# 16.2 Mathematics of waves  (Page 7/11)

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If you shake the end of a stretched spring up and down with a frequency f , you can produce a sinusoidal, transverse wave propagating down the spring. Does the wave number depend on the frequency you are shaking the spring?

The wavelength is equal to the velocity of the wave times the frequency and the wave number is equal to $k=\frac{2\pi }{\lambda },$ so yes, the wave number will depend on the frequency and also depend on the velocity of the wave propagating through the spring.

Does the vertical speed of a segment of a horizontal taut string through which a sinusoidal, transverse wave is propagating depend on the wave speed of the transverse wave?

In this section, we have considered waves that move at a constant wave speed. Does the medium accelerate?

The medium moves in simple harmonic motion as the wave propagates through the medium, continuously changing speed, therefore it accelerates. The acceleration of the medium is due to the restoring force of the medium, which acts in the opposite direction of the displacement.

If you drop a pebble in a pond you may notice that several concentric ripples are produced, not just a single ripple. Why do you think that is?

## Problems

A pulse can be described as a single wave disturbance that moves through a medium. Consider a pulse that is defined at time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ by the equation $y\left(x\right)=\frac{6.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{3}}{{x}^{2}+2.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2}}$ centered around $x=0.00\phantom{\rule{0.2em}{0ex}}\text{m}.$ The pulse moves with a velocity of $v=3.00\phantom{\rule{0.2em}{0ex}}\text{m/s}$ in the positive x -direction. (a) What is the amplitude of the pulse? (b) What is the equation of the pulse as a function of position and time? (c) Where is the pulse centered at time $t=5.00\phantom{\rule{0.2em}{0ex}}\text{s}$ ?

A transverse wave on a string is modeled with the wave function $y\left(x,t\right)=\left(0.20\phantom{\rule{0.2em}{0ex}}\text{cm}\right)\text{sin}\left(2.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x-3.00\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t+\frac{\pi }{16}\right).$ What is the height of the string with respect to the equilibrium position at a position $x=4.00\phantom{\rule{0.2em}{0ex}}\text{m}$ and a time $t=10.00\phantom{\rule{0.2em}{0ex}}\text{s}?$

$y\left(x,t\right)=-0.037\phantom{\rule{0.2em}{0ex}}\text{cm}$

Consider the wave function $y\left(x,t\right)=\left(3.00\phantom{\rule{0.2em}{0ex}}\text{cm}\right)\text{sin}\left(0.4\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x+2.00\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t+\frac{\pi }{10}\right).$ What are the period, wavelength, speed, and initial phase shift of the wave modeled by the wave function?

A pulse is defined as $y\left(x,t\right)={e}^{-2.77\phantom{\rule{0.2em}{0ex}}{\left(\frac{2.00\left(x-2.00\phantom{\rule{0.2em}{0ex}}\text{m/s}\left(t\right)\right)}{5.00\phantom{\rule{0.2em}{0ex}}\text{m}}\right)}^{2}}.$ 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 $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ and $t=3.00\phantom{\rule{0.2em}{0ex}}\text{s}$ on the same graph. Where is the pulse centered at time $t=3.00\phantom{\rule{0.2em}{0ex}}\text{s}$ ? Use your spreadsheet to check your answer.

The pulse will move $\text{Δ}x=6.00\phantom{\rule{0.2em}{0ex}}\text{m}$ .

A wave is modeled at time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ with a wave function that depends on position. The equation is $y\left(x\right)=\left(0.30\phantom{\rule{0.2em}{0ex}}\text{m}\right)\text{sin}\left(6.28\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x\right)$ . The wave travels a distance of 4.00 meters in 0.50 s in the positive x -direction. Write an equation for the wave as a function of position and time.

A wave is modeled with the function $y\left(x,t\right)=\left(0.25\phantom{\rule{0.2em}{0ex}}\text{m}\right)\text{cos}\left(0.30\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x-0.90\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t+\frac{\pi }{3}\right).$ Find the (a) amplitude, (b) wave number, (c) angular frequency, (d) wave speed, (e) phase shift, (f) wavelength, and (g) period of the wave.

a. $A=0.25\phantom{\rule{0.2em}{0ex}}\text{m};$ b. $k=0.30\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1};$ c. $\omega =0.90\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1};$ d. $v=3.0\phantom{\rule{0.2em}{0ex}}\text{m/s};$ e. $\varphi =\pi \text{/}3\phantom{\rule{0.2em}{0ex}}\text{rad};$ f. $\lambda =20.93\phantom{\rule{0.2em}{0ex}}\text{m}$ ; g. $T=6.98\phantom{\rule{0.2em}{0ex}}\text{s}$

A surface ocean wave has an amplitude of 0.60 m and the distance from trough to trough is 8.00 m. It moves at a constant wave speed of 1.50 m/s propagating in the positive x -direction. At $t=0,$ the water displacement at $x=0$ is zero, and ${v}_{y}$ is positive. (a) Assuming the wave can be modeled as a sine wave, write a wave function to model the wave. (b) Use a spreadsheet to plot the wave function at times $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ and $t=2.00\phantom{\rule{0.2em}{0ex}}\text{s}$ on the same graph. Verify that the wave moves 3.00 m in those 2.00 s.

A wave is modeled by the wave function $y\left(x,t\right)=\left(0.30\phantom{\rule{0.2em}{0ex}}\text{m}\right)\text{sin}\left[\frac{2\pi }{4.50\phantom{\rule{0.2em}{0ex}}\text{m}}\left(x-18.00\frac{\text{m}}{\text{s}}t\right)\right].$ What are the amplitude, wavelength, wave speed, period, and frequency of the wave?

$A=0.30\phantom{\rule{0.2em}{0ex}}\text{m},\phantom{\rule{0.2em}{0ex}}\lambda =4.50\phantom{\rule{0.2em}{0ex}}\text{m},\phantom{\rule{0.2em}{0ex}}v=18.00\phantom{\rule{0.2em}{0ex}}\text{m/s},\phantom{\rule{0.2em}{0ex}}f=4.00\phantom{\rule{0.2em}{0ex}}\text{Hz},\phantom{\rule{0.2em}{0ex}}T=0.25\phantom{\rule{0.2em}{0ex}}\text{s}$

A transverse wave on a string is described with the wave function $y\left(x,t\right)=\left(0.50\phantom{\rule{0.2em}{0ex}}\text{cm}\right)\text{sin}\left(1.57\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x-6.28\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right)$ . (a) What is the wave velocity of the wave? (b) What is the magnitude of the maximum velocity of the string perpendicular to the direction of the motion?

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.

$y\left(x,t\right)=0.23\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(3.49\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x-0.63\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right)$

Consider a wave described by the wave function $y\left(x,t\right)=0.3\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(2.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x-628.00\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right).$ (a) How many crests pass by an observer at a fixed location in 2.00 minutes? (b) How far has the wave traveled in that time?

Consider two waves defined by the wave functions ${y}_{1}\left(x,t\right)=0.50\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{2\pi }{3.00\phantom{\rule{0.2em}{0ex}}\text{m}}x+\frac{2\pi }{4.00\phantom{\rule{0.2em}{0ex}}\text{s}}t\right)$ and ${y}_{2}\left(x,t\right)=0.50\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{2\pi }{6.00\phantom{\rule{0.2em}{0ex}}\text{m}}x-\frac{2\pi }{4.00\phantom{\rule{0.2em}{0ex}}\text{s}}t\right).$ What are the similarities and differences between the two waves?

They have the same angular frequency, frequency, and period. They are traveling in opposite directions and ${y}_{2}\left(x,t\right)$ has twice the wavelength as ${y}_{1}\left(x,t\right)$ and is moving at half the wave speed.

Consider two waves defined by the wave functions ${y}_{1}\left(x,t\right)=0.20\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{2\pi }{6.00\phantom{\rule{0.2em}{0ex}}\text{m}}x-\frac{2\pi }{4.00\phantom{\rule{0.2em}{0ex}}\text{s}}t\right)$ and ${y}_{2}\left(x,t\right)=0.20\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\frac{2\pi }{6.00\phantom{\rule{0.2em}{0ex}}\text{m}}x-\frac{2\pi }{4.00\phantom{\rule{0.2em}{0ex}}\text{s}}t\right).$ 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?

Each particle of the medium moves a distance of 4 A each period. The period can be found by dividing the velocity by the wavelength: $t=10.42\phantom{\rule{0.2em}{0ex}}\text{s}$

Suppose the master cylinder in a hydraulic system is at a greater height than the cylinder it is controlling. Explain how this will affect the force produced at the cylinder that is being controlled.
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gtitboi
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Lindomar
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Lindomar
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omwoyo
2.54cm=1inche
omwoyo
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