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

[Ryker]

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.

Homework Equations

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!

 

[klawlor419]

Either way is fair game!

 

[vela]

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?

The only difference between f(x+vt) and f(x-vt) is the direction that the wave propagates. f(x-vt) represents a wave moving in the +x direction, and f(x+vt) represents a wave moving in the -x direction, where v>0 is the speed of the wave. Whether v=±c/b depends on the signs of b and c.

 

[Ryker]

Either way is fair game!

The only difference between f(x+vt) and f(x-vt) is the direction that the wave propagates. f(x-vt) represents a wave moving in the +x direction, and f(x+vt) represents a wave moving in the -x direction, where v>0 is the speed of the wave. Whether v=±c/b depends on the signs of b and c.

Thanks for the replies! After thinking about it some more, that's what I figured, as well, as I just couldn't justify why there would necessarily be a minus sign.

 

[paranormal]

I would say that your approach of rearranging the expression in the exponent to yield x + \frac{c}{b}t and then reading off v = \pm\frac{c}{b} is a valid method for determining the direction and speed of the wave. However, it is important to note that this method assumes that \psi(x,t) = f(x \pm vt) is the correct form of the wave equation. If we are not sure about the sign in front of vt, then we cannot determine the direction of propagation with certainty.

The alternative method you mentioned, using the equation bx + ct = C and taking the derivative with respect to time, is also a valid approach. This method is based on the fact that the wave equation can be written as a function of the form f(x \pm vt), where v is the speed and the sign in front of vt determines the direction of propagation. Therefore, by finding the derivative of this function with respect to time, we can determine the speed and direction of the wave.

Both methods are valid and can be used depending on the situation. It is important to carefully consider the form of the wave equation and the given information before deciding on the appropriate method to use.