feb 26, 2008 1john anderson: ge/cee 479/679: lecture 11 earthquake engineering ge / cee - 479/679...
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Feb 26, 2008 1 John Anderson: GE/CEE 479/679: Lecture 11
Earthquake EngineeringGE / CEE - 479/679
Topic 11. Wave Propagation 1
John G. Anderson
Professor of Geophysics
Feb 26, 2008 2 John Anderson: GE/CEE 479/679: Lecture 11
Wave Propagation• What is the physics behind propagation of
seismic waves?
• Seismic waves propagate due to the elastic properties of the medium.
• Equation of motion in a homogeneous, linear elastic medium
• Solution in terms of P- and S- waves
Feb 26, 2008 3 John Anderson: GE/CEE 479/679: Lecture 11
Derivation of the wave equation
• Starting point: F=ma
• Let u(x,t) be the infinitesimal motion of a particle in an elastic medium.
• For motion in the ith direction, the right hand side is: ( )
2
2
321
,
dt
tuddxdxdx i xρ
Feb 26, 2008 4 John Anderson: GE/CEE 479/679: Lecture 11
Stress
• Force/unit area
• Use on/in notation
• Thus is the stress on the plane normal to the unit vector in the i direction, acting in the j direction.
ijσ
xi
xj
jiσ
ijσ
Feb 26, 2008 5 John Anderson: GE/CEE 479/679: Lecture 11
Infinitesimal strain:
• By definition, sum over repeated indices.
• Consider only that part of the motion that does not include whole-body rotation.
• Define⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂
=i
j
j
iij x
u
x
ue
2
1
33
22
11
dxx
udx
x
udx
x
udx
x
udu iii
jj
ii ∂
∂+
∂∂
+∂∂
≡∂∂
=
Feb 26, 2008 6 John Anderson: GE/CEE 479/679: Lecture 11
Hooke’s Laws
• 2nd main assumption
• Stress proportional to strain
• The Lamé constants are λ and μ.
• The dilatation is
ijijij eμλθδσ 2+=
332211 eeeeii ++==θ
Feb 26, 2008 7 John Anderson: GE/CEE 479/679: Lecture 11
Combining in F=ma
• In this equation, Xi is a body force acting on the point, if any.
•
( ) iii
i Xuxt
uρμ
θμλρ +∇+
∂
∂+=
∂
∂ 22
2
23
2
22
2
21
22
x
u
x
u
x
uu
∂∂
+∂∂
+∂∂
=∇
Feb 26, 2008 8 John Anderson: GE/CEE 479/679: Lecture 11
Key Concept 1
• General description of a propagating wave:
€
f t −x
v
⎛
⎝ ⎜
⎞
⎠ ⎟
Feb 26, 2008 9 John Anderson: GE/CEE 479/679: Lecture 11
Key Concept 2
• If:
• Then:
• Where: λ is the wavelength, f is frequency, and v is wave velocity.
€
f t −x
v
⎛
⎝ ⎜
⎞
⎠ ⎟= sin 2πf t −
x
v
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
€
v = fλ
Feb 26, 2008 10 John Anderson: GE/CEE 479/679: Lecture 11
Trial Solution Number 1
• Suppose
• Then:
• And:
• And:
( ) ⎟⎠
⎞⎜⎝
⎛ −=v
xtftxxxu 1
3212 ,,,
0=θ
⎟⎠
⎞⎜⎝
⎛ −=∂∂
+∂∂
+∂∂
=∇v
xtf
vx
u
x
u
x
uu 1
223
22
22
22
21
22
22 "
1
⎟⎠
⎞⎜⎝
⎛ −=∂∂
v
xtf
t
u 12
2
"
Feb 26, 2008 11 John Anderson: GE/CEE 479/679: Lecture 11
Substitute in the equation of motion
• This is true if
• General case, define as the speed of shear waves.
• The solution describes a shear wave traveling in the x1 direction.
⎟⎠
⎞⎜⎝
⎛ −=⎟⎠
⎞⎜⎝
⎛ −v
xtf
vv
xtf 1
21 ""
μρ
ρμ
=v
ρμβ =
Feb 26, 2008 12 John Anderson: GE/CEE 479/679: Lecture 11
Feb 26, 2008 13 John Anderson: GE/CEE 479/679: Lecture 11
Trial Solution Number 2
• Suppose
• Then:
• And:
• And:
( ) ⎟⎠
⎞⎜⎝
⎛ −=v
xtftxxxu 1
3211 ,,,
⎟⎠
⎞⎜⎝
⎛ −=∂∂
+∂∂
+∂∂
=∇v
xtf
vx
u
x
u
x
uu 1
223
12
22
12
21
12
12 "
1
⎟⎠
⎞⎜⎝
⎛ −=∂∂
v
xtf
t
u 121
2
"
⎟⎠
⎞⎜⎝
⎛ −=∂∂
+∂∂
+∂∂
=v
xtf
vx
u
x
u
x
u 1
3
3
2
2
1
1 '1
θ
Feb 26, 2008 14 John Anderson: GE/CEE 479/679: Lecture 11
Substitute in the equation of motion
( ) ⎟⎠
⎞⎜⎝
⎛ −+∂∂
+=⎟⎠
⎞⎜⎝
⎛ −v
xtf
vxv
xtf 1
21
1 ""μθ
μλρ
( ) ⎟⎠
⎞⎜⎝
⎛ −+⎟⎠
⎞⎜⎝
⎛ −+=⎟⎠
⎞⎜⎝
⎛ −v
xtf
vv
xtf
vv
xtf 1
21
21 ""
1"
μμλρ
( ) ⎟⎠
⎞⎜⎝
⎛ −+=⎟⎠
⎞⎜⎝
⎛ −v
xtf
vv
xtf 1
21 "
12" μλρ
Feb 26, 2008 15 John Anderson: GE/CEE 479/679: Lecture 11
Substitute in the equation of motion
• This is true if
• General case, define as the speed of compressional waves.
• The solution describes a compressional wave traveling in the x1 direction.
ρμλ 2+
=v
ρμλα 2+
=
( ) ⎟⎠
⎞⎜⎝
⎛ −+=⎟⎠
⎞⎜⎝
⎛ −v
xtf
vv
xtf 1
21 "
12" μλρ
Feb 26, 2008 16 John Anderson: GE/CEE 479/679: Lecture 11
Notes
• If λ=μ, which is the usual assumption for crustal rocks, then
•
βα 3=
Feb 26, 2008 17 John Anderson: GE/CEE 479/679: Lecture 11
Feb 26, 2008 18 John Anderson: GE/CEE 479/679: Lecture 11
Feb 26, 2008 19 John Anderson: GE/CEE 479/679: Lecture 11
The Free Surface
• Most observations are made on the surface.• Structures are mostly built on the surface.• So it is important to understand what
happens to a seismic wave when it impacts the free surface.
• Since incoming waves cannot propagate into the air, energy is reflected back downward.
Feb 26, 2008 20 John Anderson: GE/CEE 479/679: Lecture 11
The Free Surface: Key Concept
• S-waves can have two polarizations:– SH - wave motion is
parallel to the surface. Causes only horizontal shaking.
– SV - wave motion is oriented to cause vertical motion on the surface.
SH
SV
Motion in and out of the plane of this figure - hard to draw.
Motion perpendicular to the direction of propagation causes vertical motion of the free surface.
Feb 26, 2008 21 John Anderson: GE/CEE 479/679: Lecture 11
The Free Surface
• Consider an incoming, vertically-propagating S-wave.
• There must be a downgoing wave, as the upgoing wave alone cannot satisfy the boundary conditions.
• At the free surface, the stress is zero.
• Use this boundary condition to solve for udown
x3 ⎟⎟⎠
⎞⎜⎜⎝
⎛+=β
3xtgu up
?=downu
downup uuu +=1
Shear wave
x1
Feb 26, 2008 22 John Anderson: GE/CEE 479/679: Lecture 11
The Free Surface
• At the free surface, the stress is zero.
• τ31= τ32= τ33= 0
• For this S-wave, only τ31
can be non-zero anyplace.
• θ=0
• Thus τ31=0 implies e31=0
x3
⎟⎟⎠
⎞⎜⎜⎝
⎛+=β
3xtgu up
downup uuu +=1
Shear wave
x1
ijijij eμδθλτ 2+=
Feb 26, 2008 23 John Anderson: GE/CEE 479/679: Lecture 11
The Free Surface
• e31=0• Since
• and u3=0,
• The steps at the left show how this is used to solve for udown.
• The condition is only met if g(t)=h(t) at all times.
x3
⎟⎟⎠
⎞⎜⎜⎝
⎛+=β
3xtgu up
⎟⎟⎠
⎞⎜⎜⎝
⎛−=β
3xthu down
downup uuu +=1
Shear wave
x1
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂
=3
1
1
331 2
1
x
u
x
ue
03
1 =∂∂xu
Let
⎟⎟⎠
⎞⎜⎜⎝
⎛−′−⎟⎟
⎠
⎞⎜⎜⎝
⎛+′=
∂∂
ββββ33
3
1 11 xth
xtg
x
u
At x3=0, ( ) ( ) 0=′−′ thtg
Feb 26, 2008 24 John Anderson: GE/CEE 479/679: Lecture 11
The Free Surface
• Since g(t)=h(t), at x3=0, u1=2g(t).
• For a vertically incident S-wave, the amplitude at the free surface is double the amplitude of the incoming wave.
• This result holds for vertically incident P-waves also.
x3
⎟⎟⎠
⎞⎜⎜⎝
⎛+=β
3xtgu up
⎟⎟⎠
⎞⎜⎜⎝
⎛−=β
3xtgu down
downup uuu +=1
Shear wave
x1
So
Feb 26, 2008 25 John Anderson: GE/CEE 479/679: Lecture 11
The Free Surface: Summary• At the free surface, waves are reflected back
downwards. • For a vertically incident S-wave, the amplitude at the
free surface is double the amplitude of the incoming wave.
• This holds for vertically incident P-waves also.• Waves are still approximately doubled in amplitude
when incident angles are near vertical.• If there is a structure, the boundary conditions are
changed. Some energy enters the structure instead of being reflected back.
Feb 26, 2008 26 John Anderson: GE/CEE 479/679: Lecture 11
Two Media in Contact• Observe that when
waves travel from a solid of one velocity to different velocity, the direction changes.
• This has a large impact on the nature of seismic waves, since the Earth is highly variable.
111 ,, βμρ
222 ,, βμρ
i1
i2
Feb 26, 2008 27 John Anderson: GE/CEE 479/679: Lecture 11
Two Media in Contact• Boundary conditions:
“welded contact”
• This implies that displacement is continuous across the boundary.
• Also, that stress is continuous across the boundary.
111 ,, βμρ
222 ,, βμρ
i1
i2
Feb 26, 2008 28 John Anderson: GE/CEE 479/679: Lecture 11
Two Media in Contact
• Envision a wave of frequency f. It cannot change frequency at the boundary.
• Wavefronts are drawn perpendicular to direction of wave travel.
• Note how angle of incidence, i1 and i2 are defined.
111 ,, βμρ
222 ,, βμρ
i1
i2
12 ββ >
12 λλ >
wavefront
1λ
2λ
i1
i2
wavefront
A
B
2
2
1
1
λβ
λβ
==f
Feb 26, 2008 29 John Anderson: GE/CEE 479/679: Lecture 11
Snell’s Law: Two Media in Contact• Line segment AB is
common to two right triangles.
• The geometry leads to Snell’s Law:
111 ,, βμρ
222 ,, βμρ
i1
i2
12 ββ >
12 λλ >
wavefront
1λ
2λ
i1
i2
wavefront
A
B
11 sin iAB=λ
22 sin iAB=λ
2
2
1
1
λβ
λβ
==f
2
2
1
1 sinsin1
λλii
AB==
2
2
1
1 sinsin
ββii
=
2
2
1
1 sinsin
ββii
=
Feb 26, 2008 30 John Anderson: GE/CEE 479/679: Lecture 11
Two Media in Contact
• This way of drawing is consistent with horizontal layers in the Earth.
• Lower velocities near the surface imply wave propagation direction is bent towards the vertical as the waves near the surface.
111 ,, βμρ
222 ,, βμρ
i1
i2
2
2
1
1 sinsin
ββii
=
Feb 26, 2008 31 John Anderson: GE/CEE 479/679: Lecture 11
Example of a 3-component ground motion record. Note how the S-wave is dominantly showing up on the horizontal components, and the P-wave is strongest on the vertical component.
PS
Feb 26, 2008 32 John Anderson: GE/CEE 479/679: Lecture 11
Two Media in Contact• In addition to the “refraction” of
energy into the second medium, some energy is reflected back.
• The angle of reflection is equal to the angle of incidence.
• This brings up the issue: how is the energy partitioned at the interface?
111 ,, βμρ
222 ,, βμρ
i1
i2 i2
Incoming SH
Feb 26, 2008 33 John Anderson: GE/CEE 479/679: Lecture 11
Two Media in Contact• The energy partitioning is
determined by “reflection” and “transmission” coefficients.
• The coefficients are determined by matching boundary conditions
• For incoming SH waves, the form is relatively simple.
111 ,, βμρ
222 ,, βμρ
i1
i2 i2
A
T
R
A
TTransmission coefficient
Reflection coefficientA
R
Incoming SH
Feb 26, 2008 34 John Anderson: GE/CEE 479/679: Lecture 11
Two Media in Contact• These coefficients are not a
function of frequency.
• At most, the transmitted wave has an amplitude of 2 x the amplitude of the incoming wave.
• Going from a stiffer to a softer material, the transmission coefficient is never less than 1.0.
111 ,, βμρ
222 ,, βμρ
i1
i2 i2
A
T
R
1122
222
βρβρβρ+
=AT
Incoming SH
1122
1122
βρβρβρβρ
+−
=AR
Feb 26, 2008 35 John Anderson: GE/CEE 479/679: Lecture 11
Two Media in Contact• Going from a softer to a
stiffer material, the transmission coefficient is never more than 1.0.
• If there is a large impedence contrast from softer to stiffer, the transmission coefficient approaches zero.
111 ,, βμρ
222 ,, βμρ
i1
i2 i2
A
T
R
1122
222
βρβρβρ+
=AT
Incoming SH
1122
1122
βρβρβρβρ
+−
=AR
Feb 26, 2008 36 John Anderson: GE/CEE 479/679: Lecture 11
Two Media in Contact• The reflection coefficient is always less
than 1.0.
• In the limit of the two media being identical, the transmission coefficient is 1.0 and the reflection coefficient is 0.0.
• In the limit of a reflection from a much stiffer or much softer medium, the reflection coefficient approaches 1.0.
111 ,, βμρ
222 ,, βμρ
i1
i2 i2
A
T
R
1122
222
βρβρβρ+
=AT
Incoming SH
1122
1122
βρβρβρβρ
+−
=AR
Feb 26, 2008 37 John Anderson: GE/CEE 479/679: Lecture 11
Two Media in Contact• An important case is when waves
in a soft medium contact a stiff boundary.
• In this case, the reflection coefficient is almost 1.0 (actually -1.0), meaning that the energy is trapped in the softer material.
• This applies to energy in a sedimentary basin.
111 ,, βμρ
222 ,, βμρ
i1
i2 i2
A
T
R
1122
222
βρβρβρ+
=AT
Incoming SH
1122
1122
βρβρβρβρ
+−
=AR
Feb 26, 2008 38 John Anderson: GE/CEE 479/679: Lecture 11
Two Media in Contact• For an incoming SV
wave, the situation gets even more complex.
• In this case, both P- and SV-waves are transmitted and reflected from the boundary.
• The P- and SV-waves are coupled by the deformation of the boundary.
1111 ,,, αβμρ
1222 ,,, αβμρ
i1
i2 i2
Incoming SVReflected SV
Transmitted SV
Transmitted P
Reflected Pj2
j1
Generalized Snell’s Law
2
2
1
1
2
2
1
1 sinsinsinsin
ααββjjii
===
Feb 26, 2008 39 John Anderson: GE/CEE 479/679: Lecture 11
Two Media in Contact• For an incoming P wave,
the situation is similar to incoming SV.
• In this case also, both P- and SV-waves are transmitted and reflected from the boundary.
• The P- and SV-waves are again coupled by the deformation of the boundary.
1111 ,,, αβμρ
2222 ,,, αβμρ
i1
j2 i2
Incoming PReflected SV
Transmitted SV
Transmitted P
Reflected Pj2
j1
2
2
1
1
2
2
1
1 sinsinsinsin
ααββjjii
===
Generalized Snell’s Law
Feb 26, 2008 40 John Anderson: GE/CEE 479/679: Lecture 11
Two Media in Contact
• Lower velocities near the surface also imply that waves are bent towards the horizontal at depth.
111 ,, βμρ
222 ,, βμρ
i1
i2
2
2
1
1 sinsin
ββii
=
Feb 26, 2008 41 John Anderson: GE/CEE 479/679: Lecture 11
Realistic Earth Model
• Eventually, as the velocity increases with depth, rays are bent back towards the surface.
• Waves cannot penetrate into layers where β is too large.
111 ,, βμρ
222 ,, βμρ
i1
i2
2
2
1
1 sinsin
ββii
=
βi
psin
=
p is the “ray parameter. It is constant along the ray
β increases
Feb 26, 2008 42 John Anderson: GE/CEE 479/679: Lecture 11
Body Waves: Discussion
• The travel time curves of body waves can be inverted to find the velocity structure of the path.
Feb 26, 2008 43 John Anderson: GE/CEE 479/679: Lecture 11
Feb 26, 2008 44 John Anderson: GE/CEE 479/679: Lecture 11
Realistic Earth Model
• Due to Snell’s law, energy gets trapped near the surface.
• This trapped energy organizes into surface waves.
111 ,, βμρ
222 ,, βμρ
i1
i2
2
2
1
1 sinsin
ββii
=
β increases
Feb 26, 2008 45 John Anderson: GE/CEE 479/679: Lecture 11
Four types of seismic wavesBody WavesP Waves Compressional, Primary
S Waves Shear, Secondary
Surface WavesLove Waves
Rayleigh Waves
Feb 26, 2008 46 John Anderson: GE/CEE 479/679: Lecture 11
Surface Waves
• Love waves: trapped SH energy.
• Rayleigh waves: combination of trapped P- and SV- energy.
Feb 26, 2008 47 John Anderson: GE/CEE 479/679: Lecture 11
Surface Waves
• For surface waves, geometrical spreading is changed.– For body waves, spreading is ~1/r.– For body waves, energy spreads over the
surface of a sphere, but for surface waves it spreads over the perimeter of a circle.
– Thus, for surface waves, spreading is ~1/r0.5.
Feb 26, 2008 48 John Anderson: GE/CEE 479/679: Lecture 11
Surface Waves
• Depth of motion: body waves can penetrate into the center of the Earth, but surface waves are confined to the upper 1’s to 10’s of kilometers.
Feb 26, 2008 49 John Anderson: GE/CEE 479/679: Lecture 11
Surface Waves
• Body waves are not dispersed.• Surface waves are dispersed, meaning that
different frequencies travel at different speeds.• Typically, low frequencies travel faster. These
have a longer wavelength, and penetrate deeper into the Earth, where velocities are faster.
• Typically, Love waves travel faster than Rayleigh waves.
Feb 26, 2008 50 John Anderson: GE/CEE 479/679: Lecture 11
Feb 26, 2008 51 John Anderson: GE/CEE 479/679: Lecture 11
Feb 26, 2008 52 John Anderson: GE/CEE 479/679: Lecture 11
Feb 26, 2008 53 John Anderson: GE/CEE 479/679: Lecture 11
Surface Waves
• Body waves amplitudes do not diminish so rapidly with depth in the Earth.
• Surface waves amplitudes decrease rapidly, especially below a few kilometers (depending on the period).
• Surface wave dispersion curves can be inverted to find the velocity structure of the path crossed by the surface waves.
Feb 26, 2008 54 John Anderson: GE/CEE 479/679: Lecture 11
Feb 26, 2008 55 John Anderson: GE/CEE 479/679: Lecture 11
Surface Waves• Particle motion in S-waves is normal to the direction of
propagation.
• This is also true of Love waves.
• However, Love waves would show changes in phase along the direction of propagation that would not appear in vertically propagating S waves.
Feb 26, 2008 56 John Anderson: GE/CEE 479/679: Lecture 11
Surface Waves
• Motion of Rayleigh waves is “retrograde elliptical”.
Feb 26, 2008 57 John Anderson: GE/CEE 479/679: Lecture 11
Surface Waves
• These examples have all been from surface waves seen at teleseismic distances.
• Later on, we will see examples of surface waves seen at short distances, on strong ground motion records.