feb 26, 2008 1john anderson: ge/cee 479/679: lecture 11 earthquake engineering ge / cee - 479/679...

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


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


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


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


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 ∂

∂+

∂∂

+∂∂

≡∂∂

=


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


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

∂∂

+∂∂

+∂∂

=∇


Key Concept 1

• General description of a propagating wave:

€

f t −x

v

⎛

⎝ ⎜

⎞

⎠ ⎟


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


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

"


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 =


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



( ) ⎟⎠

⎞⎜⎝

⎛ −+∂∂

+=⎟⎠

⎞⎜⎝

⎛ −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



• 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


Notes

• If λ=μ, which is the usual assumption for crustal rocks, then

•

βα 3=


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.


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.


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


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+=


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


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


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.


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


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


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


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

=



• 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

=


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


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


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


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


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


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


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


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

===


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



• 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

=


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


Body Waves: Discussion

• The travel time curves of body waves can be inverted to find the velocity structure of the path.


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


Four types of seismic wavesBody WavesP Waves Compressional, Primary

S Waves Shear, Secondary

Surface WavesLove Waves

Rayleigh Waves


Surface Waves

• Love waves: trapped SH energy.

• Rayleigh waves: combination of trapped P- and SV- energy.


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.


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.


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.


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.


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.


Surface Waves

• Motion of Rayleigh waves is “retrograde elliptical”.


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.

feb 26, 2008 1john anderson: ge/cee 479/679: lecture 11 earthquake engineering ge / cee - 479/679...

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