Question

A wave propagates on a string in the positive x-direction at a velocity ν. The shape of the string at t=t0 is given by g(x, t0)=A sin (x/a). Write the wave equation for a general time t.

Answer 1

Wavelength is Proportional to Speed of Wave

Explanation:

At $t = t_0, g(x, t_0) = Asin \frac {x}{a}$ ....(1)

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

$y = f(\frac {x}{a} - \frac {t}{T})$

Putting $t = - t_0$ and comparing with equation (1) we get -

$g(x,0) = A sin(\frac {x}{a} + \frac{t_0}{T})$

$g(x,0) = A sin(\frac {x}{a} + \frac{t_0}{T} - \frac {t}{T})$

As $T = \frac {a}{v}$  (a = wavelength and v = speed of wave)

$y = A sin(\frac {x}{a} + \frac{t_0}{\frac {a}{v}} - \frac {t}{\frac {a}{v}})$

= $A sin(\frac {x + v(t_0 - t)}{a})$

$y = A sin(\frac {x - v(t_0 - t)}{a})$