Sound Waves Exercise -1

Sound Waves  Marked Questions can be used as Revision Questions. PART - I : SUBJECTIVE QUESTIONS Section (A) : Equation of sound wave, wavelength, frequency, Pressure and Displacement amplitude A 1. The audible frequency for a normal human being is 20 Hz to 20 kHz. Find the corresponding wavelengths if the speed of sound in air 320 m/s A 2. A sound wave of frequency 80 Hz is traveling with speed 320 m/s. (a) Find the change in phase at a given position in 400 ms interval. (b) Find the phase difference between two positions separated by 20 cm at a particular instant A 3. A traveling sound wave is described by the equation y = 2 sin (4t – 5x) where y is measured in centimeter, t in seconds and x in meters. (a) Find the ratio of amplitude and wavelength of wave. (b) Find the ratio of maximum velocity of particle to wave velocity. A 4. The pressure at a point varies from 99980 Pa to 100020 Pa due to a simple harmonic sound wave. The amplitude and wavelength of the wave are 5 × 10–6 m and 40 cm respectively. Find the bulk modulus of air A 5. Find the minimum and maximum wavelengths of sound in water that is in the audible range (20 - 20000 Hz) for an average human ear. Speed of sound in water = 1500 m/s. Section (B) : Speed of sound B 1. A man stands before a large wall at a distance of 100.0 m and claps his hands at regular intervals In such way that echo of a clap merges with the next clap. If he claps 5 times during every 3 seconds, find the velocity of sound in air. B-2.  Earthquake generate sound waves inside the earth. Unlike a gas, the earth can experience both transverse(S) and longitudinal (P) sound waves. Typically the speed of 'S' wave is about 4 km/s and that of P wave is 8 km/s. A seismograph records P and S waves from an earthquake. The first P wave arrives 4 min. before the first S wave. Assuming the waves travel in straight line, the epicentre of earthquake is at 120  (in km). Find  . B 3. (a) Find the speed of sound in a mixture of 1 mol of helium and 2 mol of oxygen at 27°C. (b) If now temp. is raised by 1K from 300 K. Find the percentage change in the speed of sound in the gaseous mixture. [Note : This can be done after studying heat.] [JEE - 1995] B 4. A gas mixture has 24 % of Argon, 32 % of oxygen, and 44 % of CO 2 by mass. Find the velocity of sound in the gas mixture at 27 °C. Given R = 8.4 S.I. units. Molecular weight of Ar = 40, O 2 = 32, CO2 = 44. Ar = 5/3, O2 = 7/5, CO2= 4/3. [Note : This can be done after studying heat.] Section (C) : Intensity of sound, Decibel scale C 1. Two sound waves one in air and the other in fresh water are equal in intensity. (a) Find the ratio of pressure amplitudes of the wave in water to that of the wave in air. (b) If the pressure amplitudes of the waves are equal then what will be the ratio of the intensities of the waves. [Vsound = 340 m/s in air & density of air = 1.25 kg/m3, Vsound = 1530 m/s in water, density of water = 1000 kg/m3] C 2. A point A is located at a distance r = 1.5 m from a point source of sound of frequency n = 600 Hz. The power of the source P = 0.80 W. Neglecting the damping of the wave and assuming the velocity of 100 225 sound in air to be 340 ms–1. Find at the point A : (Use dair = kg m–3 ; 2 = ) 3  3.375 544 (a) The pressure oscillation amplitude (p)m. (b) The oscillation amplitude of particles of the medium. Section (D) : Interference Sound Waves D 1. Two point sound sources A and B each of power 25 W and frequency 850 Hz are 1 m apart. The sources are in phase (a) Determine the phase difference between the waves emitting from A and B received by detector D as shown in figure. (b) Also determine the intensity of the resultant sound wave as recorded by detector D. Velocity of sound = 340 m/s. D 2. Two identical loudspeakers are located at points A & B, 2 m apart. The loudspeakers are driven by the same amplifier (coherent and are in phase). A small detector is moved out from point B along a line perpendicular to the line connecting A & B. Taking speed of sound in air as 332 m/s, find the frequency below which there will be no position along the line BC at which destructive interference occurs. A 2m B C D 3. A sound source, detector and a movable wall are arranged as shown in figure. In this arrangement detector is detecting the maximum intensity. If the speed of sound is 330 m /s in air and frequency of source is 660 Hz, then find the minimum distance by which the wall should be moved away from source, so that detector detects minimum intensity. D 4. Two sources of sound, S1 and S2, emitting waves of equal wavelength 2 cm, are placed with a separation of 3 cm between them. A detector can be moved on a line parallel to S1S2 and at a distance of 24 cm from it. Initially, the detector is equidistant from the two sources. Assuming that the waves emitted by the sources are in phase, find the minimum distance through which the detector should be shifted to detect a minimum of sound. D 5. A sound source, detector and a cardboard are arranged as shown in figure. The wave is reflected from the cardboard at the line of symmetry of source and detector. Initially the path difference between the reflected wave and the direct wave is one third of the wavelength of sound. Find the minimum distance by which the cardboard should be moved upwards so that both waves are in phase. Section (E) : Reflection of sound equation of stationary waves E 1. A metallic rod of length 1 m is rigidly clamped at its end points. Longitudinal stationary waves are setup in the rod in such a way that there are six antinodes of displacement wave observed along the rod. The amplitude of the antinode is 2  10−6 m. Write the equations of the stationary wave and the component waves at the point 0.1 m from the one end of the rod. [Young's modulus = 7.5  1010 N/m2, density = 2500 kg/m3] E 2. The equation of a longitudinal standing wave due to superposition of the progressive waves produced by two sources of sound is s = −20 sin 10 x sin 100 t where s is the displacement from mean position measured in mm, x is in meters and t is in seconds . The specific gravity of the medium is 10−3. Density of water = 103 kg/m3. Find : (a) Wavelength, frequency and velocity of the progressive waves. (b) Bulk modulus of the medium and the pressure amplitude. (c) Minimum distance between pressure antinode and a displacement antinode. (d) Intensity at the displacement nodes. Section (F) : Organ Pipes and Resonance F 1. A closed organ pipe has length ‘ ’. The air in it is vibrating in 3rd overtone with maximum amplitude 'a '. Find the amplitude at a distance of /7 from closed end of the pipe. Sound Waves F 2. The speed of sound in an air column of 80 cm closed at one end is 320 m/s. Find the natural frequencies of air column between 20 Hz and 2000 Hz. F 3. In an organ pipe the distance between the adjacent nodes is 4 cm. Find the frequency of source if speed of sound in air is 336 m/s F 4. Two pipes P1 and P2 are closed and open respectively. P1 has a length of 0.3 m. Find the length of P2, if third harmonic of P1 is same as first harmonic of P2. F 5. Two adjacent resonance frequencies of an open organ pipe are 1800 and 2100 Hz. Find the length of the tube. The speed of sound in air is 330 m/s. F 6. A closed organ pipe of length = 100 cm is cut into two unequal pieces. The fundamental frequency of the new closed organ pipe piece is found to be same as the frequency of first overtone of the open organ pipe piece. Determine the length of the two pieces and the fundamental tone of the open pipe piece. Take velocity of sound = 320 m/s. F 7. Find the number of possible natural oscillations of air column in a pipe whose frequencies lie below f0 = 1250 Hz. The length of the pipe is = 85 cm. The velocity of sound is v = 340 m/s. Consider the two cases : (a) The pipe is closed from one end (b) The pipe is opened from both ends. The open ends of the pipe are assumed to be the antinodes of displacement. F-8. In a resonance tube experiment to determine the speed of sound in air, a pipe of diameter 5 cm is used. The air column in pipe resonates with a tuning fork of frequency 480 Hz when the minimum length of the air column is 16 cm. If the speed of sound in air at room temperature = 6 (in m/sec.) Find  [JEE - 2003, 2/60] F-9. A person hums in a well and finds strong resonance at frequencies 60Hz, 100Hz and 140Hz. What is the fundamental frequency of the well? Explain ? How deep is the well ? (velocity of sound = 344m/s). Section (G) : Beats G 1. A source of sound with adjustable frequency produces 4 beats per second with a tuning fork when its frequency is either 474 Hz. or 482 Hz. What is the frequency of the tuning fork? G 2. Two identical piano wires have a fundamental frequency of 600 vib/sec, when kept under the same tension. What fractional increase in the tension of one wire will lead to the occurrence of six beats per second when both wires vibrate simultaneously. G 3. A metal wire of diameter 1 mm, is held on two knife edges separated by a distance of 50 cm. The tension in the wire is 100 N. The wire vibrating in its fundamental frequency and a vibrating tuning fork together produces 5 beats per sec. The tension in the wire is then reduced to 81 N. When the two are excited, beats are again at the same rate. Calculate (a) The frequency of the fork (b) The density of the material of the wire. G 4. A string 25 cm long fixed at both ends and having a mass of 2.5 g is under tension. A pipe closed from one end is 40 cm long. When the string is set vibrating in its first overtone and the air in the pipe in its fundamental frequency, 8 beats per second are heard. It is observed that decreasing the tension in the string decreases the beat frequency. If the speed of sound in air is 320 m/s. Find tension in the string. Section (H) : Doppler Effect H 1. An observer rides with a sound source of frequency f and moving with velocity v towards a large vertical wall. Considering the velocity of sound waves as c, find : (i) The number of waves striking the surface of wall per second (ii) The wavelength of the reflected wave (iii) The frequency of reflected wave as observed by observer. (iv) Beat frequency heard by the observer. H 2. A stationary source emits single frequency sound. A wall approaches it with velocity u = 33 cm/s. The propagation velocity of sound in the medium is v = 330 m/s. In what way and how much, in per cent, does the wavelength of sound change on reflection from the wall ? H 3. A source of sonic oscillations with frequency f0 = 1000 Hz moves at right angles to the wall with a velocity u = 0.17 m/s. Two stationary receivers R1 and R2 are located on a straight line, coinciding with the trajectory of the source, in the following succession : R 1 − source − R2 − wall. Which receiver registers the beatings and what is the beat frequency ? The velocity of sound is equal to v = 340 m/s. Sound Waves H 4. A sound wave of frequency f propagating through air with a velocity C, is reflected from a surface which is moving away from the source with a constant speed V. Find the frequency of the reflected wave, measured by the observer at the position of the source. H 5. Two trains move towards each other with the same speed. Speed of sound is 340 ms –1. If the pitch of the tone of the whistle of one when heard on the other changes to 9/8 times, then the speed of each train is : H 6. A tuning fork P of unknown frequency gives 7 beats in 2 sec with another tuning fork Q. When Q runs towards the wall with a speed of 5 m/s it gives 5 beats per sec with its echo. On loading wax on P it gives 5 beats per second with Q. What is the original frequency of P ? Assume speed of sound = 332 m/s. PART - II : ONLY ONE OPTION CORRECT TYPE Section (A) : Equation of sound wave, wavelength, frequency, Pressure and Displacement amplitude A 1. When sound wave is refracted from air to water, which of the following will remain unchanged? (A) wave number (B) wavelength (C) wave velocity (D) frequency A 2. A piece of cork is floating on water in a small tank. The cork oscillates up and down vertically when small ripples pass over the surface of water. The velocity of the ripples being 0.21 ms –1, wave length 22 15 mm and amplitude 5 mm, the maximum velocity of the piece of cork is ( = ) 7 (A) 0.44 ms–1 A 3. (B) 0.24 ms–1 (C) 2.4 ms–1 (D) 4.4 ms–1 The frequency of a man’s voice is 300 Hz and its wavelength is 1 meter. If the wavelength of a child’s voice is 1.5 m, then the frequency of the child’s voice is: (A) 200 Hz (B) 150 Hz (C) 400 Hz (D) 350 Hz Section (B) : Speed of sound B 1. B 2. A machine gun is mounted on an armored car moving with a speed of 20 ms –1. The gun can point against the direction of motion of car. The muzzle speed of bullet is equal to speed of sound in air i.e., 340 ms–1. The time difference between bullet actually reaching and sound of firing reaching at a target 544 m away from car at the instant of firing is (A) 1.2 s (B) 0.1 s (C) 1 s (D) 10 s The ratio of speed of sound in a monoatomic gas to that in water vapours at any temperature is. (when molecular weight of gas is 40 gm/mol and for water vapours is 18 gm/mol) (A) 0.75 (B) 0.73 (C) 0.68 (D) None of these B 3. Under similar conditions of temperature and pressure, In which of the following gases the velocity of sound will be largest. (A) H2 (B) N2 (C) He (D) CO2 B 4. If vrms = root mean square speed of molecules vav = average speed of molecules vmp = most probable speed of molecules vs = speed of sound in a gas Sound Waves Then, identify the correct relation between these speeds. (A) vrms > vav > vmp > vs (B) vav > vmp > vrms > vs (C) vmp > vav > vrms > vs (D) vrms > vav > vs > vmp Section (C) : Intensity of sound, Decibel scale C 1. A sound of intensity  is greater by 3.0103 dB from another sound of intensity 10 nW cm –2. The absolute value of intensity of sound level  in Wm–2 is : (A) 2.5 × 10–4 (B) 2 × 10–4 (C) 2.0 × 10–2 (D) 2.5 × 10–2 C 2. For a sound source of intensity  W/m2, corresponding sound level is B0 decibel. If the intensity is increased to 4, new sound level becomes approximately : (A) 2B0 dB (B) (B0 + 3)dB (C) (B0 + 6)dB (D) 4B0 dB C 3. The sound intensity is 0.008 W/m2 at a distance of 10 m from an isotropic point source of sound. The power of the source is approximately : (A) 2.5 watt (B) 0.8 watt (C) 8 watt (D) 10 watt Section (D) : Interference D 1. What happens when a sound wave interferes with another wave of same frequency and constant phase difference ? (A) Energy is gained (B) Energy is lost (C) Redistribution of energy occurs changing with time (D) Redistribution of energy occurs not changing with time D 2. Sound waves from a tuning fork F reach a point P by two separate routes FAP and FBP (when FBP is greater than FAP by 12 cm there is silence at P). If the difference is 24 cm the sound becomes maximum at P but at 36 cm there is silence again and so on. If velocity of sound in air is 330 ms –1, the least frequency of tuning fork is : (A) 1537 Hz (B) 1735 Hz (C) 1400 Hz (D) 1375 Hz D 3. Sound signal is sent through a composite tube as shown in the figure. The radius of the semicircular portion of the tube is r. Speed of sound in air is v. The source of sound is capable of giving varied frequencies in the range of 1 and 2 (where 2 > 1). If n is an integer then frequency for maximum intensity is given by : (A) D 4. nv r nv r(  − 2) (C) nv r (D) nv (r − 2) An interference is observed due to two coherent sources ‘A’ & ‘B’ separated by a distance 4 along the y-axis where  is the wavelength of the source. A detector D is moved on the positive x-axis. The number of points on the x−axis excluding the points, x = 0 & x =  at which maximum will be observed is – (A) three D-5. (B) (B) four (C) two (D) infinite A person is talking in a small room and the sound intensity level is 60 dB everywhere within the room. If there are eight people talking simultaneously in the room, what is the sound intensity level ? (A) 60 dB (B) 69 dB (C) 74 dB (D) 81 dB Section (E) : Reflection of sound equation of stationary waves Sound Waves E 1. When a sound wave is reflected from a wall, the phase difference between the reflected and incident pressure wave is: (A) 0 (B)  (C) /2 (D) /4 Section (F) : Organ Pipes and Resonance F 1. If 1, 2, 3 are the wavelengths of the waves giving resonance in the fundamental, first and second overtone modes respectively in a open organ pipe, then the ratio of the wavelengths 1 : 2 : 3, is : (A) 1 : 2 : 3 (B) 1 : 3 : 5 (C) 1 : 1/2 : 1/3 (D) 1 : 1/3 : 1/5 F 2. The maximum variation of pressure in an open organ pipe of length vibrating in fundamental mode is at. L 3L (A) ends (B) middle of pipe (C) from centre (D) from centre 4 8 F 3. The fundamental frequency of a closed organ pipe is same as the first overtone frequency of an open pipe. If the length of open pipe is 50 cm, the length of closed pipe is (A) 25 cm (B) 12.5 cm (C) 100 cm (D) 200 cm F 4. Two identical tubes A and B are kept in air and water respectively as shown. If the fundamental frequency of A is f0, then the fundamental frequency of B is (A) f0 4 (B) f0 2 (C) f0 (D) 2 f0 F 5. A tube of diameter d and of length unit is open at both the ends. Its fundamental frequency of resonance is found to be 1. The velocity of sound in air is 330 m/sec. One end of tube is now closed. The  lowest frequency of resonance of tube is now 2. Taking into consideration the end correction, 2 is 1 ( + 0.6d) 1 ( + 0.3d) 1 ( + 0.6d) 1 (d + 0.3 ) (B) (C) (D) ( + 0.3d) 2 ( + 0.6d) 2 ( + 0.3d) 2 (d + 0.6 ) The second overtone of an open pipe A and a closed pipe B have the same frequencies. The ratio of fundamental frequency of A to the fundamental frequency of B is: (A) 3: 5 (B) 5: 3 (C) 5: 6 (D) 6: 5 (A) F 6. F 7. A resonance tube is resonated with tuning fork of frequency 256 Hz. If the length of first and second resonating air columns are 32 cm and 100 cm, then end correction will be (A) 1 cm (B) 2 cm (C) 4 cm (D) 6 cm Section (G) : Beats G 1. A sound source of frequency 512 Hz is producing 6 beats with a guitar. If the string of guitar is stretched slightly then beat frequency decreases. The original frequency of guitar is (A) 506 Hz (B) 512 Hz (C) 518 Hz (D) 524 Hz G 2. The number of beats heard per second if there are three sources of sound of frequencies (n – 1), n and (n + 1) of equal intensities sounded together is : (A) 2 (B) 1 (C) 4 (D) 3 G 3. A closed organ pipe and an open pipe of same length produce 4 beats when they are set into vibrations simultaneously. If the length of each of them were twice their initial lengths, the number of beats produced will be [Assume same mode of vibration in both cases] (A) 2 (B) 4 (C) 1 (D) 8 Sound Waves Section (H) : Doppler Effect H 1. Which of the following does not affect the apparent frequency in doppler effect ? (A) Speed of source (B) Speed of observer (C) Frequency of source (D) Distance between source and observer H 2. An engine driver moving towards a wall with velocity of 50 ms –1 emits a note of frequency 1.2 kHz. The frequency of note after reflection from the wall as heard by the engine driver when speed of sound in air is 350 ms–1 is : (A) 1 kHz H 3. (B) 1.8 kHz (C) 1.6 kHz (D) 1.2 kHz Source and observer both start moving simultaneously from origin, one along X-axis and the other along Y-axis with speed of source equal to twice the speed of observer. The graph between the apparent frequency (n') observed by observer and time t would be : (n is the frequency of the source) (A) (B) (C) (D) H 4. An observer moves on a circle as shown in fig. and a small sound source is at S. Let at 1 2 3 be the frequencies heard when the observer is at A, B, and C respectively. (A) 1 > 2 > 3 H-5. (B) 1 = 2 > 3 (C) 2 > 3 > 1 (D) 1 > 3 > 2 Two factories are sounding their sirens at 400 Hz each. A man walks from one factory towards the other at a speed of 2 m/s. the speed of sound is 320 m/s. The number of beats heard per second by the man is. [Olympiad 2016 stage–I] (A) 6 (B) 5 (C) 2.5 (D) 7.5 PART - III : MATCH THE COLUMN 1. Match the Column: Column- Column- (A) y = 4 sin (5x – 4t) + 3 cos (4t – 5x + /6) (p) Particles at every position are performing SHM x   (B) y = 10 cos  t −  sin (100)  330  (q) Equation of travelling wave x    t − 330    (C) y = 10 sin (2x – 120t) + 10 cos (120t + 2x) (r) Equation of standing wave (D) y = 10 sin (2x – 120 t) + 8 cos (118t – 59/30x) (s) Equation of Beats 2. S, O & W represent source of sound (of frequency f), observer & wall respectively. V O, VS, VD, V are velocity of observer, source, wall & sound (in still air) respectively. V W is the velocity of wind. They are moving as shown. Then match the following : where fr = V + Vw + VD f V + Vw − VS Sound Waves Column- (A) The wavelength of the waves coming Column- (p) (V – VW – VD)/fr towards the observer from source. (B) The wavelength of the waves incident on the wall. (q) (V – VW – VO)fr /(V – VW – VD) (C) The wavelength of the waves coming towards (r) (V – VW + VS)/f observer from the wall. (D) Frequency of the waves (as detected by O) (s) (V + VW – VS)/f coming from wall after reflection. 3. To resonate a 1m tube closed at one end, if we use different tuning forks, we get different results. Match the following according to result of using tuning fork of certain frequency. (Velocity of sound = 320 m/s) Tuning fork Result (A) 240 Hz (p) Moderate sound will be generated (B) 320 Hz (q) Violent sound will be generated (C) 400 Hz (r) Only third harmonic will be activated (D) 500 Hz (s) Only fifth harmonic will be activated Sound Waves (d) I = 20 2  200 wm–2 EXERCISE-1 Section (F) : PART - I Section (A) : A-1. 16 mm, 16 m A-2. (a) 2  ft = 64  A-3. (a) A-4. A-5. (b) 1 kA = 20 2 2fx  = v 10 (b) kA = 1 10 F-2. 100 (2n + 1) Hz where n = 0, 1, 2, 3, ...., 9 F-3. F-5. 4.2 kHz 55 cm F-6. 20, 80 cm, 200 Hz F-7. (a) vn = v (2n + 1); six oscillations 4 (b) vn = v (n + 1), also six oscillations. 2 7.5 cm, 75 m F-8. B-1. 1000 m/s 3 B-2. B-3. (a)  400.9 m/s (b) B-4. V  303.5 m/s 16 1 % 6 Section (C) : P0w = 60 C-1. (a) C-2. (a) 5Nm-2, P0a (b) Pw 1 = Pa 3600 (b) 3 m Section (D) : F-9. ( 83 Hz 4 cm D-3. ) 2 G-3. (a) 95 Hz G-4. 27.0400 N 40  103 kg/m3  fc c−v 8.5 cm y1 (x, t) =1×10–6 sin(6x+ 6 30  103t + ) at x = 0.1 6 + 6 30 103t+ ) 10 6 – 6 30  103t + ) 10 2v c–v 2u = 0.2 % (v + u) H-2. decreases by H-3. R1, fbeat = 2 f0 vu / (v2 − u2)  H-4. c−v f c+v H-6. 160 Hz 2f0u = 1.0 Hz v H-5. 20 m/s PART - II Section (A) : A 1. (a) f = 50 Hz,  = 0.2 m, v = 10 ms–1 (b) Pm = 62.8 Nm–2 = 20 Nm–2, B = 100Nm−2 (c) /4 = 0.05 m c +v c–v (iv) fbeat = f y2 (x, t) =1×10–6 sin(6x – 6 30  103t + ) E-2. (i) f  = (iii) f = f y1(0.1, t) =1×10–6 sin( (b) 2% v c v c −v (ii)  =  −   =   −   =   f f f  f  y(x, t) =2×10–6 sin 6x cos (6 30   103t+) at x = 0.1 6 y(0.1, t) = 2×10–6sin cos(6 30 103 t+) 10 y2(0.1, t) =1×10–6sin( G-2. Section (H) : Section (E) : E-1. fundamental frequency n = 20 Hz, depth of the well = 4.3 478 Hz = (25/312)2 12.5cm D-4. 20 cm Here n = 0, 1, 2, ... 56 G-1. H-1.  A − B F-4. Section (G) : (a)  (b) I = D-2. D-5. a P0  8  105 = N/m2 2S0  Section (B) : D-1. F-1. (D) A 2. (A) A 3. (A) B 2. (A) B 3. (A) Section (B) : B 1. (B) Sound Waves B 4. (A) Section (C) : C 1. (B) C 2. (C) C 3. (D) D 2. D-5. (D) (B) D 3. (B) F 2. F 5. (B) (C) F 3. F 6. (B) (B) G 2. (B) G 3. (A) H 2. H-5. (C) (B) H 3. (B) Section (D) : D 1. D 4. (D) (A) Section (E) : E 1. (A) Section (F) : F 1. F 4. F 7. (C) (C) (B) Section (G) : G 1. (A) Section (H) : H 1. H 4. (D) (D) PART - III 1. 2. 3. (A) → p, q ; (B) → q, s ; (C) → r ; (D) → s,q (A) → (r); (B) → (s); (C) → (p); (D) → (q) (A) → q, r ; (B) → p ; (C) → q, s ; (D) → p 7. 10. (2) (3) 8. (3) 9. (4)