A wave propagates on a string in the positive x-direction at a velocity v. 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

AT $t = t_{0}, {g (x, t_{0})} = A s i n (\frac{x}{a})$

For a wave travelling in the positive x-direction, the general equation is given by

$y = A s i n (\frac{x}{a} + (\frac{t}{T})$

Putting, t = - t and comparing with equation (1), we get

$\Rightarrow g (x, 0) = A s i n {(\frac{x}{a}) + (\frac{t_{0}}{T})}$

$\Rightarrow g (x, t) = A s i n [{(\frac{x}{a}) + \frac{t_{0}}{T}} - (\frac{t}{T})]$

As $T = \frac{a}{v}$

(a = wave length, v = speed of the wave)

$\Rightarrow y = A s i n [(\frac{x}{y}) + \frac{t_{0}}{(\frac{a}{v})} - \frac{t}{(\frac{a}{v})}]$

$\Rightarrow y = A s i n \frac{x + v (t_{0} - t)}{a}$

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