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Its solution have a view of ( A ϕ —if arbitrary constant)
ϕ a = A ϕ e
−ik 1 z
.
(8.15)
From here (8.5), we get that potential have a type of incoming wave
ϕ = A ϕ e
−ik 1( z−c 1 t)
(8.16)
Using this equation consistently in (8.4), (8.5) and (8.8), we get these formulas
for movement, deformation, and stresses.
u ≡ 0, w = −ik 1 A ϕ e
−ik 1( z−c 1 t)
, ε 11 = ε 13 ≡ 0, ε 33 = θ = −k
2
1 A ϕ e
−ik 1 (z−c1t)
,
σ 33 = −(λ + 2μ)k
2
1 A ϕ e
−ik 1( z−c 1 t)
= −ρω
2 A ϕ e
−ik 1 (z−c1t)
,
σ 11 = −λk
2
1 A ϕ e
−ik 1( z−c 1 t)
= −κρω
2 A ϕ e
−ik 1( z−c 1 t)
, σ 13 ≡ 0, κ =
λ
λ + 2μ
.
(8.17)
From this, accounting, that σ 33 | t=0, z=0 = p ∗ for movement and stress in incoming
wave, we get these formulas (only their amplitude values are present)
u = u ∗ ≡ 0, w = w ∗ =
ik 1 p ∗
ρω 2 e
−ik 1 z
=
i p ∗
ρc 1 ω
e
−ik 1 z
, A ϕ = −
p ∗
ρω 2 ,
σ 11 = σ 11∗ = κ p ∗ e
−ik 1 z
, σ 33 = σ 33∗ = p ∗ e
−ik 1 z
, σ 13 = σ 13∗ ≡ 0.
(8.18)
The cylindrical wave is being emitted from source place in point with coordinates O 1 (0,0, −d). Setting up cylindrical coordinates system with center in point O 1 ,
parallel to the Oz axis, and with radius
r 1 =
x 2 + (z + d)
2
(8.19)
Assuming, that ϕ = ϕ(r 1 ), from Eq. (8.12), we get following equations for this
function:
r
−1
1
r 1 ϕ
+ k
2
1 ϕ = 0
(8.20)
Its common view will be (Kostrov 1964; Ryl’ko 1977):
ϕ = A ϕ H
(2)
0 (k 1 r 1 ) + B ϕ H
(1)
0 (k 1 r 1 )
(8.21)
where H
(1)
ν (ζ ) and H
(2)
ν (ζ )—Hankel function with order ν, A ϕ and B ϕ —are arbitrary
constants.
Now, similarly to flat wave defining amplitude values of movements and stresses
in incoming wave