Plane-wave superposition

Equation (23) can be simply interpreted as plane-wave superposition. To make this clear, we first dispose of the rho filter by means of a definition.  

$$\tilde u (p, \tau ) \eq \ \it\hbox{rho} ( \tau ) \ {\rm *} \ \bar u ( p, \tau )$$ (24)

Equation (24) will be seen to be more than a definition. We will see that $\tilde u (p, \tau )$ can be interpreted as the plane-wave spectrum . Substituting the definition (24) into both (23) and (14) gives another transform pair:  

$$\begin{tabular} {\vert c\vert} \hline \\ $u(x, t)\ =\ \int\... ...\ \int u(x, \tau+px)\ dx$\space \\ \\ \hline\end{tabular}$$ (25)

To confirm that $\tilde u (p, \tau )$ may be interpreted as the plane-wave spectrum, we take $\tilde u (p, \tau )$ to be the impulse function $\delta (p\,-\,p_0 ) \, \delta ( \tau \,-\, \tau_0 )$ and substitute it into the top half of (25). The result $u(x,t) = \delta (t-p_0 x - \tau_0 )$is an impulsive plane wave, as expected.