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
obeys the ideal wave equation for all twice-differentiable functions and .