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 $f=30\text{kHz,}$ what is the wavelength of the ultrasound in bone, if the speed of sound in bone is $v=3000\text{m/s?}$

The speed of light in air is approximately $v=3.00\times {10}^8\text{m/s}$ and the speed of light in glass is $v=2.00\times {10}^8\text{m/s}$. A red laser with a wavelength of $\lambda =633.00\text{nm}$ 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 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 motorboat is traveling across a lake at a speed of $v_b=15.00\text{m/s}.$ 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?

Given the wave functions $y_1\left(x,t\right)=A\text{sin}\left(kx-\omega t\right)$ and $y_2\left(x,t\right)=A\text{sin}\left(kx-\omega t+\varphi \right)$ 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 sinusoidal wave travels down a taut, horizontal string with a linear mass density of $\mu =0.060\text{kg/m}.$ The magnitude of maximum vertical acceleration of the wave is $a_{y\text{max}}=0.90{\text{m/s}}^2$ and the amplitude of the wave is 0.40 m. The string is under a tension of $F_T=600.00\text{N}$. The wave moves in the negative x-direction. Write an equation to model the wave.

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

A 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 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?

Consider two periodic wave functions, $y_1\left(x,t\right)=A\text{sin}\left(kx-\omega t\right)$ and $y_2\left(x,t\right)=A\text{sin}\left(kx-\omega t+\varphi \right).$ (a) For what values of $\varphi$ will the wave that results from a superposition of the wave functions have an amplitude of 2A? (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, $y_1\left(x,t\right)=0.3\text{m}\text{sin}\left(4{\text{m}}^{\mathrm{-1}}x-3{\text{s}}^{\mathrm{-1}}t\right)$ and $y_2\left(x,t\right)=0.3\text{m}\text{cos}\left(4{\text{m}}^{\mathrm{-1}}x+3{\text{s}}^{\mathrm{-1}}t-\frac{\pi }{2}\right).$ 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 $\text{sin}\left(u\pm v\right)=\text{sin}u\text{cos}v\pm \text{cos}u\text{sin}v$ and $\text{cos}\left(u\pm v\right)=\text{cos}u\text{cos}v\mp \text{sin}u\text{sin}v)$

Consider what is shown below. A 20.00-kg mass rests on a frictionless ramp inclined at $45\text{°}$. A string with a linear mass density of $\mu =0.025\text{kg/m}$ 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?

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 $k_s=100\text{N/m}.$ 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 string with a length of 4 m is held under a constant tension. The string has a linear mass density of $\mu =0.006\text{kg/m}.$ 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?