Question

Which of the following functions correctly represents the traveling wave equation for finite values of x and t? 1) y=x^2-t^2. 2) y=cosx^2 sint. 3) y=log(x^2-t^2)-log(x-t). 4) e^2x sint

Answer 1

The  functions that correctly represents the traveling wave equation for finite values of x and t is :

3) y=log(x²-t²)-log(x-t)

We know that a travelling wave is represented by

When the wave is travelling along +ve X direction

y = f (x-vt)                     (eq 1)

When the wave is travelling along -ve X direction.

y = f (x+vt)                     (eq2)

Upon simplifying y=log(x²-t²)-log(x-t) , we get,

y = log ( $\frac{x^{2}-t^{2} }{x-t}$)

  = log ($\frac{(x-t)(x+t)}{x-t}$)

  = log (x+t)

Here we can see that v = 1  (from eq 1)

Therefore, y=log(x²-t²)-log(x-t) represents a travelling wave.