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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
.
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![\begin{eqnarray*}
\frac{\partial}{\partial x} y_r\left(t-\frac{x}{c}\right)
&=& -\frac{1}{c}{\dot y}_r\left(t- \frac{x}{c}\right)\\ [10pt]
\frac{\partial}{\partial x} y_l\left(t+\frac{x}{c}\right)
&=& \frac{1}{c}{\dot y}_l\left(t+ \frac{x}{c}\right).
\end{eqnarray*}](img3235.png)
![\begin{eqnarray*}
y'_r&=& -\frac{1}{c}{\dot y}_r\\ [5pt]
y'_l&=& \frac{1}{c}{\dot y}_l,
\end{eqnarray*}](img3238.png)
![\begin{eqnarray*}
y''_r&=& \left(-\frac{1}{c}\right)^2 {\ddot y}_r= \frac{1}{c^2} {\ddot y}_r\\ [5pt]
y''_l&=& \left(\frac{1}{c}\right)^2 {\ddot y}_l= \frac{1}{c^2} {\ddot y}_l.
\end{eqnarray*}](img3241.png)
