Let us consider the partial differential equation for unknown function of two variables .

First, we see this is simple to find a general solution for this equation. We have

and thus integrating with respect to x, we find:

, for some function *h*(*y*) of y only.

Now, we integrate that with respect to y to get:

, with *f*(*x*) some function of x only, and . This is the general solution to the equation.

Now, however, let us consider the change of variables given by

, ,

where c>0 is some constant. From these we find:

,

,

Then we see by the chain rule for partial derivatives that

and thus that

And thus our differential equation in terms of the new coordinates is thus:

This is the one-dimensional wave equation with speed c (see also this ‘Physics Friday’ post). Now recall that we found the general solution to our original equation to be . Now, inverting our coordinate transformation , , we see , , and so the general solution to

is , for functions of a single variable *f* and *g*.

[Note that the physical interpretation of this is that is a wave moving rightward (+x direction), and is a wave moving leftward (-x direction).]

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Tags: Coordinate Transformation, Math, Monday Math, Partial Differential Equation, waves

This entry was posted on April 28, 2008 at 4:00 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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