ABSTRACT
In the previous chapter we have seen that the dynamics of elastic, acoustic, and electromagnetic waves can be expressed in the form of wave equations. In classical linear elastodynamics without body forces we have the wave equations
∇ · [C : ∇u] = ρ u¨ ; c2p ∇2φ= φ¨ ; c2s ∇ · (∇ψ)T = ψ¨ . For a fixed frequency (ω) and in the absence of body forces the wave equations of elastodynamics have the forms
ρ−1 ∇ · [C : ∇u]+ω2 u= 0 ; c2p ∇2φ+ω2 φ= 0 ; c2s ∇ · (∇ψ)T +ω2 ψ = 0 . The wave equations in linear acoustics are
c20 ∇ · (∇v) = v¨ ; c20 ∇2 p = p¨ ; c20 ∇2φ= φ¨ . The acoustic wave equations at fixed frequency have the forms
κ ∇ · [ρ−10 ∇p]+ω2 p = 0 ; c20 ∇2 p+ω2 p = 0 ; c20 ∇2φ+ω2 φ= 0 . We have also seen that the electromagnetic wave equations can be written as
c2 ∇2φ= φ¨ ; c2 ∇2A= A¨
c2 (∇ f −∇×∇×Πe) = Π¨e ; c2 (∇ f −∇×∇×Πm) = Π¨m . At fixed frequency we have the electrodynamic wave equations
c2 ∇2E+ω2 E= 0 ; c2 ∇2H+ω2 H= 0
c2 ∇2φ+ω2 φ= 0 ; c2 ∇2A+ω2 A= 0 .
