A transverse traveling sinusoidal wave on a long stretched wire of mass per unit length ρ has frequency ω and wave speed c. The maximum amplitude is y 0 , where y 0 << λ. The wave travels toward increasing x.
Text Solution
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y = y 0 sin 
.
Every point the wave travels through undergoes simple harmonic motion. Consider an element of the wire from x to x + 
x. The mechanical energy of the element is the sum of its potential and kinetic energies and is a constant equal to the maximum of its kinetic energy. As
= ωy 0 cos 
,
the maximum vibrational velocity of the element is ωy 0 and its total mechanical energy is
m. ω 2 
= 
ρω 2 
x.
Hence the energy per unit length of the string is
E = 
ρω 2 
.
As the wave travels at a speed c, the energy that passes through a point on the string in time t is Ect. Hence the power transmitted is
ρcω 2 
.
The tension T in the string is given by c = 
.
The transverse force the mechanical device exerts on the wire at x = 0 is.
F y (t) = –T 
= ρcωy 0 cos(ωt).
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