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Question
A wave propagates on a string in the positive x-direction at a velocity \[\nu\] \[t = t_0\] is given by \[g\left( x, t_0 \right) = A \sin \left( x/a \right)\]. Write the wave equation for a general time t.
Solution
Given,
Wave velocity = \[\nu\]
Shape of the string at
\[t = t_0\]
\[g\left( x, t_0 \right) = A \sin \left( x/a \right)\]
For a wave travelling in the positive x-direction, the general equation is given by \[y = A \sin \left( \frac{x}{a} - \frac{t}{T} \right)\]
Putting t = − t and comparing with equation (i), we get:
\[g\left( x, 0 \right) = A\sin\left\{ \left( \frac{x}{a} \right) + \left( \frac{t_0}{T} \right) \right\}\]
\[ \Rightarrow g\left( x, t \right) = A\sin\left[ \left\{ \left( \frac{x}{a} \right) + \frac{t_0}{T} \right\} - \left( \frac{t}{T} \right) \right]\]
\[Now, \]
\[T = \frac{a}{\nu}\]
\[Here, \]
a = Wave length
nu = Velocity of the wave
Thus, we have:
\[y = A\sin \left[ \left( \frac{x}{a} \right) + \frac{t_0}{\left( \frac{a}{\nu} \right)} - \frac{t}{\left( \frac{a}{\nu} \right)} \right]\]
\[\Rightarrow y = A\sin \frac{x + \nu \left( t_0 - t \right)}{a}\]
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