The wavefunction of a transverse sinusoidal wave on a string has the form y(x,t)
ID: 3160811 • Letter: T
Question
The wavefunction of a transverse sinusoidal wave on a string has the form y(x,t) = A cos (kx + omega t + phi), where x and y are in m, t is in seconds and phi is the phase constant in radians. The wave has a period T = 25.4 ms and travels in the negative x-direction with a speed of 31.2 m/s. At t = 0, a particle on the string at x = 0 has a displacement of 2.20 cm and travels downward with a speed of 2.10 m/s. What is the amplitude of the wave? (For reference, see example 13.2 on p421 of the 5th edition of Serway and Jewett's "Principles of Physics") What is the initial phase constant, phi? What is the maximum traverse speed of the string?Explanation / Answer
According to the given problem,
T = 25.4*10-3 s
= 2/T = 2/25.4*10-3 = 247.37 rad/s
v = 31.2 m/s
k = /v = 247.37/31.2 = 7.93 rad/m
y(0,0) = Acos(7.93*0 + 247.37*0 + ) = 0.022
Acos = 0.022
y’(x,t) = -Asin(kx +t + )
y’(0,0) = -Asin = -247.37Asin = 2.1
Acos = 0.022, [1]
-247.37Asin = 2.1, [2]
Divide [2] by [1] :
-247.37 tan = 2.1/0.022
= tan-1{-0.3858} = -21.1°
A = 0.022/cos(-21.1) = 0.02358 m = 2.358 cm
Vmax = -A = -247.37*0.02358 = -5.833 m/s
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