Acoustic waveguide

An acoustic waveguide confines sound transversely while allowing longitudinal propagation. Rigid walls impose a homogeneous normal-velocity condition, producing Neumann transverse modes. Each transverse eigenvalue gives a longitudinal wave number: modes below their cutoff decay evanescently, while cut-on modes carry outgoing sound. An open semi-infinite guide can be studied with the Wiener-Hopf method.

For the even acoustic waveguide mode $\cos (n\pi y/h)e^{-ikx}$, set $\sin \alpha =n\pi /(k_{0}h)$ and $k=k_0\cos \alpha >0$. At leading independent-edge order for large $k_{0}h$, the two incident travelling constituents each have amplitude $(-1{)}^n/2$ at the relevant wall endpoint. Reflecting the upper-edge coordinates reverses the edge-wave sign. Their path phases are $e^{\pm i{k}_{0}h\sin \theta }$. Thus Sommerfeld half-plane diffraction gives the displayed sum, where $D(r)=\sqrt{2/(\pi k_{0}r)}{e}^{-i{k}_{0}r-i\pi /4}$. For $n>0$, the apparent poles at $\theta =\pm \alpha$ are removable in the combined expression, with limit $ikhD(r)/2$. Single-edge nonuniform far-field poles are not physical infinities. This combined far-field limit assumes a radiation distance beyond the aperture's near field and neglects further edge interactions; it is not uniform at duct cutoff.

For rigid plates $x>0$, $y=\pm b$, an even incident plane wave travelling toward the opening can be scattered using Wiener-Hopf factorization. With time dependence $e^{i\omega t}$ and transform $\Phi (k)=\int \varphi (x)e^{ikx}dx$, take $\gamma =\sqrt{k^2-k_0^2}$ on the outgoing sheet. The common normal derivative on $y=b$ is supported on $x<0$; its transform is $D^{-}=i{L}^{+}(-k_0)L^{-}(k)/(k+k_0)$. The interior transform is $D^{-}\cosh (\gamma y)/(\gamma \sinh (\gamma b))$, while the exterior transform is $-D^{-}{e}^{-\gamma (|y|-b)}/\gamma$. These follow from continuity of normal velocity and the half-line pressure jump, followed by pole subtraction in a Wiener-Hopf equation and a finite-energy edge condition.

For fixed $|y|<b$ and $x\to -\infty$, the incident pole in the scattered transform cancels the analytic continuation of the incident plane wave. Near $k=-k_0$, the remaining transform is proportional to $(k+k_0{)}^{-1/2}$, so the total outgoing field has the displayed inverse-square-root cylindrical decay. A claim of square-root growth at negative infinity is incompatible with this outgoing solution.

For a separated acoustic potential with transverse Neumann eigenvalue ${\kappa }^2$, integration by parts gives $\int |{\nabla }_{\perp }F{|}^2={\kappa }^2\int F^2$. The time-averaged kinetic energy is proportional to $(k^2+{\kappa }^2)\int F^2$, while compression energy is proportional to ${\omega }^2\int F^2/c_0^2$. The dispersion relation ${\omega }^2=c_0^2(k^2+{\kappa }^2)$ proves equality. At a ray speed below $c_0$, nondegenerate stationary phase gives dispersive amplitudes of order $t^{-1/2}$; the zero transverse mode instead preserves an undispersed pulse on the sound-speed ray.

For an even field between rigid plates at $y=\pm b$, the transverse modes are $\cos (n\pi y/b)$, with longitudinal wave numbers $k_n=\sqrt{k_0^2-(n\pi /b{)}^2}$. With time dependence $e^{i\omega t}$, outgoing modes into $x>0$ use $e^{-i{k}_{n}x}$: choose positive real $k_n$ for cut-on modes and negative imaginary $k_n$ for cut-off modes. A lower-half-plane Fourier inversion extracts these modes by residues. Their decaying components are evanescent waves, not backward-propagating waves. At an exact cutoff one takes the radiation continuation before its limit.