PHYS 1120 Waves on a String

Physics 1120 In-Class Problems: Waves

A wire of length 4.35 m and mass 137 g is under a tension of 125 N. What is the speed of a wave in this wire? If the tension is doubled, what is the speed? If the mass is doubled?
A wave on a string has the formula y = 0.030sin(0.55x − 62.8t + π/3). What is the wavelength, frequency, period, phase constant, and speed of the wave. The string has a linear density μ = 0.020 kg/m. What is the tension in the string? What is the rate of energy flow of the wave? Sketch the reference circle, the snapshot graph at t = 0, and the history graph at x = 0.
On a point on a string, a peak of a harmonic wave is observed to pass every 0.050 s. The distance between peaks is 0.75 m. The height of the peak is 0.025 m. Assume that the wave is moving to the left. What is the equation of this wave (take t = 0 at the first peak)? What is the speed of the wave? The string has a linear density μ = 0.020 kg/m. What is the tension in the string? What is the rate of energy flow of the wave? Sketch the reference circle, the snapshot graph at t = 0, and the history graph at x = 0.
Sketch the associated snapshot and history graphs for the following reference circle diagrams. What is the phase constant φ₀ in each case?
Consider the snapshot graph below for a wave moving to the right at t = 0. What is the phase constant φ₀? Sketch the reference circle at the indicated points A, B, C, and D. Where exactly does point A occur? Point B?
Consider the history graph below for a wave moving to the right at x = 0. What is the phase constant φ₀? Sketch the reference circle at the indicated points A, B, C, and D. When exactly does point A occur? Point B?

Superposition in 1D and 2D

Two out-of-phase harmonic waves are traveling on a string. They each have an amplitude of 5.0 mm. The resultant wave has an amplitude of 3.5 mm. What is the phase difference between these two waves?
Two speakers are in a line but one speaker is a distance d behind the other. You are 4.50 m in front of the nearest speaker. The sound is wavelength λ = 1.50 m.
(a) The speakers are in phase. What is the smallest non-zero d if you hear constructive interference?
(b) The speakers are in phase. What is the smallest non-zero d if you hear destructive interference?
(c) The front speaker leads the back speaker by 90°. What is the smallest non-zero d if you hear constructive interference?
(d) The front speaker leads the back speaker by 90°. What is the smallest non-zero d if you hear destructive interference?
Two loudspeakers are located 3.00 m apart on the stage of an auditorium. A detector is placed 20.0 m from one speaker and 21.2 m from the other. A signal generator drives the two speakers in phase and sweeps through the range (2 KHz - 20 KHz). Assume that the speed of sound in air is 343 m/s.
(a) What is lowest note (frequency) for which destructive interference is a maximum?
(b) What is highest note for which destructive interference is a maximum?
(c) What is the lowest note for which constructive interference is a maximum?
(d) What is the highest note for which constructive interference is a maximum?
Two stereo speakers, driven by the amplifier so that the sources are in phase, are arranged so that the sound is loudest at your position. What is the minimum distance that you are farther away from one speaker than the other? The frequency of the tone from the speakers is 750 Hz and the speed of sound in air is 340 m/s.
As shown in the diagram below, a speaker plays a single frequency sound. On its way to you, some of the sound echoes (reflects) off a wall. As a result, the sound is quieter than it should be.
(a) If the wall is flexible, what is the lowest frequency that creates the lowest sound intensity at your position?
(b) If the wall is stiff, what is the lowest frequency that creates the lowest sound intensity at your position?

Doppler Shift

A driver travels north on a highway at a speed of 25 m/s. A police car, driving south at a speed of 40 m/s, approaches with its siren sounding at a base frequency of 2500 Hz. (a) What frequency is heard by the driver as the police car approaches? (b) What frequency is heard by the driver after the police car passes him? If the driver had been travelling south, what would your results have been for (a) and (b)? The speed of sound in air is v = 340 m/s.
In sonar, an intermittent high frequency sound pulse is broadcast in all directions. The sound is reflected from solid objects and returns to broadcaster. The time it took for the echo to return and the direction from which the echo came are used to locate nearby objects. This is Echo Location. By measuring the Doppler Shift of the echo, the speed of the object can be found. There is an added complication in that the Doppler Shift occurs twice, once from the source to the receiver, and then from the receiver (now a source of the echo) back to the original source (which is now a receiver of the echo). The echo will have a frequency

f´ = f₀ [(1 ± u_r/v)/(1 ± u_s/v][(1 ± u_s/v)/(1 ± u_r/v)] .

A submarine traveling at 17 km/h sends out pulses at 38.7 MHz. The delay in the echo off a second sub has been rapidly decreasing and is currently 75 ms. How far apart are the two subs? If the second sub is moving at 22 km/h, what is the frequency of the returned echo? The speed of sound in seawater is 1.54 km/s.
14. A siren with a frequency f₀ = 2000 Hz is attached to a block. The siren and block together have a mass of 2.00 kg. The block is attached to a spring of unknown spring constant K. The spring is compressed an unknown distance and then released. The siren and block oscillate back and forth. A listener hears the source as emitting a varying frequency since the siren is moving. The highest frequency that the listener hears is 2060 Hz. The listener also determines that he hears this highest frequency repeat every 1.50 seconds. The speed of sound in air is 343 m/s. (a) How fast is the block moving when the listener hears the highest frequency? (b) Where in its motion is the block when the listener hears the highest frequency? (c) What is the lowest frequency that the listener would hear? (d) Where in its motion is the block when the listener hears the lowest frequency? (e) What is the period and angular frequency of the block? (f) What is the amplitude of the displacement of the block? (g) What is the spring constant of the spring?

Beats

The beat frequency between an unknown tuning fork and a 500 Hz tuning fork is 12 Hz. Compared with a 504 Hz tuning fork, the beat frequency is 16 Hz. What is the frequency of the unknown tuning fork?
You have two 400-Hz tuning forks which you ring together. You drop one fork down a well. Before the fork hits bottom and stops ringing you hear a beat frequency of 15 Hz. How deep is the well?

Questions?mike.coombes@kwantlen.ca