Is f(x,t)=Acos(K(x-vt)+phi) a wave equation solution?

[leoflc]

Can someone show me that $$f(x, t) = A\cos(K(x-vt) + \phi)$$ is in fact a solution of the wave equation?
I kind of know how to show it by using calculus, but is there other way to show it?

Thank you very much!

 

[Fredrik]

Any function f that can be expressed as

$$f(x,t)=g(x-vt)$$

satisfies the wave equation. You don't even have to know what g is to show it.

 

[leoflc]

Could you please show me a little more? I don't really get it why that will satisfie the wave equation...
Thank you very much!

 

[Fredrik]

We're supposed to give hints here, not complete answers, but if you compute the second-order partial derivative

$$\frac{\partial^2 f(x,t)}{\partial t^2}$$

using the formula

$$f(x,t)=g(x-vt)$$

you're almost there. Does the result look anything like any other second-order partial derivative that appears in the wave equation?

 

[leoflc]

Thanks, I'll give it a try.
But I'm just wondering, is there any other way to show it beside using calculus?
Thanks again!

 

[Fredrik]

I don't think so. The wave equation is a partial differential equation, so any explanation would have to involve derivatives in some way.

I'm pretty sure that there's no easier way to understand the wave equation than the way I suggested. You should note that the graphs of the functions $$h_t$$, defined by

$$h_t(x)=g(x-vt)$$

can be thought of as the individual frames of a "movie" that shows the graph of g moving with velocity v.