Q: Given the wave equation $$y = 0.1 \sin(2x - 8t)$$, what is the wavelength $$\lambda of $$the wave?

$$\lambda = \dfrac{2\pi}{2} = \pi\ \text{m}$$

Q: If the speed of a wave is $$v = 340\ \text{m/s}$$ and its wavelength is $$\lambda = 0.5\ \text{m}$$, what is its frequency?

$$f = \dfrac{v}{\lambda} = 680\ \text{Hz}$$

Q: A wave is described by $$y = 0.2 \sin(3x - 6t). $$What is the period $$T of $$the wave?

$$T = \dfrac{2\pi}{6} = \dfrac{\pi}{3}\ \text{s}$$

Q: If a wave has a frequency of $$50\ \text{Hz}$$ and a wavelength of $$0.2\ \text{m}$$, what is its speed?

$$v = f\lambda = 10\ \text{m/s}$$

Q: A string vibrates with a wave described by $$y = 0.1 \sin(5x - 10t). $$What is the maximum speed of a particle on the string?

$$v_{\text{max}} = A\omega = 0.1 \times 10 = 1.0\ \text{m/s}$$

Q: Given a wave equation $$y = 0.3 \sin(2x - 4t)$$, what is the displacement of the medium at $$x = 1\ \text{m}$$ and $$t = 2\ \text{s}$$?

$$y = 0.3 \sin(2 \times 1 - 4 \times 2) = 0.3 \sin(2 - 8) = 0.3 \sin(-6)$$

Q: A sound wave in air has a speed of $$343\ \text{m/s}$$ and a frequency of $$686\ \text{Hz}. $$What is its wavelength?

$$\lambda = \dfrac{v}{f} = \dfrac{343}{686} = 0.5\ \text{m}$$

Q: If the displacement of a wave is given by $$y = 0.2 \sin(2x - 6t)$$, what is the phase difference between two points separated by $$0.5\ \text{m}$$?

$$\Delta \phi = k \Delta x = 2 \times 0.5 = 1\ \text{rad}$$

Question 1

Which of the following equations best describes the displacement $$y of a $$point on a string at position $$x$$ and time $$t$$ for a sinusoidal wave traveling in the positive $$x-$$direction?

Accepted Answer

$$y = A \sin(kx - \omega t)$$

Question 2

A wave on a string is described by $$y = 0.05 \sin(4x - 20t)$$, where $$y$$ and $$x$$ are in meters and $$t is in $$seconds. What is the amplitude of this wave?

Accepted Answer

$$0.05\ 	ext{m}$$

Question 3

Given the wave equation $$y = 0.1 \sin(2x - 8t)$$, what is the wavelength $$\lambda of $$the wave?

Accepted Answer

$$\lambda = \dfrac{2\pi}{2} = \pi\ 	ext{m}$$

Question 4

If the speed of a wave is $$v = 340\ 	ext{m/s}$$ and its wavelength is $$\lambda = 0.5\ 	ext{m}$$, what is its frequency?

Accepted Answer

$$f = \dfrac{v}{\lambda} = 680\ 	ext{Hz}$$

Question 5

A snapshot graph of a wave shows the displacement $$y$$ versus position $$x at$$ $$t = 0. $$What information can you determine from this graph?

Accepted Answer

Amplitude and wavelength

Question 6

The equation $$v = f\lambda$$ relates which of the following quantities?

Accepted Answer

Wave speed, frequency, and wavelength

Question 7

A wave is described by $$y = 0.2 \sin(3x - 6t). $$What is the period $$T of $$the wave?

Accepted Answer

$$T = \dfrac{2\pi}{6} = \dfrac{\pi}{3}\ 	ext{s}$$

Question 8

Which of the following best distinguishes between wave speed and the speed of particles in the medium?

Accepted Answer

Wave speed is the rate at which the disturbance travels, while particle speed is the rate at which individual particles oscillate.

Question 9

If a wave has a frequency of $$50\ 	ext{Hz}$$ and a wavelength of $$0.2\ 	ext{m}$$, what is its speed?

Accepted Answer

$$v = f\lambda = 10\ 	ext{m/s}$$

Question 10

A string vibrates with a wave described by $$y = 0.1 \sin(5x - 10t). $$What is the maximum speed of a particle on the string?

Accepted Answer

$$v_{	ext{max}} = A\omega = 0.1 	imes 10 = 1.0\ 	ext{m/s}$$

Question 11

Given a wave equation $$y = 0.3 \sin(2x - 4t)$$, what is the displacement of the medium at $$x = 1\ 	ext{m}$$ and $$t = 2\ 	ext{s}$$?

Accepted Answer

$$y = 0.3 \sin(2 	imes 1 - 4 	imes 2) = 0.3 \sin(2 - 8) = 0.3 \sin(-6)$$

Question 12

A sound wave in air has a speed of $$343\ 	ext{m/s}$$ and a frequency of $$686\ 	ext{Hz}. $$What is its wavelength?

Accepted Answer

$$\lambda = \dfrac{v}{f} = \dfrac{343}{686} = 0.5\ 	ext{m}$$

Question 13

If the displacement of a wave is given by $$y = 0.2 \sin(2x - 6t)$$, what is the phase difference between two points separated by $$0.5\ 	ext{m}$$?

Accepted Answer

$$\Delta \phi = k \Delta x = 2 	imes 0.5 = 1\ 	ext{rad}$$

Question 14

A wave is traveling along a string with the equation $$y = 0.15 \sin(4x - 12t). $$What is the frequency of the wave?

Accepted Answer

$$f = \dfrac{\omega}{2\pi} = \dfrac{12}{2\pi} = 1.91\ 	ext{Hz}$$

Question 15

Which of the following statements is true regarding the amplitude of a wave?

Accepted Answer

Amplitude is the maximum displacement from equilibrium.

Question 16

A wave on a string is described by $$y = 0.2 \sin(2x - 8t). $$What is the speed of the wave?

Accepted Answer

$$v = \dfrac{\omega}{k} = \dfrac{8}{2} = 4\ 	ext{m/s}$$

Question 17

If a wave has a period of $$0.01\ 	ext{s}$$, what is its frequency?

Accepted Answer

$$f = \dfrac{1}{T} = 100\ 	ext{Hz}$$

Question 18

A wave is described by $$y = 0.1 \sin(2x - 6t). $$What is the displacement at $$x = 0.5\ 	ext{m}$$ and $$t = 1\ 	ext{s}$$?

Accepted Answer

$$y = 0.1 \sin(2 	imes 0.5 - 6 	imes 1) = 0.1 \sin(1 - 6) = 0.1 \sin(-5)$$

Chapter 15: Traveling Waves and Sound – Study Guide

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