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Boundary conditions turn the governing wave equations into a particular scattering problem. They state whether an interface can sustain pressure, whether it can move, and which field quantities pass continuously between adjacent media. A fixed-rigid surface, a pressure-release surface, and a penetrable fluid target can therefore scatter very differently even when they have the same geometry.
The conditions must act on the total field at an exterior boundary. The incident and scattered fields are separated to solve the exterior problem, but neither field generally satisfies the boundary condition alone (Colton and Kress 2013).
See Notation and Symbols and the Acoustic Scattering Primer.
A fixed-rigid boundary cannot move in its normal direction. The normal fluid velocity at the surface must therefore vanish:
\[ v_{n,1}=\mathbf{v}_1\mathbin{\cdot}\mathbf{n}=0 \qquad\text{on }\Gamma. \]
The same limit can be expressed through the specific surface impedance:
\[ Z_s=\frac{p_1^{\mathrm{tot}}}{v_{n,1}}, \qquad |Z_s|\longrightarrow\infty. \]
Substitution of the phasor momentum relation gives the Neumann boundary condition:
\[ \partial_n p_1^{\mathrm{tot}}=0 \qquad\text{on }\Gamma. \]
In terms of the incident and scattered fields, the prescribed data for the scattered field are:
\[ \partial_n p_1^{\mathrm{scat}} =-\partial_n p_1^{\mathrm{inc}} \qquad\text{on }\Gamma. \]
The pressure itself need not vanish. Instead, the reflected field cancels the incident normal velocity at the surface. In partial-wave formulations, the Neumann condition fixes each scattering coefficient by excluding any total field that would produce radial surface motion. At high frequency, a smooth rigid surface supports a locally specular reflection without the pressure phase reversal of a pressure-release surface.
A pressure-release surface cannot sustain an acoustic pressure fluctuation. It is the zero-impedance limit:
\[ Z_s=\frac{p_1^{\mathrm{tot}}}{v_{n,1}}, \qquad |Z_s|\longrightarrow0. \]
For finite normal velocity, the total pressure obeys the Dirichlet condition:
\[ p_1^{\mathrm{tot}}=0 \qquad\text{on }\Gamma. \]
The corresponding condition on the scattered field is:
\[ p_1^{\mathrm{scat}}=-p_1^{\mathrm{inc}} \qquad\text{on }\Gamma. \]
Normal motion is not constrained to vanish. The idealization is appropriate when the material on the other side has negligible acoustic impedance relative to the exterior fluid, as for an idealized gas boundary away from effects that require an explicit interior solution. A locally reflected pressure wave undergoes a phase reversal. In a modal solution, zero surface pressure replaces the zero-normal-velocity condition used for a rigid target.
A penetrable fluid target supports acoustic pressure on both sides of its boundary. The exterior total field and interior field satisfy (Anderson 1950):
\[ \nabla^2p_1^{\mathrm{tot}}+k_1^2p_1^{\mathrm{tot}}=0, \qquad \nabla^2p_2+k_2^2p_2=0. \]
An inviscid fluid has Cauchy stress \(\boldsymbol{\sigma}_j=-p_j\mathbf{I}\). Continuity of normal traction therefore requires pressure continuity:
\[ p_1^{\mathrm{tot}}=p_2 \qquad\text{on }\Gamma. \]
The interface also admits neither separation nor interpenetration, so normal particle velocity is continuous:
\[ \mathbf{v}_1\mathbin{\cdot}\mathbf{n} =\mathbf{v}_2\mathbin{\cdot}\mathbf{n} \qquad\text{on }\Gamma. \]
Using the momentum equation gives the derivative form of the second transmission condition:
\[ \frac{1}{\rho_1}\partial_n p_1^{\mathrm{tot}} =\frac{1}{\rho_2}\partial_n p_2 \qquad\text{on }\Gamma. \]
Pressure and normal motion are both generally nonzero. Their continuity couples the exterior and interior solutions, while density and compressibility contrasts determine the strength and phase of the reflected and transmitted fields (Medwin and Clay 1998). Tangential velocity need not be continuous across an ideal inviscid fluid-fluid interface because neither fluid transmits shear traction.
An acoustic shell is a finite fluid layer, not an immobile shell and
not an elastic solid. Medium 2 occupies the region between
\(\Gamma_a\) and \(\Gamma_b\). It supports compressional
pressure waves but no shear stress. The shell field satisfies:
\[ \nabla^2p_2+k_2^2p_2=0 \qquad\text{between }\Gamma_a\text{ and }\Gamma_b. \]
At the outer interface, pressure and normal velocity are continuous:
\[ p_1^{\mathrm{tot}}=p_2, \qquad \frac{1}{\rho_1}\partial_n p_1^{\mathrm{tot}} =\frac{1}{\rho_2}\partial_n p_2 \qquad\text{on }\Gamma_a. \]
For a spherical shell, both independent radial solutions are admissible because the layer does not include the origin. An axisymmetric shell field can be expanded as:
\[ p_2(r,\theta) =\sum_{m=0}^{\infty} \left[ B_m\,j_m(k_2r)+C_m\,y_m(k_2r) \right]P_m(\cos\theta), \qquad b<r<a. \]
Here, \(j_m\) and \(y_m\) are spherical Bessel functions of the first and second kinds. The singularity of \(y_m(k_2r)\) at the origin is irrelevant because \(r=0\) is outside the shell layer. The condition at \(\Gamma_b\) determines the relative weights of these two branches.
For a pressure-release core, the shell pressure vanishes at the inner interface:
\[ p_2=0 \qquad\text{on }\Gamma_b. \]
In a spherical modal solution, a radial combination that satisfies this inner condition identically is:
\[ R_m^{\mathrm{pr}}(r) =j_m(k_2r)y_m(k_2b)-y_m(k_2r)j_m(k_2b). \]
Its value at the inner radius is:
\[ R_m^{\mathrm{pr}}(b)=0. \]
The outer pressure and velocity conditions then couple this admissible shell field to \(p_1^{\mathrm{tot}}\). The shell can carry normal acoustic motion even though the pressure is zero at its inner boundary. Its finite thickness and wavenumber add propagation phase between the two interfaces. The resulting response can contain cavity-like resonances that are absent from a single pressure-release surface.
If medium 3 is a fluid or gas core, it supports its own
Helmholtz field:
\[ \nabla^2p_3+k_3^2p_3=0 \qquad\text{inside }\Gamma_b. \]
Pressure and normal velocity are continuous at the inner interface:
\[ p_2=p_3, \qquad \frac{1}{\rho_2}\partial_n p_2 =\frac{1}{\rho_3}\partial_n p_3 \qquad\text{on }\Gamma_b. \]
For a spherical core containing the origin, regularity excludes the spherical Bessel function of the second kind. The core expansion is therefore:
\[ p_3(r,\theta) =\sum_{m=0}^{\infty}D_m\,j_m(k_3r)P_m(\cos\theta), \qquad 0\leq r<b. \]
Together, the two conditions at \(\Gamma_a\) and the two at \(\Gamma_b\) determine the exterior scattering field, the two shell branches, and the regular core field for each mode. The shell is not assigned infinite impedance, and its normal velocity is not set to zero. Resonance and interference arise from compressional propagation through the shell and core rather than from elastic shear or bending waves.
An elastic shell supports longitudinal and transverse motion. Its displacement \(\mathbf{u}(\mathbf{x},t)\) follows from balance of linear momentum and Hooke’s law. For a homogeneous isotropic shell, the time-domain Navier equation is (Achenbach 1973):
\[ (\lambda+2\mu)\nabla(\nabla\mathbin{\cdot}\mathbf{u}) -\mu\nabla\mathbin{\times}(\nabla\mathbin{\times}\mathbf{u}) -\rho_s\frac{\partial^2\mathbf{u}}{\partial t^2} =0. \]
For a displacement phasor with \(e^{-i\omega t}\) dependence, this becomes:
\[ (\lambda+2\mu)\nabla(\nabla\mathbin{\cdot}\mathbf{u}) -\mu\nabla\mathbin{\times}(\nabla\mathbin{\times}\mathbf{u}) +\rho_s\omega^2\mathbf{u} =0. \]
The Helmholtz decomposition separates dilatational and equivoluminal motion:
\[ \mathbf{u}=\nabla\Phi+\nabla\mathbin{\times}\mathbf{\Psi}, \qquad \nabla\mathbin{\cdot}\mathbf{\Psi}=0. \]
The scalar and vector potentials satisfy separate Helmholtz equations:
\[ \nabla^2\Phi+k_L^2\Phi=0, \qquad \nabla^2\mathbf{\Psi}+k_T^2\mathbf{\Psi}=0. \]
Their wavenumbers and wave speeds are:
\[ k_L=\frac{\omega}{c_L}, \qquad k_T=\frac{\omega}{c_T}, \qquad c_L=\sqrt{\frac{\lambda+2\mu}{\rho_s}}, \qquad c_T=\sqrt{\frac{\mu}{\rho_s}}. \]
The infinitesimal strain and Cauchy stress tensors are:
\[ \boldsymbol{\varepsilon} =\frac{1}{2}\left[ \nabla\mathbf{u}+(\nabla\mathbf{u})^{\mathsf T} \right], \qquad \boldsymbol{\sigma} =\lambda(\nabla\mathbin{\cdot}\mathbf{u})\mathbf{I} +2\mu\boldsymbol{\varepsilon}. \]
At either fluid-solid interface, let \(\mathbf{n}_s\) point from the solid shell into the adjacent fluid. The fluid stress is \(\boldsymbol{\sigma}_f=-p_f\mathbf{I}\). Continuity of traction is then the vector condition:
\[ \boldsymbol{\sigma}\mathbf{n}_s =-p_f\mathbf{n}_s. \]
Separating that relation into normal and tangential parts gives:
\[ \mathbf{n}_s\mathbin{\cdot} \boldsymbol{\sigma}\mathbf{n}_s=-p_f, \qquad (\mathbf{I}-\mathbf{n}_s\mathbf{n}_s^{\mathsf T}) \boldsymbol{\sigma}\mathbf{n}_s=\mathbf{0}. \]
The second condition states that an inviscid fluid applies no tangential traction. It does not require the tangential solid displacement to vanish. Continuity of normal velocity supplies the remaining kinematic condition:
\[ \mathbf{v}_f\mathbin{\cdot}\mathbf{n}_s =-i\omega\mathbf{u}\mathbin{\cdot}\mathbf{n}_s. \]
Written in terms of fluid pressure, the same condition is:
\[ \frac{1}{i\omega\rho_f}\partial_{n_s}p_f =-i\omega u_n, \qquad u_n=\mathbf{u}\mathbin{\cdot}\mathbf{n}_s. \]
At the outer shell surface, \(p_f=p_1^{\mathrm{tot}}\). At the inner surface, \(p_f=p_3\). The normal \(\mathbf{n}_s\) points radially outward at \(r=a\) and radially inward at \(r=b\). Stating the normal this way removes any ambiguity in the pressure-traction sign.
For a solid elastic target with no core, these conditions are imposed only at the exterior surface. Regularity at the origin excludes singular longitudinal and transverse radial solutions. The resulting fluid-solid coupling is the starting point for classical elastic-sphere and elastic-cylinder scattering theory (Faran 1951).
For an axisymmetric shell, normal traction, normal velocity, and one tangential traction condition are imposed at each radius. These six scalar conditions couple the exterior acoustic field, both elastic potentials in the shell, and the interior acoustic field (Goodman and Stern 1962). They admit resonances governed by shell thickness, density, longitudinal and transverse wave speeds, and fluid loading. This is the essential distinction from an acoustic shell, which has no transverse potential and cannot support shear or bending motion (Stanton 1990).
A compressible Newtonian viscous fluid supports deviatoric stress as well as pressure. If \(\eta\) is the shear viscosity, \(\zeta\) is the bulk viscosity, and \(\mathbf{D}\) is the rate-of-deformation tensor, its Cauchy stress is (Pierce 1989):
\[ \boldsymbol{\sigma}_v =-p_v\mathbf{I} +2\eta\mathbf{D} +\left(\zeta-\frac{2}{3}\eta\right) (\nabla\mathbin{\cdot}\mathbf{v}_v)\mathbf{I}. \]
The rate-of-deformation tensor is:
\[ \mathbf{D} =\frac{1}{2}\left[ \nabla\mathbf{v}_v+(\nabla\mathbf{v}_v)^{\mathsf T} \right]. \]
At an interface with an inviscid acoustic fluid, let \(\mathbf{n}\) point from the viscous medium into the inviscid fluid. Normal traction and normal velocity are continuous:
\[ \mathbf{n}\mathbin{\cdot}\boldsymbol{\sigma}_v\mathbf{n} =-p_f, \qquad \mathbf{v}_v\mathbin{\cdot}\mathbf{n} =\mathbf{v}_f\mathbin{\cdot}\mathbf{n}. \]
Because the inviscid fluid cannot apply shear traction, the tangential traction on the viscous side vanishes:
\[ (\mathbf{I}-\mathbf{n}\mathbf{n}^{\mathsf T}) \boldsymbol{\sigma}_v\mathbf{n}=\mathbf{0}. \]
At a bonded viscous-fluid and elastic-solid interface, both velocity and traction are continuous. With \(\mathbf{n}\) used consistently on both sides, the phasor conditions are:
\[ \mathbf{v}_v=-i\omega\mathbf{u}_s, \qquad \boldsymbol{\sigma}_v\mathbf{n} =\boldsymbol{\sigma}_s\mathbf{n}. \]
These vector conditions couple normal and tangential motion. Viscous compressional and shear branches are generally attenuating, so the interface can broaden and damp resonances that would be sharper in a lossless elastic shell (Feuillade and Nero 1998).
Boundary conditions do not introduce circumferential waves as separate terms. Instead, they determine which wave families the complete field solution can support and how those waves couple to the exterior fluid. Geometry, frequency, material properties, and attenuation then control their phase, propagation distance, and resonance structure (Überall 1973).
| Phenomenon | Boundary setting | Physical interpretation |
|---|---|---|
| Franz or creeping waves | Smooth convex rigid or pressure-release boundaries | Exterior diffracted waves that follow the surface into the geometrical shadow while continually radiating into the surrounding fluid. |
| Rayleigh-type waves | Fluid-loaded elastic solids and sufficiently thick elastic shells | Surface-localized elastic motion formed from coupled longitudinal and transverse fields. |
| Lamb-type waves | Thin or moderately thin elastic shells | Dispersive guided shell waves whose motion depends on the conditions at both shell surfaces. |
| Whispering-gallery waves | Penetrable fluid or elastic targets | High-order internal or surface-guided waves retained near the boundary by repeated refraction or reflection. |
| Glory | Spherical or nearly axisymmetric targets | A directional enhancement produced when circumferential contributions converge coherently near a symmetry direction. It is an interference or focusing effect, not a separate wave family. |
These labels describe identifiable contributions within an exact modal solution. They are not additional boundary conditions. For example, the longitudinal and transverse potentials of an elastic target already contain its surface and guided-wave contributions once the traction and velocity conditions have been enforced. Separating those contributions generally requires modal phase, resonance-pole, time-domain, or asymptotic analysis. Surface-elastic-wave interpretations of elastic-sphere and elastic-cylinder backscatter provide a classical example (Marston 1988).
Weak fluid-like scattering is an approximation regime, not a replacement for the fluid-fluid transmission conditions. For spatially varying density and compressibility, the source-free pressure equation can be written as:
\[ \nabla\mathbin{\cdot} \left(\frac{1}{\rho}\nabla p\right) +\omega^2\kappa p=0, \qquad \kappa=\frac{1}{\rho c^2}. \]
When medium 2 differs only slightly from medium
1, its total internal field may be replaced at first order
by the incident field:
\[ p_2(\mathbf{x})\approx p_1^{\mathrm{inc}}(\mathbf{x}). \]
The density and sound-speed ratios must remain close to unity:
\[ g_{21}=\frac{\rho_2}{\rho_1}\approx1, \qquad h_{21}=\frac{c_2}{c_1}\approx1. \]
The corresponding density and compressibility perturbations are:
\[ \gamma_\rho =\frac{\rho_2-\rho_1}{\rho_2} =1-\frac{1}{g_{21}}, \qquad \gamma_\kappa =\frac{\kappa_2-\kappa_1}{\kappa_1} =\frac{1}{g_{21}h_{21}^2}-1. \]
The weak-contrast assumption requires:
\[ |\gamma_\rho|\ll1, \qquad |\gamma_\kappa|\ll1. \]
The scattered field is then the first-order response to the residual density and compressibility contrasts (Chu and Ye 1999). Strong impedance jumps, internal multiple scattering, and resonant behavior violate this approximation even though the exact fluid-fluid boundary conditions remain valid.
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