. d 9. c
PW 7 FINAL STEP EXERCISE (OBJECTIVE). In case of superposition of waves (at x = 0): y = 4 sin (06t) and y = sin (04t). Then choose the correct statement 5. A piezo-electric quartz plate of thickness d vibrating in resonant conditions. The Young s Modulus of elasticity of quartz Y and density Its fundamental frequency The frequency of resulting wave 50 Hz The amplitude of resulting wave varies at frequency 3 Hz and the frequency of beats 6 Hz. The ratio of maximum to minimum intensity 9 d d Y Y d d Y 3Y all are correct. A glass tube of.0 m length filled with water; the water can be drained out slowly at the bottom of the tube. If a vibrating tuning fork of frequency 500 Hz brought at the upper end of the tube and the velocity of sound 330 m/s, then the total number of resonances obtained will be 4 3 3. Two stretched wires are in unon. If he tension one of the wires increased by %, 3 beats are produced in s. The initial frequency of each wire 50 Hz 00 Hz 300 Hz 450 Hz 4. A copper wire of Young s Modulus Y, coefficient of linear expansion and densiry held at the two ends by rigid supports. At initial temperature, the wire just taut, with negligible tension. The speed of transverse waves in th wire if the change in temperature t given by 6. Two identical piano wires have a fundamental frequency of 600 Hz when kept under the same tension. The fractional increase in the tension of one wire will lead to the occurrence of 6 beats/s when both wires oscillate 0.0 0.03 0.04 0.05 7. The velocity of sound in the gaseous mixture consts of mol of He gas and moles of nitrogen gas at the temperature of 7 0 C 400 m/s 450 m/s 550 m/s none of these 8. The pressure for 6 0 3 m 3 of certain liquid decreased from 0 m of water to 9 m of water. Due to the th the fractional increase in volume of the liquid 0 6. The density of the liquid 540 kg/m 3. The velocity of sound in th liquid 500 m/s 000 m/s 500 m/s none Yt Yt ANSWERS (FINAL STEP EXERCISE) Yt 3Yt. d 5. a. b 6. a 3. c 7. d 4. a 8. d
PW 8 TEST YOURSELF. When a wave travels in a medium, the particle dplacement are given by y(x, t) = 0.03 sin (t 0.0 x) where y and x are in metres and t in seconds. The wavelength of the wave 0 m 0 m 00 m 00 m. A sonometer wire, with a suspended mass of M = kg, in resonance with a given tuning fork. The apparatus taken on the moon where the acceleration due to gravity /6 that on earth. To obtain resonance on the moon, the value of M should be kg 6 kg 6 kg 36 kg 3. Particle dplacement (in cm) in a standing wave are given by y(x, t) = sin (0. x) cos (00 t). The dtance between a node and the next anti-node.5 cm 5.0 cm 7.5 cm 0.0 cm 4. If the Young s modulus of the material of a rod 0 N m and its density 8000 kg m 3, the time taken by a sound wave to traverse m of the rod will be 0 4 s 0 4 s 0 s 0 s 5. An observer moves towards a stationary source of sound with a velocity one-tenth the velocity of sound. The apparent increase in frequency zero 5% 0% 0. % 6. Two parts of a sonometer wire, divided by a movable knife-edge, differ in length by cm and produce beat per second when sounded together. If the total length of the wire 00 cm, the frequencies of the two parts of the wire are 5 Hz, 50 Hz 50.5 Hz, 49.5 Hz 49 Hz, 48 Hz 49.5 Hz, 48.5 Hz 7. A stone hangs from the free end of a sonometer wire whose vibrating length, when turned to a tuning fork, 40 cm. When the stone hangs wholly immersed in water, the resonant length reduced to 30 cm. The relative density of the stone 6/9 6/7 6/5 6/3 8. A pipe closed at one end and open at the other will give all the harmonics all even harmonics all odd harmonics none of the harmonics 9. A cylindrical tube, open at both ends, has fundamental frequency n. If one of the ends closed. the fundamental frequency will become n/ n n 4 n 0. Two sources A and B are sounding notes of frequency 680 Hz. A ltener moves from A to B with a constant velocity u. If the speed of sound 340 ms, what must be the value of u so that he hears 0 beats per second?.0 ms.5 ms 3.0 ms 3.5 ms. d. c 3. b 4. b 5. c 6. b 7. b 8. c 9. a 0. b ANSWERS