Question

A wave propagetes on a string with a speed v towords the positive X-axis. This shape of the string at t = 0 is given by g(x) = Asin(x/a) , where A and a are constants. ​

Answer 1

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Sol. At t = t0, g(x, t0) = A sin (x/a) …(1) For a wave traveling in the positive x-direction, the general equation is given by y = f(x/a – t/T) Putting t = –t0 and comparing with equation (1), we get ⇒ g(x, 0) = A sin {(x/a) + (t0/T)} ⇒ g(x, t) = A sin {(x/a) + (t0/T) – (t/T)} As T = a/v (a = wave length, v = speed of the wave) ⇒ y = A sin (x/a + t0/(a/v) – t/(a/v)) = A sin (x + v(t0 - t)/a) ⇒ Y = A sin [x – v(t – t0/a)]

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Answer 2

General equation of a wave propagating on a string with speed v, amplitude A and angular frequency $\omega$ towards the positive x-axis:

$y = Asin(\omega (\frac{x}{v} - t) + \phi)$

At t = 0:

$y = Asin(\frac{\omega}{v}x + \phi) = Asin(\frac{x}{a})$

Implies: $\phi = 0$, $\frac{\omega}{v} = \frac{1}{a}$

$a = \frac{v}{\omega}$

a is the inverse of the wave number (generally denoted by k).