1. Prove that y(x,t) = Ae-i(kx-?t) is a solution to the wave equation. In other
ID: 2101145 • Letter: 1
Question
1. Prove that y(x,t) = Ae-i(kx-?t) is a solution to the wave equation. In other words, you should end with v = ?/k.
2. For the function y(x,t) = 7cm sin[?(x/4cm + t/6s) find the following:
a. What are the frequency, wavelength, and amplitude?
b. What is the velocity of the wave?
c. What is the direction of propagation?
3. Explicitly write out the standard form for a wave that is traveling in the +x direction and has a frequency of 200 Hz, a wave velocity of 400m/s, and an amplitude of 0.1m.
4. With what tension must a rope with length 2.50 m and mass 0.12 kg be stretched for transverse waves of frequency 40.0 Hz to have a wavelength of 0.75 m?
Explanation / Answer
y(x,t) = Ae-i(kx-?t)
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?y/?t = Ai?e-i(kx-?t)
?2y/?t2 = A(i?)(i?)e-i(kx-?t) = -A?2 e-i(kx-?t)
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?y/?x = -Aike-i(kx-?t)
?2y/?x2 = A(-ik)(-ik)e-i(kx-?t) = -Ak2 e-i(kx-?t)
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wave equation:
?2y/?t2 = v2 ?2y/?x2
>>>> -A?2 e-i(kx-?t) = v2 (-Ak2 e-i(kx-?t))
>>>> ?2 = v2 k2
>>>> ? = v k
>>>> v = ?/k
>>>> proved!
2.A.The frequency of the wave is
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