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MathGroup Archive 2004

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Re: The Wave Equation : Mathematica vs. Mathworld

  • To: mathgroup at smc.vnet.net
  • Subject: [mg47902] Re: The Wave Equation : Mathematica vs. Mathworld
  • From: adam.smith at hillsdale.edu (Adam Smith)
  • Date: Thu, 29 Apr 2004 19:39:53 -0400 (EDT)
  • References: <c6qas9$s8r$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

There are very many reasons that your Mathematica code does not
produce the result listed in Mathworld.  I think you need to study and
understand the analytical way of getting the solution before trying to
reproduce this in Mathematica.

Much of the problem appears to stem from not understanding the steps
to get to Eq(39).  Note that the Am of Eq(39) contains an integral
given by Eq(37).  In many ways Eq(37) is the most important result. 
Getting to this requires understanding the steps 22-37 of the
mathworld argument.  Steps 22-36 leading to Eq[37] take a completely
different approach using the separation of variables technique, where
instead of f[t+v x] + g[t-v x] the form of the solution is taken as a
product of a function only of x and a function only of t: X(x)T(t). 
DSolve uses the logic up to Eq(14) and Mathematica reproduces this
result as shown below.
   

Having said that there is a syntax/typo problems in your attempt.  I
am assuming that you meant the constant c to equal 1/v^2

First the wave equation involves 2nd derivatives.  So your command

DSolve[D[ Psi[x, t], x] - c*D[ Psi[x, t], t] == 0, Psi[x, t], {x, t}]

Should read

DSolve[D[ Psi[x, t], {x,2}] - c*D[ Psi[x, t], {t,2}] == 0, Psi[x, t],
{x, t}]

This gives the solution:

C[1][t+Sqrt[c]x] + C[2][t-Sqrt[c]x]

Which is equivalent to Eq(14) of the mathworld site.  Instead of f and
g Mathematic uses C[1] and C[2] for the unspecified functions.

Adam Smith

gilmar.rodriguez at nwfwmd.state.fl.us (Gilmar Rodr?guez Pierluissi) wrote in message news:<c6qas9$s8r$1 at smc.vnet.net>...
> *******************************************************************************
> 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?
> 
> Thank you!


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