peat8002 - seismology lecture 5: surface waves and dispersion

PEAT8002 - SEISMOLOGYLecture 5: Surface waves and dispersion

Nick Rawlinson

Research School of Earth SciencesAustralian National University

Surface wavesIntroduction

The high frequency component of a seismic wavetraingenerated by an earthquake is usually dominated by thearrivals of P and S waves, with later arrivals often causedby multipathing/scattering as a result of heterogeneousstructure.However, most broadband seismograms are dominated bylarge, much longer period (lower frequency) waves thatarrive after the P and S waves. These waves are calledsurface waves, and propagate along the surface of theEarth.Surface waves arise from the interaction of elastic waveswith the free surface, and are composed of P and S wavesin a linear combination, the amplitude of which decays withdepth within a medium.


As a result of geometric spreading in 2-D, the energycarried by surface waves decays with distance r from thesource as 1/r ; this is in contrast to 1/r2 for body waves,which is why they are usually much less prominent on aseismogram.Two types of surface waves, known as Love waves andRayleigh waves after their discoverers, propagate near theEarth’s surface.Rayleigh waves result from a combination of P and SVmotions, while Love waves result from SH waves trappednear the surface.Due to their slower rate of decay, surface waves can circlethe globe many times following a large earthquake.

Surface wavesRayleigh waves

In order to quantify the properties of Rayleigh waves,suppose that the wave propagates in the x-direction andthat displacement is modulated by some functionaldependence on depth defined by f (z):

ui(x , z, t) = f (z) exp[i(kx − ωt)]

If u(x , z, t) satisfies the P or S wave equation with phasevelocity c = α or c = β:

∂2u∂t2 = c2∇2u

then substitution of u(x , z, t) = f (z) exp[i(kx − ωt)] yields:

−ω2f (z) = c2[−k2f (z) + f ′′(z)]


This can be re-written as:

f ′′(z) = κc2f (z)

where

κc2 = k2

[1−

( ω

kc

)2]

The above second order homogeneous linear differentialequation has a general solution of the formf (z) = c1 exp(κcz) + c2 exp(−κcz), where c1 and c2 areconstants.This means that f (z) is an exponential function of depthwith scale length 1/κc .


The phase speed of the surface wave is by definitioncs = ω/k , so from before:

κc2 = k2

[1−

(cs

c

)2]

If cs < c, then κc is real. In this case, we set c1 = 0 andtake the positive value of κc (z increases with depth); thenegative value implies exponential growth with increasingdepth, which cannot occur.If, on the other hand, cs > c, then κc is imaginary, implyingthat the wave propagates down and away from the surfacewith increasing time i.e. it is a body wave rather than asurface wave (see Lecture 4).


A necessary requirement for the propagation of surfacewaves is that its apparent velocity along the surface mustbe less than either of the P- or S-wave components ofwhich it is constructed.The only type of surface wave which propagates along thesurface of a uniform medium is a Rayleigh wave. Rayleighwaves can be represented using the potential functions Φand Ψ:

Φ = −ia exp[καz + i(kx − ωt)]

Ψ = b exp[κβz + i(kx − ωt)]

If there is no component of SH displacement, and the waveonly propagates in the x-direction, then the vector b isparallel to the y -axis, and we can write its nonzerocomponent as the scalar b.


The relative amplitudes of a and b will be determined bythe free-surface boundary conditions.The displacement is related to the potentials byu(x , z, t) = ∇Φ +∇×Ψ, so

ux(x , z, t) = ka exp[καz+i(kx−ωt)]−κβb exp[κβz+i(kx−ωt)]

uz(x , z, t) = −iκαa exp[καz+i(kx−ωt)]+ikb exp[κβz+i(kx−ωt)]

At the free surface z = 0, the stress-free boundaryconditions apply. In order to utilise these constraints, thestrain components need to be computed.


∂ux

∂x= ik2a exp[καz + i(kx − ωt)]− ikκβb exp[κβz + i(kx − ωt)]

∂uz

∂x= kκαa exp[καz + i(kx − ωt)]− k2b exp[κβz + i(kx − ωt)]

∂ux

∂z= kκαa exp[καz + i(kx − ωt)]− κβ

2b exp[κβz + i(kx − ωt)]

∂uz

∂z= −iκα

2a exp[καz + i(kx−ωt)]+ ikκβb exp[κβz + i(kx−ωt)]

At the surface z = 0, the tangential stress condition is:

σxz = µ

(∂ux

∂z+

∂uz

∂x

)= 0


Substitution of the displacement gradient terms produces(noting that z = 0):

2kκαa− (k2 + κβ2)b = 0 (1)

The normal stress condition at z = 0 is

σzz = λ

(∂ux

∂x+

∂uz

∂z

)+ 2µ

∂uz

∂z= 0

Since ρα2 = λ + 2µ and ρβ2 = µ, this can be written:

(α2 − 2β2)∂ux

∂x+ α2 ∂uz

∂z= 0


Substitution of the displacement gradient terms yields(again noting that z = 0):

(α2 − 2β2)(ik2a− ikκβb) + α2(−iκα2a + ikκβb) = 0

which can be re-organised as:

[(α2 − 2β2)k2 − α2κα2]a + 2β2kκβb = 0

This can be further simplified by using the definitionsκα

2 = k2 − (ω/α)2 and κβ2 = k2 − (ω/β)2, which produce:[

−2k2 +

(ω

β

)2]

a + 2kκβb = 0


Finally, we have that:

(k2 + κβ2)a− 2kκβb = 0 (2)

Equations (1) and (2) comprise a pair of simultaneoushomogeneous equations:[

2kκα −(k2 + κβ2)

(k2 + κβ2) −2kκβ

] [ab

]= 0

and a non-trivial solution occurs only if the determinant ofthe coefficient matrix is equal to zero.Therefore,

(k2 + κβ2)2 = 4k2κακβ


Substitution for κα and κβ yields:(1− ω2

2k2β2

)4

=

(1− ω2

k2α2

) (1− ω2

k2β2

)The above equation enables us to obtain a constraint onthe possible combinations of wavenumber and frequency,the so-called dispersion relation.Substitution of ζ = ω2/k2β2 = cR

2/β2 (where cR is thephase velocity of the Rayleigh wave) and η = β2/α2 intothe above expression results in:(

1− ζ

2

)4

= (1− ζη)(1− ζ)


This can be expanded to produce:

ζ3 − 8ζ2 + 8ζ(3− 2η)− 16(1− η) = 0

Note that we have divided through by ζ and have thusneglected the solution ζ = 0. Thus, a cubic equationremains to be solved.For a Poisson solid (λ = µ, which is often a reasonableassumption for the Earth), η = λ/(λ + 2µ) = 1/3, and theonly real root of the cubic is ζ = 0.8453.The phase velocity of Rayleigh waves across ahomogeneous medium is then simply proportional to theshear wave velocity β, and is independent of frequency:

cR =ω

k= β

√0.8453


If the dispersion relation is satisfied, the relative amplitudeof the P and SV components is then obtained fromEquations (1) or (2):

b =k2 + κβ

2

2kκβa =

2kκα

k2 + κβ2 a

At z = 0, the real parts of the x and z components ofdisplacement are:

ux(x , t) = [ak−bκβ] cos(kx−ωt) = a(

ω2

2kβ2

)cos(kx−ωt)

uz(x , t) = [καa− kb] sin(kx − ωt)

= a(

2kκα

k2 + κβ2

) (ω2

2kβ2

)sin(kx − ωt)


Note that we have again used κα2 = k2 − (ω/α)2 and

κβ2 = k2 − (ω/β)2 to simplify.

Comparison of the two components of displacement atz = 0 shows that:

when ux is positive and maximum, uz is zero, butdecreasing (upward displacement) with increasing t .when uz is positive and maximum, ux is zero, but increasing(in the +ve x direction) with increasing t .

If each component had the same amplitude, the motionwould be circular, but the ratio of amplitudes2kκα/(k2 + κβ

2) defines the ellipticity.The motion of the Rayleigh wave is therefore retrogradeelliptical, because at the top of the orbit, it is opposite tothe direction of propagation.


Retrogradeparticlemotion

Rayleighwaveparticlemotion

particlemotion

Prograde

Dep

th

Propagation direction


Note that since κα > κβ, the P-wave component of themotion decays faster than the SV motion.Thus, at sufficiently great depth, SV motion dominates, andthe particle motion is prograde.Stonely waves are similar to Rayleigh waves, but maypropagate along the interface between a solid and a fluid.They are also non-dispersive.Note that Rayleigh waves also exist when the medium ismore complicated than a homogeneous half-space. In thiscase, rather than having a single apparent velocity for allfrequencies, cR is a function of frequency.

Surface wavesLove waves

Unlike Rayleigh waves, which have coupled P-SV typedisplacement, Love waves contain only SH motion.Love waves require a velocity structure that varies withdepth, and cannot exist in a uniform half-space. This isbecause the free-surface stress condition is incompatiblewith the propagation of an SH-type surface wave in thismedium.The simplest circumstance in which we can quantitativelyanalyse the propagation of Love waves is the case of aplane layer (density ρ1, shear velocity β1) overlying ahalf-space (ρ2, β2).


From before, we showed that a general type of surfacewave is characterised by a depth dependent function f (z)whose solutions are exponential functions of depth:

f (z) = A exp(κLz) + B exp(−κLz)

where

κL2 = k2

(1− cL

2

β2

)For the previous analysis of Rayleigh waves in ahalf-space, we eliminated one of the solutions which growsexponentially with depth, but in the case of a layer, bothsolutions must be retained.


For a wave propagating in the x direction, a generalexpression for the Love wave displacement within a layer istherefore:

uy (x , z, t) = [A exp(κ1z) + B exp(−κ1z)] exp[i(kx − ωt)]

noting that if the wave comprises only SH motion, only ay -component of displacement is present.In the half-space below the layer, the growing exponentialterm is omitted, which yields:

uy (x , z, t) = C exp(−κ2z) exp[i(kx − ωt)]

The vertical length scale terms are given by:

κ12 = k2

(1− cL

2

β12

)and κ2

2 = k2(

1− cL2

β22

)


If the displacement is to decay exponentially with depth inthe half-space (which is a condition for a surface wave),then κ2 is real and positive. Therefore, cL = ω/k < β2, theS-wavespeed in the half-space.As in the analysis for Rayleigh waves, we must matchtraction and displacement conditions at the upper freesurface. In this case, we must also ensure thatdisplacement and traction are continuous where the layerjoins the half-space.Only the tangential components of traction anddisplacement are non-zero for SH-type displacement.


Free stress at the upper surface (z = 0) implies thatσzy = 0, and therefore ∂uy/∂z = 0. From our expressionfor displacement within the layer:

∂uy

∂z= κ1[A exp(κ1z)− B exp(−κ1z)] exp[i(kx − ωt)]

which means that A = B.From our expression for displacement within thehalf-space, the displacement gradient is:

∂uy

∂z= κ2C exp(−κ2z) exp[i(kx − ωt)]


Continuity of displacement uy at the interface (z = H) gives

A[exp(κ1H) + exp(−κ1H)] = C exp(−κ2H)

which can be written as:

2A cos(iκ1H)− C exp(−κ2H) = 0 (1)

Continuity of shear stress at the interface (rememberingthat σyz = 2µeyz) z = H gives

µ1Aκ1[exp(κ1H)− exp(−κ1H)] = −µ2Cκ2 exp(−κ2H)

which can be written as:

2Aiµ1κ1 sin(iκ1H)− µ2Cκ2 exp(−κ2H) = 0 (2)


Equations (1) and (2) constitute a system of two linearhomogeneous equations for two unknowns A and C. Thesystem has a non-trivial solution if the determinant of thecoefficients of A and C is zero.[

2 cos(iκ1H) −exp(−κ2H)2iµ1κ1 sin(iκ1H) −µ2κ2 exp(−κ2H)

] [AC

]= 0

Taking the determinant,

−2µ2κ2 cos(iκ1H) exp(−κ2H)+2iµ1κ1 sin(iκ1H) exp(−κ2H) = 0

which can be expressed:

tan(iκ1H) =µ2κ2

iµ1κ1


If we now substitute

κ12 = k2

(1− cL

2

β12

)and κ2

2 = k2(

1− cL2

β22

)into the above equation and note that the conditionβ1 < cL < β2 must be enforced to ensure that all terms arereal, we get:

tan(ωH√

β1−2 − c−2

L ) =µ2

µ1

√c−2

L − β2−2√

β1−2 − c−2

L

In order to solve this equation for the unknown phase

velocity cL, we introduce the variable ζ = H√

β1−2 − c−2

L .


This gives

tan(ωζ) =µ2

µ1

√H2(β1

−2 − β2−2 − ζ2

ζ(3)

In order to determine where solutions exist, we can ploteach function on either side of the equality sign. A finitenumber of solutions for cL exist, depending on ω, β1, β2, ρ1,ρ2 and H.Where the two curves intersect, we get a value for ζ, andhence for cL.The solutions are called modes; for a given frequency,there are several modes, each with different apparentvelocity.

Surface wavesLove wave dispersion


In the previous figure, the leftmost solution, with the lowestcx , is called the fundamental model; the others are highermodes or overtonesThe fundamental mode is the lowest frequency mode, andis typically most important in surface waves generated bylarge earthquakes at teleseismic distances.Equation (3) is the dispersion relation, an implicit equationfor the phase velocity of the Love wave in terms offrequency ω.Love waves travel faster than Rayleigh waves, but becauseω appears in Equation (3), the phase velocity is frequencydependent, so Love waves are dispersive. Consequently, along train of waves can be observed from a teleseismicevent.

Surface wavesGroup velocity

As demonstrated earlier, Love waves are dispersive,because its apparent velocity along the surface varied withfrequency.When dispersive waves originate at a common source inspace and time, their arrival time depends on the phasevelocity at each frequency. The group velocity refers to thespeed at which the whole group of waves travel.Consider the sum of two harmonic waves with slightlydifferent angular frequencies and wavenumbers:

u(x , t) = cos(k1x − ω1t) + cos(k2x − ω2t)


If we assume that

ω1 = ω + δω ω2 = ω − δω ω >> δω

andk1 = k + δk k2 = k − δk k >> δk

Substitution into the previous equation yields:

u(x , t) = cos(kx +δkx−ωt−δωt)+cos(kx−δkx−ωt +δωt)

which can be simplified to

u(x , t) = 2 cos(kx − ωt) cos(δkx − δωt)


Thus, the sum of two harmonic waves is a product of twocosine functions. The second term has much lowerfrequency and wavenumber, and thus varies more slowlywith time and space.Therefore, we have a carrier wave with angular frequencyω and wavenumber k , on which a slower varying envelopewith angular frequency δω and wavenumber δk issuperimposed.Examination of when the phase of each term remainsconstant shows that each describes waves travelling atdifferent speed.The envelope or beat pattern propagates at the groupvelocity U = δω, while the carrier wave moves at the phasevelocity c = ω/k .


In general, it turns out that the group velocity can bewritten:

U =dω

dkSince ω = ck ,

U =dω

dk=

dckdk

= c + kdcdk

Alternatively, using wavelength η = 1/k ,

U = c − ηdcdη

sincedcdk

=dcdη

dη

dk.

peat8002 - seismology lecture 5: surface waves and dispersion

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