Solving the Wave Equation Using Separation of Variables

[pt176900]

From Griffiths: Problem 9.4 Obtain Eq. 9.20 directly from the wave equation, by separation of variables.

Eq. 9.20: f(z,t)~ = integral [-inf, inf] A~(k)e^i(kz-wt) dk

where ~ denotes the complex conjugate

the wave equation: f''(z) = (1/v^2) f''(t)

I'm a little confused on how I can separate the variables. Can I assume that I can represent f(z,t) as A sin(kz)cos(kvt) and then calculate the aforementioned 2nd derivatives.

I think my reasoning is correct but my math... well my math isn't up to par.

 

[dextercioby]

You want to solve the the 1D linear PDE

\frac{\partial^{2} f}{\partial z^{2}}=\frac{1}{v^{2}}\frac{\partial^{2} f}{\partial t^{2}}

with proper boundary conditions (field-type)...?

Use the Fourier transform...

Daniel.

 

[pt176900]

What are the boundary conditions for an electromagnetic wave?

 

[dextercioby]

There are none.Simply the solution be writible as a Fourier series (discrete spectrum of frequencies),or as Fourier integral...

Daniel.