Solution
= Solution
For a progressive <plane wave>, write
$$
p'=P e^{i(\mathbf k\cdot\mathbf x-\omega t)},
\qquad
\mathbf u=U\widehat{\mathbf k}e^{i(\mathbf k\cdot\mathbf x-\omega t)}.
$$
The momentum equation gives $\rho_0\omega U=kP$, while the <dispersion relation> is $\omega=c_0k$. Therefore the pressure and longitudinal velocity amplitudes obey
$$
\boxed{\frac{P}{U}=\rho_0c_0=:Z.}
$$
The constant $Z$ is the <acoustic impedance>.