Theory of Elastic Waves
Part 3 - Body Waves
Daniel Köhn
based on the lecture notes by Gerhard Müller
November 18, 2014
Theory of Elastic Waves Part 3 - Body Waves
1
Plane body waves
2
Spherical waves from explosive point sources
3
Spherical waves from single force and dipole point sources
Spherical waves from single force point sources
Spherical waves from dipole force point sources
4
References
Plane body waves
1D wave equations
1D equations for longitudinal and transversal waves
Longitudinal (P-)waves:
∂ 2 ux
1 ∂ 2 ux
=
,
∂x 2
α2 ∂t 2
α2 =
λ + 2µ
,
ρ
Transversal (SH-)waves:
∂ 2 uy
1 ∂ 2 uy
=
,
∂x 2
β 2 ∂t 2
β2 =
µ
.
ρ
General 1D wave equation
∂2u
1 ∂2u
=
.
∂x 2
c 2 ∂t 2
(1)
Plane body waves
Solution of the general 1D wave equation
u(x, t) =
F (x − ct)
| {z }
wave propagating in positive x-direction
+
G (x + ct)
| {z }
wave propagating in negative x-direction
where F(x) and G(x) are any twice differentiable functions,
c is positive and time t grows.
Another form is
x
x
u(x, t) = F t −
+G t +
.
c
c
,
Plane body waves
Initial value problem for plane waves
We search solutions of the 1D wave equation (1), which satisfy the
initial conditions:
u(x, 0) = f (x) for the displacement and
∂u
(x, 0) = g (x) for the displacement velocities
∂t
Solution of the initial value problem for plane waves
Z x+ct
1
1
u(x, t) = {f (x − ct) + f (x + ct)} +
g (ξ)dξ.
2
2c x−ct
Plane body waves
u(x, t) = f (x − ct)
Plane body waves
u(x, t) = f (x + ct)
Plane body waves
u(x, t) =
1
{f (x − ct) + f (x + ct)}
2
Spherical waves from explosive
point sources
Spherical waves from explosive point sources
Wave equation in spherical coordinates in case of radial symmetry
∂ 2 (r Φ)
1 ∂ 2 (r Φ)
=
∂r 2
α2 ∂t 2
with the non-zero compressional potential
Z r
Φ(r , t) =
U(r 0 , t)dr 0
and no contribution from the shear potential
→
−
Ψ = (0, 0, 0)
Spherical waves from explosive point sources
Solution for waves propagating in positive r-direction
1 r
Φ(r , t) = F t −
r
α
or in terms of the displacement in r-direction
U(r , t) =
∂Φ
1 r
1
r
= − 2F t −
− F0 t −
∂r
|r
{z α } |r α {z α }
near-field term
far-field term
→ geometrical spreading of the spherical waves
Spherical waves from explosive point sources
Spherical waves
Spherical waves from explosive point sources
Near-field vs. Far-field contribution
Spherical waves from single
force and dipole point sources
Spherical waves from single force point sources
Single force point sources
So far we neglected any source terms in the wave equation
1 ∂2Φ
=0
α2 ∂t 2
→
−
→
−
1 ∂2 Ψ
−rot rot Ψ − 2
=0
β ∂t 2
∇2 Φ −
Now we introduce source terms:
ϕ
1 ∂2Φ
=−
α2 ∂t 2
λ + 2µ
→
−
→
−
→
−
1 ∂2 Ψ
ψ
−rot rot Ψ − 2
=−
β ∂t 2
µ
∇2 Φ −
(2)
Spherical waves from single force point sources
Single force point sources
z
K(t)
x
y
→
−
f = (0, 0, δ(x) δ(y ) δ(z) K (t))
which can be decomposed in terms of potentials:
→
−
→
−
f = grad ϕ + rot ψ
Spherical waves from single force point sources
Single force point sources
ϕ(x, y , z, t)
1
4π
ZZZ
+∞
=
ZZZ
−∞
+∞
=
1
4π
=
−
→
ψ (x, y , z, t)
−∞
K (t)z
,
4πr 3
1
(z − ζ)δ(ξ)δ(η)δ(ζ)K (t)dξdηdζ
r 03
r2 = x2 + y2 + z2
→
−
→ −
f × r0
dξdηdζ
r 03
1
4π
ZZZ
+∞
=
ZZZ
−∞
+∞
=
1
4π
=
K (t)
(−y , x, 0)
4πr 3
−∞
→
−
→ −
f · r0
dξdηdζ
r 03
1
{−(y − η), x − ξ, 0} δ(ξ)δ(η)δ(ζ)K (t)dξdηdζ
r 03
Spherical waves from single force point sources
Single force point sources
Substitution into wave equations (2):
1 ∂2Φ
α2 ∂t 2
1 ∂ 2 Ψx
∇2 Ψx − 2
β ∂t 2
1 ∂ 2 Ψy
∇2 Ψy − 2
β ∂t 2
∇2 Φ −
K (t)z
4πρα2 r 3
K (t)y
=
4πρβ 2 r 3
K (t)x
= −
4πρβ 2 r 3
= −
Spherical waves from single force point sources
Single force point sources
Solutions for the inhomogenous wave equation
∇2 a −
1 ∂2a
= f (x, y , z, t)
c 2 ∂t 2
for vanishing initial conditions
Z Z Z +∞
1
1
r0
a(x, y , z, t) = −
f
ξ,
η,
ζ,
t
−
dξdηdζ
0
4π
c
−∞ r
with
r 02 = (x − ξ)2 + (y − η)2 + (z − ζ)2 .
Kirchhoff Equation
Spherical waves from single force point sources
Single force point sources
Details of the solution for Φ are described in [Müller, 2007] pp. 51
- 53. We can write the potentials for the single force point source
as

R αr

z

K
(t
−
τ
)τ
dτ
Φ(x, y , z, t) = 4πρr

3 0








r

R

y
β
Ψx (x, y , z, t) = − 4πρr 3 0 K (t − τ )τ dτ 






(3)
r
Rβ

x

Ψy (x, y , z, t) = 4πρr
3 0 K (t − τ )τ dτ












Ψz (x, y , z, t) = 0





2
2
2
2

r
= x +y +z
Spherical waves from single force point sources
Single force point sources
Introduce spherical coordinates
z
K(t)
P
υ r
x
λ
x = r sin ϑ cos λ
y = r sin ϑ sin λ
z = r cos ϑ
y
Spherical waves from single force point sources
Single force point sources
The shear potential has no ϑ- and r-component.
According to p. 54 in [Müller, 2007] we get:
=
cos ϑ
4πρr 2
R
r
α
Ψλ (r , ϑ, t) =
sin ϑ
4πρr 2
R
r
β
Φ(r , ϑ, t)
0
0
K (t − τ )τ dτ
K (t − τ )τ dτ
Spherical waves from single force point sources
Single force point sources
Finally the displacement vector is calculated according to
→
−
→
−
u = gradΦ + rot Ψ :
ur
=
uϑ =
∂Φ
∂r
+
1 ∂Φ
r ∂ϑ
1
∂
r sin ϑ ∂ϑ (sin ϑΨλ )
−
1 ∂
r ∂r (r Ψλ )
uλ = 0
ur and uϑ depend on both compressional and shear potentials.
Spherical waves from single force point sources
Single force point sources
Far-field contributions:
ur
'
uϑ '
cos ϑ
K
4πρα2 r
− sin ϑ
K
4πρβ 2 r
t−
t−
r
α

(longitudinal P − wave) 



r
β



(transversal S − wave) 
(4)
Spherical waves from single force point sources
Single force point sources
Polar plot of the far-field contributions
90
120
S−Abstrahlung
60
P−Abstrahlung
30
150
O
180
0
θ
P1
P2
210
330
240
300
270
Polarisationsrichtung der S−Welle
Spherical waves from dipole force point sources
Dipole force point sources
K(t)
11
00
11
00
111
000
00
11
00
11
00
11
ε
1
0
1
0
0
1
1
0
0
1
1
0
0
1
−K(t)
Dipole with moment (single couple) and 2 dipoles without joint
momentum (double couple)
Spherical waves from dipole force point sources
Single couple point sources
z
P(x,y,z)
y
r
K(t)
υ
x0 ε
λ
x
-K(t)
We can calculate the far-field displacements ux , uy , uz for K(t)
from Eq. (4)
Spherical waves from dipole force point sources
Single couple point sources
z
P(x,y,z)
y
r
K(t)
υ
x0 ε
λ
x
-K(t)
Far-field displacements ux0 , uy0 , uz0 for -K(t) are estimated by Taylor
series expansion:
∂ux
0
ux = − ux +
...
∂x0
Spherical waves from dipole force point sources
Single couple point sources
z
P(x,y,z)
y
r
K(t)
υ
x0 ε
λ
x
-K(t)
... and superposition of the solutions:
ux00 = ux + ux0 = −
∂ux
∂x0
Spherical waves from dipole force point sources
Single couple point sources
Far-field solution in polar coordinates
ur
2ϑ cos λ 0
= − sin8πρα
t−
3r M
uϑ =
sin2 ϑ cos λ 0
M
4πρβ 3 r
t−
uλ = 0
with the moment function
lim K (t) = M(t)
→0
r
β











r
α










(5)
Spherical waves from dipole force point sources
Single couple point sources
Polar plot of the far-field contributions
90
120
60
150
30
P
− +
180
0
+ −
210
330
S
240
300
270
Spherical waves from dipole force point sources
Double couple point sources
Far-field solution in polar coordinates
ur
uϑ
uλ



t−
=









r
2ϑ cos λ 0
= − cos4πρβ
M
t
−
3r
β










cos ϑ sin λ 0
r

= 4πρβ 3 r M t − β
2ϑ cos λ 0
− sin4πρα
3r M
r
α
(6)
Spherical waves from dipole force point sources
Double couple point sources
Polar plot of the far-field contributions
90
120
60
150
30
+
−
180
P
0
−
+
210
330
S
240
300
270
Next lecture:
26th of November 2014
References
Müller, G. (2007).
Theory of elastic waves.
available at
https://www.geophysics.zmaw.de/fileadmin/documents/teaching/Skripten/tew 022013.pdf.