### 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

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

*second*partial derivatives in are

*linear operator*, establish that

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