Acoustic energy flux

For harmonic acoustic amplitudes, $⟨\mathbf{I}⟩=\frac{1}{2}\mathrm{Re}(p{\mathbf{u}}^{\ast })$.

With the time convention $e^{-i\omega t}$, pressure amplitude $p$ and background density ${\rho }_0$, the acoustic energy flux is $I_x=(2{\rho }_0\omega {)}^{-1}\mathrm{Im}(\overline{p}{\partial }_{x}p)$. A transverse plane wave mode with positive axial wavenumber $\kappa$ carries flux $\kappa |p{|}^2/(2{\rho }_0\omega )$. For the reduced pressure $p=e^{ikx}E$, the parabolic wave equation gives $\kappa =k-{\nu }^2/(2k)$, approximated by $k$ at leading paraxial order. Stationary illumination uses the infinite-aperture spectral normalization: replace $|p{|}^2$ by the corresponding spectral power per unit transverse length. Other acoustic amplitude conventions change the dimensional prefactor.

For a finite duct segment of phase length $\lambda$ joined to a semi-infinite medium, continuity of pressure and velocity gives the outgoing potential amplitudeAt half-integer $\lambda /\pi$, a large impedance mismatch can strongly enhance transmitted flux for a prescribed piston displacement.