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# The Wave Equation

This is a bogus "derivation" in that we start with a solution to the WAVE EQUATION and then show what sort of differential equation it satisfies. Of course, once we have the equation we can work in the other direction, so this is not so bad . . . .

Suppose we know that we have a traveling wave  .

At a fixed position  ( const)  we see SHM in time:

 (14.8)

(Read: "The second partial derivative of with respect to time [i.e. the acceleration of ] with held fixed is equal to times itself.") I.e. we must have a linear restoring force.

Similarly, if we take a "snapshot" (hold fixed) and look at the spatial variation of , we find the oscillatory behaviour analogous to SHM,

 (14.9)

(Read: "The second partial derivative of with respect to position [i.e. the curvature of ] with held fixed is equal to times itself.")

Thus

If we multiply both sides by , we get

But    so   ,  giving the WAVE EQUATION:
 (14.10)

In words, the curvature of is equal to times the acceleration of at any point (what we call an event in spacetime).

Whenever you see this differential equation governing some quantity , i.e. where the acceleration of is proportional to its curvature, you know that will exhibit wave motion!

Next: Wavy Strings Up: WAVES Previous: Speed of Propagation
Jess H. Brewer - Last modified: Sun Nov 15 17:59:18 PST 2015