### Additional Problems

Ultrasound equipment used in the medical profession uses sound waves of a frequency above the range of human hearing. If the frequency of the sound produced by the ultrasound machine is $f=30\phantom{\rule{0.2em}{0ex}}\text{kHz,}$ what is the wavelength of the ultrasound in bone, if the speed of sound in bone is $v=3000\phantom{\rule{0.2em}{0ex}}\text{m/s?}$

Shown below is the plot of a wave function that models a wave at time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ and $t=2.00\phantom{\rule{0.2em}{0ex}}\text{s}$. The dotted line is the wave function at time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ and the solid line is the function at time $t=2.00\phantom{\rule{0.2em}{0ex}}\text{s}$. Estimate the amplitude, wavelength, velocity, and period of the wave.

The speed of light in air is approximately $v=3.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{8}\phantom{\rule{0.2em}{0ex}}\text{m/s}$ and the speed of light in glass is $v=2.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{8}\phantom{\rule{0.2em}{0ex}}\text{m/s}$. A red laser with a wavelength of $\lambda =633.00\phantom{\rule{0.2em}{0ex}}\text{nm}$ shines light incident of the glass, and some of the red light is transmitted to the glass. The frequency of the light is the same for the air and the glass. (a) What is the frequency of the light? (b) What is the wavelength of the light in the glass?

A radio station broadcasts radio waves at a frequency of 101.7 MHz. The radio waves move through the air at approximately the speed of light in a vacuum. What is the wavelength of the radio waves?

A sunbather stands waist deep in the ocean and observes that six crests of periodic surface waves pass each minute. The crests are 16.00 meters apart. What is the wavelength, frequency, period, and speed of the waves?

A tuning fork vibrates producing sound at a frequency of 512 Hz. The speed of sound of sound in air is $v=343.00\phantom{\rule{0.2em}{0ex}}\text{m/s}$ if the air is at a temperature of $20.00\text{\xb0}\text{C}$. What is the wavelength of the sound?

A motorboat is traveling across a lake at a speed of ${v}_{b}=15.00\phantom{\rule{0.2em}{0ex}}\text{m/s}.$ The boat bounces up and down every 0.50 s as it travels in the same direction as a wave. It bounces up and down every 0.30 s as it travels in a direction opposite the direction of the waves. What is the speed and wavelength of the wave?

Use the linear wave equation to show that the wave speed of a wave modeled with the wave function $y\left(x,t\right)=0.20\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(3.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x+6.00\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right)$ is $v=2.00\phantom{\rule{0.2em}{0ex}}\text{m/s}.$ What are the wavelength and the speed of the wave?

Given the wave functions ${y}_{1}\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t\right)$ and ${y}_{2}\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t+\varphi \right)$ with $\varphi \ne \frac{\pi}{2}$, show that ${y}_{1}\left(x,t\right)+{y}_{2}\left(x,t\right)$ is a solution to the linear wave equation with a wave velocity of $v=\frac{\omega}{k}.$

A transverse wave on a string is modeled with the wave function $y\left(x,t\right)=0.10\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(0.15\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x+1.50\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t+0.20\right)$. (a) Find the wave velocity. (b) Find the position in the *y*-direction, the velocity perpendicular to the motion of the wave, and the acceleration perpendicular to the motion of the wave, of a small segment of the string centered at $x=0.40\phantom{\rule{0.2em}{0ex}}\text{m}$ at time $t=5.00\phantom{\rule{0.2em}{0ex}}\text{s}.$

A sinusoidal wave travels down a taut, horizontal string with a linear mass density of $\mu =0.060\phantom{\rule{0.2em}{0ex}}\text{kg/m}.$ The magnitude of maximum vertical acceleration of the wave is ${a}_{y\phantom{\rule{0.2em}{0ex}}\text{max}}=0.90\phantom{\rule{0.2em}{0ex}}{\text{cm/s}}^{2}$ and the amplitude of the wave is 0.40 m. The string is under a tension of ${F}_{T}=600.00\phantom{\rule{0.2em}{0ex}}\text{N}$. The wave moves in the negative *x*-direction. Write an equation to model the wave.

A transverse wave on a string $\left(\mu =0.0030\phantom{\rule{0.2em}{0ex}}\text{kg/m}\right)$ is described with the equation $y\left(x,t\right)=0.30\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{2\pi}{4.00\phantom{\rule{0.2em}{0ex}}\text{m}}\left(x-16.00\frac{\text{m}}{\text{s}}t\right)\right).$ What is the tension under which the string is held taut?

A transverse wave on a horizontal string $\left(\mu =0.0060\phantom{\rule{0.2em}{0ex}}\text{kg/m}\right)$ is described with the equation $y\left(x,t\right)=0.30\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{2\pi}{4.00\phantom{\rule{0.2em}{0ex}}\text{m}}\left(x-{v}_{w}t\right)\right).$ The string is under a tension of 300.00 N. What are the wave speed, wave number, and angular frequency of the wave?

A student holds an inexpensive sonic range finder and uses the range finder to find the distance to the wall. The sonic range finder emits a sound wave. The sound wave reflects off the wall and returns to the range finder. The round trip takes 0.012 s. The range finder was calibrated for use at room temperature $T=20\text{\xb0}\text{C}$, but the temperature in the room is actually $T=23\text{\xb0}\text{C}.$ Assuming that the timing mechanism is perfect, what percentage of error can the student expect due to the calibration?

A wave on a string is driven by a string vibrator, which oscillates at a frequency of 100.00 Hz and an amplitude of 1.00 cm. The string vibrator operates at a voltage of 12.00 V and a current of 0.20 A. The power consumed by the string vibrator is $P=IV$. Assume that the string vibrator is $90\text{\%}$ efficient at converting electrical energy into the energy associated with the vibrations of the string. The string is 3.00 m long, and is under a tension of 60.00 N. What is the linear mass density of the string?

A traveling wave on a string is modeled by the wave equation $y\left(x,t\right)=3.00\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(8.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x+100.00\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right).$ The string is under a tension of 50.00 N and has a linear mass density of $\mu =0.008\phantom{\rule{0.2em}{0ex}}\text{kg/m}.$ What is the average power transferred by the wave on the string?

A transverse wave on a string has a wavelength of 5.0 m, a period of 0.02 s, and an amplitude of 1.5 cm. The average power transferred by the wave is 5.00 W. What is the tension in the string?

(a) What is the intensity of a laser beam used to burn away cancerous tissue that, when $90.0\text{\%}$ absorbed, puts 500.0 J of energy into a circular spot 2.00 mm in diameter in 4.00 s? (b) Discuss how this intensity compares to the average intensity of sunlight (about $1\text{kW}/{\text{m}}^{2}$) and the implications that would have if the laser beam entered your eye. Note how your answer depends on the time duration of the exposure.

Consider two periodic wave functions, ${y}_{1}\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t\right)$ and ${y}_{2}\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t+\varphi \right).$ (a) For what values of $\varphi $ will the wave that results from a superposition of the wave functions have an amplitude of 2*A*? (b) For what values of $\varphi $ will the wave that results from a superposition of the wave functions have an amplitude of zero?

Consider two periodic wave functions, ${y}_{1}\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t\right)$ and ${y}_{2}\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{cos}\left(kx-\omega t+\varphi \right)$. (a) For what values of $\varphi $ will the wave that results from a superposition of the wave functions have an amplitude of 2*A*? (b) For what values of $\varphi $ will the wave that results from a superposition of the wave functions have an amplitude of zero?

A trough with dimensions 10.00 meters by 0.10 meters by 0.10 meters is partially filled with water. Small-amplitude surface water waves are produced from both ends of the trough by paddles oscillating in simple harmonic motion. The height of the water waves are modeled with two sinusoidal wave equations, ${y}_{1}\left(x,t\right)=0.3\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(4\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-3\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right)$ and ${y}_{2}\left(x,t\right)=0.3\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(4\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x+3\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t-\frac{\pi}{2}\right).$ What is the wave function of the resulting wave after the waves reach one another and before they reach the end of the trough (i.e., assume that there are only two waves in the trough and ignore reflections)? Use a spreadsheet to check your results. (*Hint:* Use the trig identities $\text{sin}\left(u\pm v\right)=\text{sin}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}v\pm \text{cos}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}v$ and $\text{cos}\left(u\pm v\right)=\text{cos}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}v\mp \text{sin}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}v)$

A seismograph records the S- and P-waves from an earthquake 20.00 s apart. If they traveled the same path at constant wave speeds of ${v}_{S}=4.00\phantom{\rule{0.2em}{0ex}}\text{km/s}$ and ${v}_{P}=7.50\phantom{\rule{0.2em}{0ex}}\text{km/s},$ how far away is the epicenter of the earthquake?

Consider what is shown below. A 20.00-kg mass rests on a frictionless ramp inclined at $45\text{\xb0}$. A string with a linear mass density of $\mu =0.025\phantom{\rule{0.2em}{0ex}}\text{kg/m}$ is attached to the 20.00-kg mass. The string passes over a frictionless pulley of negligible mass and is attached to a hanging mass (*m*). The system is in static equilibrium. A wave is induced on the string and travels up the ramp. (a) What is the mass of the hanging mass (*m*)? (b) At what wave speed does the wave travel up the string?

Consider the superposition of three wave functions $y\left(x,t\right)=3.00\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(2\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-3\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right),$ $y\left(x,t\right)=3.00\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(6\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x+3\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right),$ and $y\left(x,t\right)=3.00\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(2\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-4\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right).$ What is the height of the resulting wave at position $x=3.00\phantom{\rule{0.2em}{0ex}}\text{m}$ at time $t=10.0\phantom{\rule{0.2em}{0ex}}\text{s?}$

A string has a mass of 150 g and a length of 3.4 m. One end of the string is fixed to a lab stand and the other is attached to a spring with a spring constant of ${k}_{s}=100\phantom{\rule{0.2em}{0ex}}\text{N/m}.$ The free end of the spring is attached to another lab pole. The tension in the string is maintained by the spring. The lab poles are separated by a distance that stretches the spring 2.00 cm. The string is plucked and a pulse travels along the string. What is the propagation speed of the pulse?

A standing wave is produced on a string under a tension of 70.0 N by two sinusoidal transverse waves that are identical, but moving in opposite directions. The string is fixed at $x=0.00\phantom{\rule{0.2em}{0ex}}\text{m}$ and $x=10.00\phantom{\rule{0.2em}{0ex}}\text{m}.$ Nodes appear at $x=0.00\phantom{\rule{0.2em}{0ex}}\text{m,}$ 2.00 m, 4.00 m, 6.00 m, 8.00 m, and 10.00 m. The amplitude of the standing wave is 3.00 cm. It takes 0.10 s for the antinodes to make one complete oscillation. (a) What are the wave functions of the two sine waves that produce the standing wave? (b) What are the maximum velocity and acceleration of the string, perpendicular to the direction of motion of the transverse waves, at the antinodes?

A string with a length of 4 m is held under a constant tension. The string has a linear mass density of $\mu =0.006\phantom{\rule{0.2em}{0ex}}\text{kg/m}.$ Two resonant frequencies of the string are 400 Hz and 480 Hz. There are no resonant frequencies between the two frequencies. (a) What are the wavelengths of the two resonant modes? (b) What is the tension in the string?