# Determine the direction and speed of the wave from a given wave equation

## Homework Statement

Given an equation for a wave $\psi(x,t) = A e^{-a(bx+ct)^{2}}$ determine the direction of its propagation if you know $\psi(x,t) = f(x \pm vt)$ and use this to find its speed.

## The Attempt at a Solution

I figured I would just rearrange the expression in the exponent, so as to yield $x + \frac{c}{b}t$, and then just read off $v = \pm\frac{c}{b}$. However, if we don't know whether $\psi(x,t) = f(x + vt)$ or $\psi(x,t) = f(x - vt)$, can we really determine the direction of its propagation?

Also, I found somewhere the answer to this question would uniquely be $v = -\frac{c}{b}$ by letting $bx + ct = C$, and then after solving for x, $x = \frac{C}{b} - \frac{c}{b}$, taking the derivative with respect to time, yielding the above unique solution with the minus sign. Is this the proper way of doing things instead of just rearranging the expression like I did?

Thanks!

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Either way is fair game!

vela
Staff Emeritus
I figured I would just rearrange the expression in the exponent, so as to yield $x + \frac{c}{b}t$, and then just read off $v = \pm\frac{c}{b}$. However, if we don't know whether $\psi(x,t) = f(x + vt)$ or $\psi(x,t) = f(x - vt)$, can we really determine the direction of its propagation?