solution of wave equation, 2nd partial derivatives of time/position


by mathnerd15
Tags: derivatives, equation, partial, solution, time or position, wave
mathnerd15
mathnerd15 is offline
#1
Apr1-13, 04:45 PM
P: 108
f(z,t)=\frac{A}{b(z-vt)^{2}+1}...

\frac{\partial^{2} f(z,t)v^{2} }{\partial z^2}=\frac{-2Abv^{2}}{[b(z-vt)^{2}+1]^{2}}+\frac{8Ab^{2}v^{2}(z-vt)^{2}}{[b(z-vt)+1]^{3}}=\frac{\partial^2 f}{\partial t^2}
\frac{-2Abv^{2}}{[b(z-vt)^{2}+1]^{2}}+\frac{8Ab^{2}v^{2}(z-vt)^{2}}{[b(z-vt)+1]^{3}}

this is a solution of the wave equation, but it can be written with the Laplacian. is this also a hyperbolic partial differential equation. Alembert derived the solution that 1D waves are the addition of right and left moving functions

what is the meaning of the 2nd partial derivatives in respect to time and position which differ by v^2? (I wrote this on online Latex editor, the differentiation is in the attachment)
thanks very much!
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mathnerd15
mathnerd15 is offline
#2
Apr9-13, 10:22 AM
P: 108
so you can derive the speed c from the Maxwell equations which for an electromagnetic wave is the Weber/Kohrausch ratio 1/(epsilono*muo)^(1/2)....


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