Is a given expression a travelling wave?

[shanepitts]

How can one compute and/or tell if a given expression is a traveling wave? I thought of using the characteristic wave equation δ^(2)ψ/δt^(2)=v^(2)δ^(2)ψ/δx^(2).

An example of an expression would be ψ(x,t)=A(x-t). Should I partially derive this expression twice with respect to x and then to t and see if they match with the wave equation? Moreover, would I be correct in assuming that if they do match then this expression is a traveling wave?

(This is not a homework assignment)

 

[Simon Bridge]

Characteristic wave equation: $$\frac{\partial^2}{\partial t^2}\psi = v^2\frac{\partial^2}{\partial x^2}\psi$$
... just checking.

You can compute if a particular expression is a traveling wave by following the definition of a traveling wave.
i.e. if it holds it's basic shape while changing position - then it's a traveling wave.

If you start out with some expression, then you can prove that it is a wave by seeing if it is a solution to the wave equation.
To see if it is a traveling wave - you need to use the definition of a traveling wave.

Similarly you can check to see if your example is a general form of a traveling wave.

 

[rumborak]

I think the OP question is more generic. That is, how do you mathematically establish whether a given solution is a traveling wave? There are a lot of solutions to the above wave equation that I wouldn't call a wave

I think the most generic definition of a traveling wave is if it (within certain bounds) obeys the equation A(x,t) = A(x+deltaX, t+deltaT). Meaning, after a time of deltaT the shape is the same but has moved by deltaX.
For the regular sine function (the most basic solution to the above mentioned wave equation) this equation for example should hold, as the sine function "travels" over time.