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136. Problem 19.44 (RHK) We consider a standing wave that is the sum of two waves
travelling in opposite directions but otherwise identical. We have to show that
the maximum kinetic energy in each loop of the standing wave is
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Solution: Click For PDF Version Let us represent the two identical sinusoidal waves travelling in opposite directions by the functions
Standing wave formed by the superposition of these waves is described by the function
Speed of vibration at position x of the string is
given by the time derivative of
At a given x, the maximum speed of oscillation is
We have expressed
End points of the first loop are
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. That is
in terms
of f,
.
and
.
Therefore, the maximum kinetic energy in each loop can be calculated by
integrating maximum KE of element of mass
at
position x over the length of the loop. We have