Writing down expressions for traveling waves

[rmjmu507]

Homework Statement

There is no specific problem - this is more of a broad question...given a wave equation and asked to write down/guess an expression/general solution for a traveling wave, it is sufficient to say the following:

1) For $\frac{\partial^2 E}{\partial z^2} - \frac{1}{c^2}*\frac{\partial^2 E}{\partial t^2}=0$

when asked to right down an expression for a traveling wave moving in the positive z direction with amplitude A is it sufficient to write $E=A*sin(kz-\omega t)$

2) For $\frac{\partial^2 E}{\partial z^2} - \frac{\epsilon}{c^2}*\frac{\partial^2 E}{\partial t^2}-\sigma*\mu*\frac{\partial E}{\partial t}=0$

when asked to guess a general solution for a traveling wave, it is sufficient to write $E=A*sin(kz-\omega t)$

Please let me know if this sounds reasonable. Thank you for your time and help

Homework Equations

The Attempt at a Solution

 

[Simon Bridge]

Those may be sufficient guesses if the rest of the context supports them - however, those choices are making assumptions about the boundary conditions and initial values. i.e. what if $E(0,0)\neq 0$ ?

 

[rmjmu507]

So a more all-inclusive/comprehensive/safer guess would be something like:

$E=A*exp^{i(wt-kz)}$

Thus, if the initial conditions are $E(0,0)=0$ the sine function is returned and if $E(0,0)\neq0$ then the cosine function is returned?

Please let me know if this is a better guess to make when dealing with traveling waves.

Thank you!

 

[olivermsun]

I don't think you need to worry about the boundary conditions when you are finding the general solution.

Also, I don't think you need to assume the exponential (or sinusoidal) form in the solution. There should be two solutions (since the equation is 2nd order) but they can have any shape!

 

[rmjmu507]

So the initial guess is sufficient?

Is there anything wrong with the exponential guess, or will both serve as solutions to the wave equations given?

 

[olivermsun]

Well that depends if the question is asking for "the" general solutions or if some working guesses would be enough.

Also, the form of the solutions you guessed is fine… but check them against the original equations. There are no $k$ or $z$, only $c$. How can you express your "guess" solutions in terms of $c$?

 

[Simon Bridge]

Just because $E(0,0)\neq 0$, it does not mean that $E(x,t)$ is a cosine wave.
I'm trying to get you to rethink the assumptions you are making about traveling waves.

Any $E(x,t)=f(x-ct)$, where $f$ is an arbitrary function, would be your most general possible guess - but it is not all that helpful so you need to use the specifics of the situation to select what sort of $f$ to choose. There is no "best" choice that works for everything.