Solving the Wave Equation Using Separation of Variables

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The discussion focuses on solving the wave equation using separation of variables, specifically deriving Eq. 9.20 from Griffiths. Participants express confusion about separating variables and suggest representing the function f(z,t) as a product of sine and cosine functions. The wave equation in question is a 1D linear PDE, and there is a mention of applying the Fourier transform for a solution. Boundary conditions for electromagnetic waves are noted to be non-existent, allowing for solutions to be expressed as a Fourier series or integral. The conversation emphasizes the importance of proper mathematical representation in solving the wave equation.
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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.
 
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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.
 
What are the boundary conditions for an electromagnetic wave?
 
There are none.Simply the solution be writible as a Fourier series (discrete spectrum of frequencies),or as Fourier integral...

Daniel.
 
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