8 Interaction of Harmonic Waves of Different Types …
115
u =
∂ϕ
∂ x
−
∂ψ
∂z
, w =
∂ϕ
∂z
+
∂ψ
∂ x
.
(8.8)
Only harmonic waves with the frequency ω are considered
ϕ = ϕ a e
iωt
, ψ = ψ a e
iωt
. . .
(8.9)
Assuming the propagating waves be harmonic only, we obtain the following
dynamic equations:
ρω
2 u +
∂σ 11
∂ x
+ +
∂σ 13
∂z
= 0,
ρω
2 w +
∂σ 31
∂ x
+ +
∂σ 33
∂z
= 0,
(8.10)
Lame’s equation
ρω
2 u + (λ + μ)
∂θ
∂ x
+ μμu = 0,
ρω
2 w(λ + μ)
∂θ
∂z
+ μμw = 0.
(8.11)
Finally, for the scalar and vector potentials, we obtain
ϕ + k
2
1 ϕ = 0, ,ψ + k
2
2 ψ = 0, k j = ω/c j
(8.12)
Since the medium «2» is not bounded by the coordinate z, then the Somerfield
radiation condition acts as a boundary condition (Gorshkov et al. 2004).
∂ϕ
∂r
+ ik 1 ϕ = o
1
r
,
∂ψ
∂r
+ ik 2 ψ = o
1
r
, r → ∞
(8.13)
8.4 Incoming Wave
To describe plane harmonic wave, the flat one-dimensional stretch-compression wave
(ψ ≡ 0) (Gorshkov et al. 2004), spreading along positive direction of Oz axis with
amplitude of normal pressure, is being reviewed. In this case in first of ten Eq. (8.12),
we assume ϕ = ϕ(z). As a result, we have an equation, which depends on potential
amplitude, which solution satisfies the conditions (8.13).
ϕ
a + k
2
1 ϕ a = 0.
(8.14)