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MW Assignments: Mechanical Waves Questions:1. The figure at right shows three waves that are separately sent along a string that is stretched under a certain tension along the x-axis. Rank the waves according to their (a) wavelengths, (b) speeds, and (c) angular frequencies, greatest first.(HR W 7e Q 16.1 , i.e. Ch. 16, question #1 on p. 436)

2. Two waves travel on the same string. Is it possible for them to have (a) different frequencies; (b) different wavelengths; (c) different speeds; (d) different amplitudes; (e) the same frequency but different wavelengths? Explain your reasoning. (YF 12e Q1 5.1 o n p. 518 in our text) 3. The four strings on a violin have different thicknesses, but all are under approximately the same tension. Do the waves travel faster on the thick strings or the thin strings? Why? How does the fundamental vibration frequency compare for the thick versus the thin strings? (YF12e Q15.12) 4. A long rope with mass m is suspended from the ceiling and hangs vertically. A wave pulse is produced at the lower end of the rope, and the pulse travels up the rope. Does the speed of the wave pulse change as it moves up the rope? If not, why not? If so, does it increase or decrease? (YF12e Q15.15)

Problems:1. The following wave functions represent traveling waves: (a) y2 (x , t ) = Acos [k(x + 34t)] (b) y3 (x , t ) = Ae-k (x - 20 t) (c) y1 (x , t ) = B/[C + (x - 10t)2 ] where x is in meters, t is in seconds, and A, k, B, and C are constants that have the proper units for y to be in meters. Give the direction of propagation and the speed of the wave for each wave function. (T13.6) 2. A transverse traveling wave has the equation: y = (6.0 cm) sin (0.02Bx + 4Bt), where x and y are in centimeters and t is in seconds. Find (a) the amplitude, (b) the wavelength, (c) the frequency, (d) the speed, (e) the direction, and (f) the maximum transverse speed of the wave. (HRW6e17.6; see also YF12e15.5 & 8 )

3. A continuous sinusoidal wave is traveling on a string with velocity 80 cm/s. The displacement of the particles of the string at x = 10 cm is found to vary with time according to the equation: y = (5.0 cm) sin(1.0 - 4.0 t). The linear density of the string is 4.0 g/cm. What is (a) the frequency of the wave, (b) the wavelength of the wave, (c) the general equation of the wave, and (d) the tension in the string? (HR19.10)

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4. Two connecting wires with linear mass densities that are related by :1 = 3:2 are under the same tension. When the wires oscillate at a frequency of 120 Hz, waves of wavelength 10 cm travel down the first wire with the linear density of :1 . (a) What is the wave speed in the first wire? (b) What is the wave speed in the second wire? (c) What is the wavelength in the second wire? (T13.58) 5. A wire 10.0 m long and having a mass of 100 g is stretched under a tension of 250 N. If two pulses, separated in time by 30.0 ms, are generated, one at each end of the wire, where will the pulses first meet? (HRW6e 17.21) 6. The wave function for a certain standing wave on a string fixed at both ends is given by: y(x,t) = 0.5 sin (0.025x) cos (500t) where y and x are in centimeters and t is in seconds. (a) Find the speed and amplitude of the two traveling waves that result in this standing wave. (b) What is the distance between successive nodes on the string? (c) What is the shortest possible length of the string? (T13.36) 7. In the figure at right, an aluminum wire, of length L1 = 60.0 cm, cross-sectional area 1.00 10-2 cm2 , and density 2.60 g/cm3 , is joined to a steel wire of density 7.80 g/cm3 and the same cross-sectional area. The compound wire, loaded with a block of mass m = 10.0 kg, is arranged so that the distance L2 from the joint to the supporting pulley is 86.6 cm. Transverse waves are set up on the wire by an external source of variable frequency; a node is located at the pulley. (a) Find the lowest frequency that generates a standing wave having the joint as one of the nodes. (b) How many nodes are observed at this frequency? (HRW7e 16.55; see also YF12e 15.15 & 40 & 68)

See Also Problem Summary: Ch. 15 - problems 5, 8, 15, 40, & 68 in the text (YF12e) - pp. 519...524(Brief solutions to the see also problems are posted on the class Blackboard site.)

Brief Answers to Questions:1. (a) 3 > 2 = 1 for wavelength (b) speed is the same for all: since all waves travel on the same string & tension F is constant (c) 6 frequency is larger when wavelength is smaller. So: 1 = 2 > 3 for frequency f and angular frequency T. 2. 3. 4. (a) yes (b) yes (c) no (d) yes (e) no

Speed of waves is larger on thinner wires. The fundamental frequency is the lower on the thicker wires. Speed increases as the pulse moves up the rope.

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SW Assignments: Sound Waves Questions:1. In the figure at right, two point sources S1 & S2 , which are in phase, emit identical sound waves of wavelength 2.0 m. In terms of wavelengths, what is the phase difference between the waves arriving at point P on the far right if (a) L1 = 38 m and L2 = 34 m, and (b) L1 = 39 m and L2 = 36 m? (c) Assuming that the source separation is much smaller than L1 & L2 , what type of interference occurs at P in situations (a) & (b)? (HRW7e Q 17.1)

2. The figure at right shows a stretched string of length L and pipes a, b, c, & d of lengths L, 2L, L/2, & L/2 respectively. The strings tension is adjusted until the speed of the waves on the string equals the speed of sound waves in air. The fundamental mode of oscillation is then set up on the string. In which pipe will the sound produced by the string cause resonance, and what oscillation mode will that sound set up? (HRW7e Q 17.7)

3. If you wait at a railroad crossing as a train approaches and passes, you hear a Doppler shift in the sound emitted by the trains whistle. But if you listen closely, you hear that the change in frequency is continuous; it does not suddenly go from one high frequency to another low frequency. Instead the frequency smoothly (but rather quickly) changes from high to low as the train passes. Why is the change smooth rather than sudden? (YF12eQ16.21 reworded)

Problems:1. You are at a large outdoor concert, seated 300 m from the speaker system. The concert is also being broadcast live via satellite (at the speed of light). Consider a listener 5000 km away who receives the broadcast. Who hears the music first, you or the listener, and by what time difference? (HRW 6e 18.2)

2. A stone is dropped into a well. The sound of the splash is heard 3.00 s later. What is the depth of the well?(HRW6e 18.7P)

3. Two loudspeakers are driven in phase by an audio amplifier at a frequency of 600 Hz. The speakers are on the y-axis, one at y = +1.00 m and the other at y = - 1.00 m. A listener begins at y = 0 and walks along a line parallel to the y-axis at a very large distance D away. (See diagram at right.) (a) At what angle 2 will the person first hear a minimum in the sound intensity? (b) At what angle will a maximum first be heard (after 2 = 0)? (c) How many maxima will be heard if the person keeps walking in the same direction? (T14-80; seealso Y F12 e 16.33; HINT: D is very large, so you may assume that the two lines from the two loudspeakers are parallel. In that case the path difference between them is (2.00 m)sin 2.)

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4. The water level in a vertical glass tube 1.0 m long can be adjusted to any position in the tube. A tuning fork vibrating at 686 Hz is held just over the open top end of the tube. At what positions of the water level will there be resonance? (HRW 6e 18.32; see also YF1 2e 16.27 & 28)

5. Two identical piano wires have a fundamental frequency of f1 = 600 Hz when kept under the same tension. What fractional increase in the tension of one wire will lead to 6 beats per second when both wires vibrate?(HRW6e 18.45)

6. Trooper B is chasing speeder A along a straight stretch of road. Both are moving at a speed of 100 mi/h. Trooper B, failing to catch up, sounds his siren again. Take the speed of sound in air to be 1100 ft/s and the frequency of the source to be 500 Hz. What is the Doppler shift in the frequency heard by speeder A? (HRW6e18.46E - modified)

7. Two students with vibrating 440-Hz tuning forks walk away from each other with equal speeds. How fast must they walk to hear a beat frequency of 2-Hz? (T14.67; see also YF12e 40)

8. A bat is flitting about in a cave, navigating via ultrasonic bleeps. Assume that the sound emission frequency of the bat is 39,000 Hz. During one fast swoop directly toward a flat wall surface, the bat is moving at 0.025 times the speed of sound in air. What frequency does the bat hear reflected off the wall? (HRW6e 18.54P; see also YF12e16.43)

9. A girl is sitting near the open window of a train that is moving at a velocity of 10.00 m/s to the east. The girls uncle stands near the tracks and watches the train move away. The locomotive whistle emits sound at frequency 500.0 Hz. The air is still. (a) What frequency does the uncle hear? (b) What frequency does the girl hear? A wind begins to blow from the east at 10.00 m/s. (c) What frequency does the uncle now hear? (d) What frequency does the girl now hear? (HRW6e 18.55; see also YF12e 50)

See Also Problem Summary: Ch. 16 (YF12e) - problems 27, 28, 33, 40, 43, 50(Brief solutions to the see also problems are posted on the class Blackboard site.)

Brief Answers to Questions:1. (a) zero wavelength phase difference (peak to peak) 6 constructive interference (b) half a wavelength phase difference (peak to trough) 6 destructive interference Pipe d - fundamental mode The Doppler frequency shift for a stationary listener & moving source depends on vs - where vs is the line of sight velocity of the source (along a straight line from the listener to the source of the s