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A sinusoidal wave traveling in the negative x direction (to the left) has an amplitude of 20.0 cm, a wavelength of 35.0 cm, and a frequency of 12.0 Hz. The transverse position of an element of the medium at t = 0, x = 0 is y = −3.00 cm, and the element has a positive velocity here. We wish to find an expression for the wave function describing this wave. (a) Sketch the wave at t = 0. (b) Find the angular wave number k from the wavelength. (c) Find the period T from the frequency. Find (d) the angular frequency ω and (e) the wave speed v. (f) From the information about t = 0, find the phase constant ϕ. (g) Write an expression for the wave function y(x, t).
(a)
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The sinusoidal wave at time
Answer to Problem 8P
Therefore, the sinusoidal wave at time
Figure I
Explanation of Solution
The wave is traveling in the negative
The general expression for the wave function of a sinusoidal wave is,
Here,
Substitute
At
Substitute
At
The graph for the above equation (III) is shown below.
Figure I
Conclusion:
Therefore, the sinusoidal wave at time
(b)
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The angular wave number.
Answer to Problem 8P
The angular wave number is
Explanation of Solution
Formula to calculate the angular wave number is,
Substitute
Conclusion:
Therefore, the angular wave number is
(c)
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The period of the wave from frequency.
Answer to Problem 8P
The period of the wave from frequency is
Explanation of Solution
Formula to calculate the period is,
Substitute
Conclusion:
Therefore, the period of the wave from frequency is
(d)
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The angular frequency of the wave.
Answer to Problem 8P
The angular frequency of the wave is
Explanation of Solution
Formula to calculate the angular frequency is,
Substitute
Conclusion:
Therefore, the angular frequency of the wave is
(e)
![Check Mark](/static/check-mark.png)
The speed of the wave.
Answer to Problem 8P
The speed of the wave is
Explanation of Solution
Formula to calculate the speed of the wave is,
Substitute
Conclusion:
Therefore, the speed of the wave is
(f)
![Check Mark](/static/check-mark.png)
The phase constant
Answer to Problem 8P
The phase constant
Explanation of Solution
As calculated in part (a), the phase constant is,
Conclusion:
Therefore, the phase constant
(g)
![Check Mark](/static/check-mark.png)
The expression for the wave function
Answer to Problem 8P
The expression for the wave function is
Explanation of Solution
The expression for the wave function from equation (II) is,
Or it may be written as,
Here,
Conclusion:
Therefore, the expression for the wave function
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Chapter 16 Solutions
Physics for Scientists and Engineers with Modern Physics
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