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Re: The Wave Equation : Mathematica vs. Mathworld

• To: mathgroup at smc.vnet.net
• Subject: [mg47897] Re: The Wave Equation : Mathematica vs. Mathworld
• From: Thomas E Burton <tburton at brahea.com>
• Date: Thu, 29 Apr 2004 19:39:45 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

The traveling-wave solution f[x + c t] to the wave equation in one 
dimension is about as "classical" as classical physics gets! It 
represents a wave propagating to the left (if c is positive) with speed 
|c| and shape f.

You are apparently attempting to satisfy Dirichlet boundary conditions 
(f[0]===f[L]===0) with this solution, which will require something like 
a method of multiple reflections. If you then expand the function f in 
a Fourier series, convert traveling waves Exp[+/-I omega t] to standing 
waves Sin[omega t] & Cos[omega t], you'll arrive at a standing-wave 
solution somewhat resembling equation (39) in Mathworld's wave equation 
intro, with v = c. Note: you have not transcribed equation (39) 
accurately into your notebook.

Tom Burton

You had written:

I'm attempting to duplicate the analysis found in:

http://mathworld.wolfram.com/WaveEquation.html

about how to derive a solution to the Wave Equation. I want to get to 
that solution using Mathematica. The solution to the Wave Equation is 
given by Eq(39) displayed on the above website. To see my attempt to 
solve the Wave Equation using Mathematica v.5, please double-click the 
following internet link, and save this notebook into a directory of 
your choice:

http://www.tilmarlily.netfirms.com/download/wave.nb

The solution given by Out[2](in the notebook) is very different from 
Eq(39).

My question is, why is the solution given by Mathematica so different 
than the classical result given by Eq(39)in the Mathworld website?