Wave packet

wormhole

here is wave equation:



I know that any function of x-ct variable will solve the above equation, where c is wave velocity.

Below is an example of wave written as wave packet with Ai - Airy function.
the first part of equation describes propagation of wave' envelope with group velocity and second part, Airy function, describes propagation of wave itself with phase velocity.



I noticed that both parts can't be written as
part1(x-Vg*t) and part2(x-Vph*t) .... so are these parts still represent traveling waves?

If so then is it true that any function f(x,t) with x appearing only once in power 1, will describe travelling wave?

thanks

Kalimaa23

As far I'm aware, the general solution of a travelling wave as you call it is the general solution to the following wave equation :
[tex] \frac{\partial ^{2}\phi}{\partial ^{2} x^{2}}+ \frac{1}{v^{2}}\frac {\partial ^{2} \phi}{\partial ^{2} t^{2}}=0 [/tex]

wormhole

I know that... but this is not what i asked

turin

No. The most obvious counterexample that comes to mind is f(x,t) = Ax. Unless you would want to call that a travelling wave.

wormhole

I will reformulate like this:

Is it true to say that any function of form

F( (x-V(t)) / G(t) )

representing some form of travelling wave?

thanks

turin

If V(t) is arbitrary, then I would suspect that this is not generally a travelling wave, based on similar arguement. You will probably need to be a little bit more specific/exclusive. BTW, I'm just entertaining this question with a guess. It will probably boil down to a matter of definition.

wormhole

In my first post i said that this question is related to the form of wave packet:



as you can see the wave envelope and the wave itself is not a function of x-c*t but of (x-V(t))/G(t)
it was one of the examples we studied during "wave and optics" course

Stingray

Some introductory books will define a "traveling wave" to only be something satisfying the equation you gave. Nobody really uses that definition past 1st year courses though (it really should never be presented that way). A traveling wave is not a perfectly well defined concept. The general usage is that the term is applied to anything that "looks wavy." Your Airy function example qualifies.

Basically, there are many more things that can rightfully be called wave equations other than what you gave, so restrictive definitions are pointless.

wormhole

ok thanks
i got the idea...

turin

Should a traveling wave also be required to transmit energy, or is that positting a physical feature prematurely on the definition?