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L. Igumnov et al.
8.8 Computing of the Fourier Coefficients for the Potentials
in Ambient Media
The wave equations in the potentials are also represented through the Fourier
coefficients (8.30):
∂
2
ϕ
(l)
n
∂z 2 + ϕ
(l)
n
k
2
1 − λ
2
n
= 0,
(8.34)
∂
2
ψ
(l)
i n
∂z 2 + ψ
(l)
in
k
2
1 − λ
2
n
= 0, i = 1, 2.
Solution of these equations must satisfy the Somerfield condition written in the
Fourier coefficients as (8.30).
∂
2
ϕ
(l)
n
∂z 2 + sign(k 1 − λ n )κ
2
1n ϕ
(l)
n = 0 (n ≥ 1),
∂
2
ψ
(l)
in
∂z 2 + sign(k 2 − λ n )κ
2
2n ψ
(l)
in = 0 (n ≥ 0),
κ jn =
k
2
j − λ 2
n
.
(8.35)
General solution of the wave Eq. (8.34) is given by the following way
ϕ
(1)
n (z, ω) = C 11n (ω)
e
iκ 1n (ω
2
)z H (k 1 − λ n ) + e
κ 1n (ω
2
)z H (λ n − k 1 )
,
ψ
(1)
n (z, ω) = C 21n (ω)
e
iκ 2n (ω
2
)z H (k 2 − λ n ) + e
κ 2n (ω
2
)z H (λ n − k 2 )
,
ϕ
(2)
n (z, ω) = C 12n (ω)
e
−iκ 1n (ω
2
)z H (k 1 − λ n ) + e
−κ 1n (ω
2
)z H (λ n − k 1 )
,
ψ
(2)
n (z, ω) = C 22n (ω)
e
−iκ 2n (ω
2
)z H (k 2 − λ n ) + e
−κ 2n (ω
2
)z H (λ n − k 2 )
, (8.36)
To determine the constants C 11n (ω), C 21n (ω), C 12n (ω), C 22nm (ω), we have to use
the conditions of the contact between the plate and the medium (8.26), (8.27) for
the plane wave and (8.28), (8.29) for the cylindrical wave. This requires that the
stresses, deformations, and displacements will be expressed in term of potentials.
Then, the normal and tangential displacements of the medium are found based on
Eq. (8.8). Further, the vibration acceleration (8.1), (8.2) and displacements (8.8) are
determined.