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The equation of the vibration of a wire ...

The equation of the vibration of a wire is ` y = 5 "cos"(pi x)/(3) sin 40 pi t ` , where x and y are given in cm and t is given in s. calculate the
amplitudes and velocities of the two waves which on superposition, from the above-mentioned vibration,

Text Solution

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` y = 5 "cos"(pir)/(3) sin 40 pi t = (5)/(2) * 2 sin 40 pi t cos (pir)/(3)`
` = (5)/(2) [ sin ( 40 pi t + (pir)/(3))+ sin ( 40 pi t - (pir)/(3))]`
`= (5)/(2) sin 40 pi (t + (x)/(120))+(5)/(2)sin 40 pi (t - (x)/(120))`
` y _(1) + y_(2)`
So, the resultant vibration is produced due to the superposition of two waves ` y _(1) and y_(2)` . comparing these two waves with the general equation, y ` = A sin omega (t - (x)/(V)) ` .
Amplitude , `A_(1) = A_(2) = (5)/(2) = 2 . 5 cm ` :
wave velocity, ` V_(1) = 120 cm * s^(-1)` (towards negative x direction) and `V_(2) = 120 cm*s^(-1)` (towards positive x -direction).
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