Problems

Waves on a swimming pool propagate at 0.75 m/s. You splash the water at one end of the pool and observe the wave go to the opposite end, reflect, and return in 30.00 s. How far away is the other end of the pool?

How many times a minute does a boat bob up and down on ocean waves that have a wavelength of 40.0 m and a propagation speed of 5.00 m/s?

What is the wavelength of the waves you create in a swimming pool if you splash your hand at a rate of 2.00 Hz and the waves propagate at a wave speed of 0.800 m/s?

Radio waves transmitted through empty space at the speed of light $\left(v=c=3.00\times {10}^8\text{m/s}\right)$ by the Voyager spacecraft have a wavelength of 0.120 m. What is their frequency?

(a) Seismographs measure the arrival times of earthquakes with a precision of 0.100 s. To get the distance to the epicenter of the quake, geologists compare the arrival times of S- and P-waves, which travel at different speeds. If S- and P-waves arrive at 4.00 and 7.20 km/s, respectively, in the region considered, how precisely can the distance to the source of the earthquake be determined? (b) Seismic waves from underground detonations of nuclear bombs can be used to locate the test site and detect violations of test bans. Discuss whether your answer to (a) implies a serious limit to such detection. (Note also that the uncertainty is greater if there is an uncertainty in the propagation speeds of the S- and P-waves.)

A quality assurance engineer at a frying pan company is asked to qualify a new line of nonstick-coated frying pans. The coating needs to be 1.00 mm thick. One method to test the thickness is for the engineer to pick a percentage of the pans manufactured, strip off the coating, and measure the thickness using a micrometer. This method is a destructive testing method. Instead, the engineer decides that every frying pan will be tested using a nondestructive method. An ultrasonic transducer is used that produces sound waves with a frequency of $f=25\text{kHz}.$ The sound waves are sent through the coating and are reflected by the interface between the coating and the metal pan, and the time is recorded. The wavelength of the ultrasonic waves in the coating is 0.076 m. What should be the time recorded if the coating is the correct thickness (1.00 mm)?

A transverse wave on a string is modeled with the wave function $y\left(x,t\right)=(0.20\text{cm})\text{sin}\left(2.00{\text{m}}^{\mathrm{-1}}x-3.00{\text{s}}^{\mathrm{-1}}t+\frac{\pi }{16}\right).$ What is the height of the string with respect to the equilibrium position at a position $x=4.00\text{m}$ and a time $t=10.00\text{s}?$

A pulse is defined as $y\left(x,t\right)=e^{\mathrm{-2.77}{\left(\frac{2.00\left(x-2.00\text{m/s}\left(t\right)\right)}{5.00\text{m}}\right)}^2}.$ Use a spreadsheet, or other computer program, to plot the pulse as the height of medium y as a function of position x. Plot the pulse at times $t=0.00\text{s}$ and $t=3.00\text{s}$ on the same graph. Where is the pulse centered at time $t=3.00\text{s}$? Use your spreadsheet to check your answer.

A wave is modeled with the function $y\left(x,t\right)=(0.25\text{m})\text{cos}\left(0.30{\text{m}}^{\mathrm{-1}}x-0.90{\text{s}}^{\mathrm{-1}}t+\frac{\pi }{3}\right).$ Find the (a) amplitude, (b) wave number, (c) angular frequency, (d) wave speed, (e) initial phase shift, (f) wavelength, and (g) period of the wave.

A wave is modeled by the wave function $y\left(x,t\right)=(0.30\text{m})\text{sin}\left[\frac{2\pi }{4.50\text{m}}\left(x-18.00\frac{\text{m}}{\text{s}}t\right)\right].$ What are the amplitude, wavelength, wave speed, period, and frequency of the wave?

A swimmer in the ocean observes one day that the ocean surface waves are periodic and resemble a sine wave. The swimmer estimates that the vertical distance between the crest and the trough of each wave is approximately 0.45 m, and the distance between each crest is approximately 1.8 m. The swimmer counts that 12 waves pass every two minutes. Determine the simple harmonic wave function that would describes these waves.

Consider two waves defined by the wave functions $y_1\left(x,t\right)=0.50\text{m}\text{sin}\left(\frac{2\pi }{3.00\text{m}}x+\frac{2\pi }{4.00\text{s}}t\right)$ and $y_2\left(x,t\right)=0.50\text{m}\text{sin}\left(\frac{2\pi }{6.00\text{m}}x-\frac{2\pi }{4.00\text{s}}t\right).$ What are the similarities and differences between the two waves?

The speed of a transverse wave on a string is 300.00 m/s, its wavelength is 0.50 m, and the amplitude is 20.00 cm. How much time is required for a particle on the string to move through a distance of 5.00 km?

A copper wire has a density of $\rho =8920{\text{kg/m}}^3,$ a radius of 1.20 mm, and a length L. The wire is held under a tension of 10.00 N. Transverse waves are sent down the wire. (a) What is the linear mass density of the wire? (b) What is the speed of the waves through the wire?

A string with a linear mass density of $\mu =0.0060\text{kg/m}$ is tied to the ceiling. A 20-kg mass is tied to the free end of the string. The string is plucked, sending a pulse down the string. Estimate the speed of the pulse as it moves down the string.

A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?

Two strings are attached to poles, however the first string is twice the linear mass density mu of the second. If both strings have the same tension, what is the ratio of the speed of the pulse of the wave from the first string to the second string?

Two strings are attached between two poles separated by a distance of 2.00 m as shown below, both under the same tension of 600.00 N. String 1 has a linear density of ${\mu }_1=0.0025\text{kg/m}$ and string 2 has a linear mass density of ${\mu }_2=0.0035\text{kg/m}.$ Transverse wave pulses are generated simultaneously at opposite ends of the strings. How much time passes before the pulses pass one another?

The note $E_4$ is played on a piano and has a frequency of $f=393.88.$ If the linear mass density of this string of the piano is $\mu =0.012\text{kg/m}$ and the string is under a tension of 1000.00 N, what is the speed of the wave on the string and the wavelength of the wave?

A sinusoidal wave travels down a taut, horizontal string with a linear mass density of $\mu =0.060\text{kg/m}$. The maximum vertical speed of the wave is $v_{y\text{max}}=0.30\text{cm/s}.$ The wave is modeled with the wave equation $y\left(x,t\right)=A\text{sin}\left(6.00{\text{m}}^{\mathrm{-1}}x-24.00{\text{s}}^{\mathrm{-1}}t\right).$ (a) What is the amplitude of the wave? (b) What is the tension in the string?

A string of length 5 m and a mass of 90 g is held under a tension of 100 N. A wave travels down the string that is modeled as $y\left(x,t\right)=0.01\text{m}\text{sin}\left(15.7{\text{m}}^{\mathrm{-1}}x-1170.12{\text{s}}^{\mathrm{-1}}t\right).$ What is the power over one wavelength?

The low-frequency speaker of a stereo set has a surface area of $A=0.05{\text{m}}^2$ and produces 1 W of acoustical power. (a) What is the intensity at the speaker? (b) If the speaker projects sound uniformly in all directions, at what distance from the speaker is the intensity $0.1{\text{W/m}}^2$?

A device called an insolation meter is used to measure the intensity of sunlight. It has an area of $100{\text{cm}}^2$ and registers 6.50 W. What is the intensity in ${\text{W/m}}^2$?

Suppose you have a device that extracts energy from ocean breakers in direct proportion to their intensity. If the device produces 10.0 kW of power on a day when the breakers are 1.20 m high, how much will it produce when they are 0.600 m high?

A microphone receiving a pure sound tone feeds an oscilloscope, producing a wave on its screen. If the sound intensity is originally $2.00\times {10}^{\mathrm{-5}}{\text{W/m}}^2$, but is turned up until the amplitude increases by $30.0\text{\%}$, what is the new intensity?

The power versus time for a point on a string $\left(\mu =0.05\text{kg/m}\right)$ in which a sinusoidal traveling wave is induced. The wave is modeled with the wave equation $y\left(x,t\right)=A\text{sin}\left(20.93{\text{m}}^{\mathrm{-1}}x-\omega t\right)$. What is the frequency and amplitude of the wave?

A 250-Hz tuning fork is struck and the intensity at the source is $I_1$ at a distance of one meter from the source. (a) What is the intensity at a distance of 4.00 m from the source? (b) How far from the tuning fork is the intensity a tenth of the intensity at the source?

The energy of a ripple on a pond is proportional to the amplitude squared. If the amplitude of the ripple is 0.1 cm at a distance from the source of 6.00 meters, what was the amplitude at a distance of 2.00 meters from the source?

Consider two sinusoidal sine waves traveling along a string, modeled as $y_1\left(x,t\right)=0.3\text{m}\text{sin}\left(4{\text{m}}^{\mathrm{-1}}x+3{\text{s}}^{\mathrm{-1}}t+\frac{\pi }{3}\right)$ and $y_2\left(x,t\right)=0.6\text{m}\text{sin}\left(8{\text{m}}^{\mathrm{-1}}x-6{\text{s}}^{\mathrm{-1}}t\right).$ What is the height of the resultant wave formed by the interference of the two waves at the position $x=1.0\text{m}$ at time $t=3.0\text{s?}$

Two sinusoidal waves are moving through a medium in the same direction, both having amplitudes of 3.00 cm, a wavelength of 5.20 m, and a period of 6.52 s, but one has a phase shift of an angle $\varphi$. What is the phase shift if the resultant wave has an amplitude of 5.00 cm? [Hint: Use the trig identity $\text{sin}u+\text{sin}v=2\text{sin}\left(\frac{u+v}{2}\right)\text{cos}\left(\frac{u-v}{2}\right)$

Two sinusoidal waves are moving through a medium in the positive x-direction, both having amplitudes of 7.00 cm, a wave number of $k=3.00{\text{m}}^{\mathrm{-1}},$ an angular frequency of $\omega =2.50{\text{s}}^{\mathrm{-1}},$ and a period of 6.00 s, but one has a phase shift of an angle $\varphi =\frac{\pi }{12}\text{rad}.$ What is the height of the resultant wave at a time $t=2.00\text{s}$ and a position $x=0.53\text{m?}$

Two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is $y_R\left(x,t\right)=0.70\text{m}\text{sin}(3.00{\text{m}}^{\mathrm{-1}}x-6.28{\text{s}}^{\mathrm{-1}}t+$
$\pi \text{/}16\text{rad}).$ What are the angular frequency, wave number, amplitude, and phase shift of the individual waves?

Consider two wave functions, $y_1\left(x,t\right)=4.00\text{m}\text{sin}\left(\pi {\text{m}}^{\mathrm{-1}}x-\pi {\text{s}}^{\mathrm{-1}}t\right)$ and $y_2\left(x,t\right)=4.00\text{m}\text{sin}\left(\pi {\text{m}}^{\mathrm{-1}}x-\pi {\text{s}}^{\mathrm{-1}}t+\frac{\pi }{3}\right).$ (a) Using a spreadsheet, plot the two wave functions and the wave that results from the superposition of the two wave functions as a function of position $\left(0.00\le x\le 6.00\text{m}\right)$ for the time $t=0.00\text{s}.$ (b) What are the wavelength and amplitude of the two original waves? (c) What are the wavelength and amplitude of the resulting wave?

Consider two wave functions that differ only by a phase shift, $y_1\left(x,t\right)=A\text{cos}\left(kx-\omega t\right)$ and $y_2\left(x,t\right)=A\text{cos}\left(kx-\omega t+\varphi \right).$ Use the trigonometric identities $\text{cos}u+\text{cos}v=2\text{cos}\left(\frac{u-v}{2}\right)\text{cos}\left(\frac{u+v}{2}\right)$ and $\text{cos}\left(\text{−}\theta \right)=\text{cos}\left(\theta \right)$ to find a wave equation for the wave resulting from the superposition of the two waves. Does the resulting wave function come as a surprise to you?

A 2-m long string is stretched between two supports with a tension that produces a wave speed equal to $v_w=50.00\text{m/s}.$ What are the wavelength and frequency of the first three modes that resonate on the string?

A cable with a linear density of $\mu =0.02\text{kg/m}$ is hung from telephone poles. The tension in the cable is 500.00 N. The distance between poles is 20 meters. The wind blows across the line, causing the cable resonate. A standing waves pattern is produced that has 4.5 wavelengths between the two poles. The speed of sound at the current temperature $T=20\text{°}\text{C}$ is $343.00\text{m/s}$. What are the frequency and wavelength of the hum?

Consider two wave functions $y\left(x,t\right)=0.30\text{cm}\text{sin}\left(3{\text{m}}^{\mathrm{-1}}x-4{\text{s}}^{\mathrm{-1}}t\right)$ and $y\left(x,t\right)=0.30\text{cm}\text{sin}\left(3{\text{m}}^{\mathrm{-1}}x+4{\text{s}}^{\mathrm{-1}}t\right)$. Write a wave function for the resulting standing wave.

A string with a linear mass density of 0.0062 kg/m and a length of 3.00 m is set into the $n=100$ mode of resonance. The tension in the string is 20.00 N. What is the wavelength and frequency of the wave?

Two sinusoidal waves with identical wavelengths and amplitudes travel in opposite directions along a string producing a standing wave. The linear mass density of the string is $\mu =0.075\text{kg/m}$ and the tension in the string is $F_T=5.00\text{N}.$ The time interval between instances of total destructive interference is $\text{Δ}t=0.13\text{s}.$ What is the wavelength of the waves?

A string is fixed at both end. The mass of the string is 0.0090 kg and the length is 3.00 m. The string is under a tension of 200.00 N. The string is driven by a variable frequency source to produce standing waves on the string. Find the wavelengths and frequency of the first four modes of standing waves.

A string is fixed at both ends to supports 3.50 m apart and has a linear mass density of $\mu =0.005\text{kg/m}.$ The string is under a tension of 90.00 N. A standing wave is produced on the string with six nodes and five antinodes. What are the wave speed, wavelength, frequency, and period of the standing wave?