16.6 Standing Waves and Resonance

A truck manufacturer finds that a strut in the engine is failing prematurely. A sound engineer determines that the strut resonates at the frequency of the engine and suspects that this could be the problem. What are two possible characteristics of the strut can be modified to correct the problem?

Wine glasses can be set into resonance by moistening your finger and rubbing it around the rim of the glass. Why?

Consider a standing wave modeled as $y \left(x , t\right) = 4.00 \text{cm} \text{sin} \left(3 \text{m}^{−1} x\right) \text{cos} \left(4 \text{s}^{−1} t\right) .$ Is there a node or an antinode at $x = 0.00 \text{m} ?$ What about a standing wave modeled as $y \left(x , t\right) = 4.00 \text{cm} \text{sin} \left(3 \text{m}^{−1} x + \frac{\pi}{2}\right) \text{cos} \left(4 \text{s}^{−1} t\right) ?$ Is there a node or an antinode at the $x = 0.00 \text{m}$ position?

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

A cable with a linear density of $\mu = 0.02 \text{kg}/\text{m}$ 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 $T = 20 ° \text{C}$ is $343.00 \text{m}/\text{s}$. What are the frequency and wavelength of the hum?

Consider two wave functions $y \left(x , t\right) = 0.30 \text{cm} \text{sin} \left(3 \text{m}^{−1} x - 4 \text{s}^{−1} t\right)$ and $y \left(x , t\right) = 0.30 \text{cm} \text{sin} \left(3 \text{m}^{−1} x + 4 \text{s}^{−1} t\right)$. 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 $\mu = 0.075 \text{kg}/\text{m}$ and the tension in the string is $F_{T} = 5.00 \text{N} .$ The time interval between instances of total destructive interference is $\Delta t = 0.13 \text{s} .$ 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 $\mu = 0.005 \text{kg}/\text{m} .$ 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?

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 $v = 3000 \text{m}/\text{s}?$

The speed of light in air is approximately $v = 3.00 \times 10^{8} \text{m}/\text{s}$ and the speed of light in glass is $v = 2.00 \times 10^{8} \text{m}/\text{s}$. 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 $v_{b} = 15.00 \text{m}/\text{s} .$ 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 $y_{1} \left(x , t\right) = A \text{sin} \left(k x - \omega t\right)$ and $y_{2} \left(x , t\right) = A \text{sin} \left(k x - \omega t + \phi\right)$ with $\phi \neq \frac{\pi}{2}$, 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 $\mu = 0.060 \text{kg}/\text{m} .$ The magnitude of maximum vertical acceleration of the wave is $a_{y \text{max}} = 0.90 \text{m}/\text{s}^{2}$ 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 $\left(\mu = 0.0060 \text{kg}/\text{m}\right)$ 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 $P = I V$. Assume that the string vibrator is $90 \%$ efficient at converting electrical energy into the energy associated with the vibrations of the string. The stri…

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, $y_{1} \left(x , t\right) = A \text{sin} \left(k x - \omega t\right)$ and $y_{2} \left(x , t\right) = A \text{sin} \left(k x - \omega t + \phi\right) .$ (a) For what values of $\phi$ will the wave that results from a superposition of the wave functions have an amplitude of 2A? (b) For what values of $\phi$ 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, $y_{1} \left(x , t\right) = 0.3 \text{m} \text{sin} \left(4 \text{m}^{−1} x - 3 \text{s}^{−1} t\right)$ and $y_{2} \left(x , t\right) = 0.3 \text{m} \text{cos} \left(4 \text{m}^{−1} x + 3 \text{s}^{−1} t - \frac{\pi}{2}\right) .$ What is the wave function of the resulting wave after the waves reach one another and…

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 $k_{s} = 100 \text{N}/\text{m} .$ 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 al…

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

Consider two wave functions $y_{1} \left(x , t\right) = A \text{sin} \left(k x - \omega t\right)$ and $y_{2} \left(x , t\right) = A \text{sin} \left(k x + \omega t + \phi\right)$. What is the wave function resulting from the interference of the two wave? (Hint: $\text{sin} \left(\alpha \pm \beta\right) = \text{sin} \alpha \text{cos} \beta \pm \text{cos} \alpha \text{sin} \beta$ and $\phi = \frac{\phi}{2} + \frac{\phi}{2}$.)