### Traveling-Wave Partial Derivatives

Because we have defined our traveling-wave components
and
as having arguments in units of *time*, the partial
derivatives with respect to time are identical to simple
derivatives of these functions. Let
and
denote the
(partial) derivatives with respect to time of and ,
respectively. In contrast, the partial derivatives with respect to
are

Denoting the *spatial*
partial derivatives by and
, respectively, we can write more succinctly

where this argument-free notation assumes the same and for all terms in each equation, and the subscript or determines whether the omitted argument is or .

Now we can see that the *second* partial derivatives in are

These relations, together with the fact that partial differention is a
*linear operator*, establish that

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