A wave propagates on a string in the positive $x -$ direction at a velocity $u$ . The shape of the string at $t = t_{0}$ is given by $g (x, t_{0}) = A S i n (\frac{x}{a})$ . Write the wave equation for a general time $t$ .

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Solution

Step 1. Given Data,
A wave propagets in positive $x -$ direction at a velocity $u$ .
The shape of the string at $t = t_{0} i s g (x, t_{0}) = A S i n (\frac{x}{a}) .$
We have to write the wave equation for a general time $t .$
Step 2. Write the wave equation for a general time $t,$
We know that the general wave equation for positive $x -$ direction,
$Y = A S i n [(\frac{x}{a}) - (\frac{t}{T})]$ $($ where, $T = \frac{a}{v})$
$\Rightarrow Y = A S i n [(\frac{x}{a}) - (\frac{v t}{a})] \dots (1)$
Since, $g (x . t_{0}) = A S i n (\frac{x}{a}) = A S i n [(\frac{x}{a}) + (\frac{v t_{0}}{a})]$
Comparing above equation with equation $(1)$ we get,
$\Rightarrow Y = A S i n [(\frac{x}{a}) - (\frac{v t}{a}) + (\frac{v t_{0}}{a})]$
Hence $Y = A S i n [(\frac{x}{a}) - (\frac{v t}{a}) + (\frac{v t_{0}}{a})]$ is the wave equation of time.

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