336 T. Chouwhere ρ is the bulk fluid density, ρg ≡−ρgẑ is the external body force density due togravity, p is the dynamic pressure, and η(r,t) is the surface deformation as a function<strong>of</strong> surface coordinate r ≡ (x, y). We neglect all dynamical effects <strong>of</strong> the upper fluid,air. By assuming irrotational flows, the velocity is potential, v ≡ ∇ϕ. Hence,∇·v=∇ 2 ϕ(r,z,t)=(∇ 2 ⊥+∂z)ϕ(r,z,t)=0, 2 z < η(r), (2.2)where ∇ 2 ⊥ ≡ ∂2 x + ∂y 2 is the two-dimensional Laplacian in the coordinates <strong>of</strong> figure 1.The linearized kinematic conditions at the free interface and the impenetrable bottom(ẑ · v(z = −h) =0)arelim ∂ zϕ(r,z,t)=∂ t η (2.3)z→0 −andlim ∂ zϕ(r,z,t)=0, (2.4)z→−h +respectively. The linearized normal fluid stress at the interface isP nn ≡ lim ρ [∂z→0tϕ + gη(r,t)]≃P− zz , (2.5)where we have assumed the absence <strong>of</strong> externally imposed stream flow and that theonly disturbances are in the form <strong>of</strong> dynamic surface waves. The fluid stress P nnis balanced by material restoring forces such as those due to surface bending andstretching. In the absence <strong>of</strong> fluid viscous stresses, the application <strong>of</strong> surface stressesmust be carefully applied to ensure conservation <strong>of</strong> ẑ-component forces, torques, andbending moments. These forces and the energy are manifestly conserved when thesurface stresses are derived from an energy functional for the bulk fluid plus interface.For example, P zz and gradients <strong>of</strong> σ(r) have to be considered in inviscid fluids toconserve wave energy. Thus, the net ẑ-component surface stress balance reads[∇⊥ · (σ(r)∇ ⊥ ) − ∇ 2 ⊥(D(r)∇ 2 ⊥) ] η(r,t)=P zz (2.6)which upon taking the time derivative and using (2.3) becomes[lim ∇⊥ · (σ(r)∇ ⊥ ) − ∇ 2 ⊥(D(r)∇ 2 ⊥) ] ∂ z ϕ(r,z,t)=∂ t P zz , (2.7)z→0 −where D(r) and σ(r) are the surface flexural rigidity and surface tension respectively. In(2.7), we have neglected the rotational inertial terms in the surface bending stresses,valid when ω 2 ≪ Eρ −1s /λ 2 (ρ s is the bulk mass density <strong>of</strong> the surface material).Within thin plate theory, D = Ed 3 /12(1 − s 2 ) where E and s are Young’s modulusand Poisson’s ratio <strong>of</strong> the surface material in its bulk phase.Consider dynamical variables with a time dependence <strong>of</strong> the form e −iωt . In thefrequency domain, we combine the time derivative <strong>of</strong> the ẑ-component stresses anduse (2.3) and (2.5) to obtain[lim ρω 2 ϕ(r,z)− ( ρg − ∇ ⊥ · (σ(r)∇ ⊥ )+∇ 2 ⊥(D(r)∇ 2 ⊥) ) ∂ z ϕ(r,z) ]z→0 − []≡ ρω 2 lim −L(r,ω) ϕ(r,z)=0. (2.8)z→0 −Equations (2.8) and (2.2) determine the velocity potential with frequency ω. Theeffects <strong>of</strong> spatially varying surface properties are implicit in the boundary condition(2.8). In the limit <strong>of</strong> uniform σ(r) =σand D(r,t)=D,ϕ∝e ±ik·r cosh k(h + z), fromwhich we obtain the standard gravity–capillary–flexural wave dispersion relationω 2 =(gk + σ ρ k3 + D )ρ k5 tanh kh, (2.9)
Band structure <strong>of</strong> surface flexural–gravity waves 337where k = |k| =2π/λ. This relation is valid only for uniform surfaces or very far(many wavelengths) from localized spatial inhomogeneities <strong>of</strong> the surface parametersσ(r) and D(r).However, when σ and/or D are not uniform, surface waves can diffract or refractfrom the regions <strong>of</strong> varying surface properties and the Fourier modes <strong>of</strong> the velocitypotential at the interface mix with those <strong>of</strong> the surface variations. The remainder<strong>of</strong> this study deals with periodic variations in σ(r) and D(r) where the boundarycondition (2.8) is to be used to solve ∇ 2 ϕ(z
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