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.

Answer 1

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: 

\[y = A\sin  \left[ \left( \frac{x}{a} \right) + \frac{t_0}{\left( \frac{a}{\nu} \right)} - \frac{t}{\left( \frac{a}{\nu} \right)} \right]\]

\[\Rightarrow y = A\sin  \frac{x + \nu  \left( t_0 - t \right)}{a}\]