Waves

3-1

A travelling wave on a string is described by

$$y= 2 \sin \left[ 2\pi \left( {t\over 0.40} + {x\over 80}\right)\right],$$

where $x$ and $y$ are in centimetres and $t$ is in seconds (a) for $t=0$, plot $y$ as a function of $x$ for $0\le x\le 160  \rm cm$. (b) Repeat (a) for $t=0.05 \rm s$ and $t=0.01 \rm s$. (c) From your graphs, what is the wave speed, and in which direction ($+x$ or $-x$) is the wave travelling?

3-2

What is the fastest transverse wave that can be sent along a steel wire? Allowing for a reasonable safety factor, the maximum tensile stress to which stress to which steel wires should be subject is $7.0\times 10^8  \rm N/m^2$. The density of steel is $7800  \rm kg/m^3$. Show that your answer does not depend on the diameter of the wire.

3-3

A series of pulses, each of amplitude 0.150 m, are sent down a string that is attached to a post at one end. The pulses are reflected at the post and travel back along the string without loss of amplitude. What is the displacement at a point on the string where two pulses are crossing (a) if the string is rigidly attached to the post? (b) If the end at which reflection occurs is free to slide up and down?

3-4

A transverse sinusoidal wave on a string has a period $T=25.0 \rm ms$ and travels in the negative $x$ direction with a speed of $30.0 \rm m/s$. At $t=0$, a particle on a string at $x=0$ has a displacement of $2.00  \rm cm$ and travels downward with a speed of $2.00 \rm m/s$. (a) What is the amplitude of the wave? (b) What is the initial phase angle? (c) What is the maximum transverse speed of the string? (d) Write the wave function for the wave.

3-5

An acoustic microscope can form a remarkably detailed image of microelectronic circuits such as transistors. The sound waves have a frequency of $4.2 \rm GHz$. The speed of such waves in the liquid helium in which the specimen is immersed is $240 \rm m/s$. (a) What is the wavelength of these ultra-high-frequency acoustic waves? (b) You wish to take images of circuits of conductors that are approximately $2 \rm\mu m$ wide. To how many wavelengths does this correspond?

3-6

A sinusoidal sound wave is described by the displacement

$$s (x,t) = (2.00  {\rm\mu m})\cos [ ( 15.7  {\rm m^{-1}})x - (858 {\rm s^{-1}})t]$$

(a) find the amplitude, wavelength and speed of this wave. (b) Determine the instantaneous displacement of the molecules at the position $x=0.00500 \rm m$ at $t=3.00  \rm ms$. (c) Determine the maximum speed of the molecule's oscillator motion.

3-7

Two identical travelling waves, moving in the same direction, are out of phase by $90 \deg$. What is the amplitude of the combined wave in terms of the common amplitude $y_m$ of the two combining waves?

3-8

Two loudspeakers, $S_1$ and $S_2$, are 7.0 m apart. Each emits a sound frequency of 200 Hz, uniformly in all directions. $S_1$ has an acoustic output of $1.2\times 10^{-3} \rm W$ and $S_2$ one of $1.8\times 10^{-3} \rm W$. $S_1$ and $S_2$ vibrate in phase. Consider a point $P$ that is 4.0 m from $S_1$ and 3.0 m from $S_2$. (a) How are the phases of the two waves arriving at $P$ related? (b) What is the intensity of sound at $P$ with both $S_1$ and $S_2$ on? (c) What is the intensity of sound at $P$ if $S_1$ is turned off ($S_2$ on)? (d) What is the intensity of sound at $P$ if $S_2$ is turned off ($S_1$ on)?

3-9

A 15-cm violin string, fixed at both ends, is vibrating in its $n=1$ mode. The speed of waves in this wire is 250 m/s, and the speed of sound in air is 348 m/s. What are (a) the frequency and (b) the wavelength of the emitted sound wave?

3-10

A 75 cm string is stretched between fixed supports. It is observed to have resonant frequencies of 420 and 315 Hz, and no other resonant frequencies between the two. (a) What is the lowest resonant frequency for this string? (b) What is the wave speed for this string?

3-11

A tuning fork of unknown frequency makes 3 beats per second with a standard fork of frequency 384 Hz. The beat frequency decreases when a small piece of wax is put on a prong of the first fork. What is the frequency of this fork?

3-12

A string 2.7 m long has a mass of 260 g. The tension in the string is 36 N. What must be the frequency of travelling waves of amplitude 7.7 mm in order that the average transmitted power be 85 W?

3-13

A vacuum cleaner has a measured sound level of 70 dB. (a) What is the intensity of this sound in watts per square meter? (b) What is the pressure amplitude of this sound?

3-14

A whistle used to call a dog has a frequency of 30 kHz. The dog, however, ignores it. The owner of the dog, who cannot hear sounds above 20 kHz, wants to use the Doppler effect to make certain that the whistle is working. She asks a friend to blow the whistle from a moving car, while the owner remains stationary and listens. (a) How fast must the car move and in what direction? Is the experiment practical? (b) Repeat for a whistle frequency of 22 kHz instead of 30 kHz.

3-15

The ``red shift'' of radiation from a distant galaxy consists of the light $\rm H_\gamma$, known to have a wavelength of 434 nm when observed in the laboratory, appearing to have a wavelength of 462 nm. (a) What is the speed of the galaxy in the line of sight relative to the earth? (b) Is it approaching or receding?