Plane waves in isotropic fluids and solids - Propagation of Sound in Porous Media - page 12

or, using the notation ∇ for the gradient operator

u = ∇ ϕ + ∇ ∧ ψ (1.67)

The rotation vector in Equation (1.17) is then equal to

1
2 ∇ ∧ ∇ ∧ ψ

= (1.68)

Therefore, the scalar potential involves dilatation while the vector potential describes
infinitesimal rotations.
In the absence of body forces, the displacement equation of motion (1.46) is

ρ ∂ 2 u
∂t 2

2 u

= (λ + µ) ∇∇ · u + µ ∇ (1.69)

Substitution of the displacement representation given by Equation (1.67) into Equation
(1.69) yields

2 [ ∇ ϕ + ∇ ∧ ψ ] + (λ + µ) ∇∇ · [ ∇ ϕ + ∇ ∧ ψ ] = ρ ∂ 2

µ ∇

∂t 2 [ ∇ ϕ + ∇ ∧ ψ ]

(1.70)

2 ϕ ,

In Equation (1.70), ∇ · ∇ ϕ can be replaced by ∇ ∇ · ∇ ∧ ψ = 0, allowing this

equation to reduce to

µ

∂t 2

ρ ∂ 2 ρ ∂ 2

2 2 ϕ 2 ϕ 2

µ ∇ ∇ ϕ + λ ∇ ∇ + µ ∇ ∇ −

∂t 2 ∇

ϕ + ∇ − ∇ ∧ ψ = 0 (1.71)

2 2 ϕ and 2 2

By using the relations ∇ ∇ ϕ = ∇ ∇ ∇ ∇ ∧ ψ = ∇ ∧∇ ψ , Equation (1.71)

can be rewritten

(λ

∂t 2

µ

∂t 2

ρ ∂ 2 ϕ ρ ∂ 2

ψ

2 ϕ 2

∇ + 2 µ) ∇ − + ∇ ∧ ∇

ψ −

= 0 (1.72)

From this, we obtain two equations containing, respectively, the scalar and the vector
potential

∂ 2 ϕ
∂t 2

ρ

2 ϕ

∇ = (1.73)

λ + 2 µ

∂ 2

ρ
µ

ψ
∂t 2

2

∇

ψ =

(1.74)

Equation (1.73) describes the propagation of irrotational waves travelling with a wave
number vector k equal to

2 µ)) 1 / 2

k = ω(ρ/(λ +

(1.75)

The phase velocity c is always related to the wave number k by Equation (1.62). The
quantity K c defined as

K c = λ + 2 µ (1.76)