Seismic Waves

SEISMIC WAVES(Lecture NoteIEQ-02 HRWASON)

Earthquakes produce different types of seismic waves These waves travel through rock and provide an effective way to see events and structures deep inside the earth

Seismic waves are the waves of energy caused by the sudden breaking of rock within the earth or an explosion They are recorded on Seismographs Seismic waves produced by explosions have been used to map salt domes and other oil ndash bearing rocks faults (cracks in deep rock ) rock types etc

TYPES OF SEISMIC WAVESTwo basic types of elastic waves or seismic waves are generated by an earthquake these are body waves and surface waves These waves cause shaking that is felt and cause damage in various ways These waves are similar in many important ways to the

familiar waves in air generated by a hand clap or in water generated by a stone thrown into water1 Body WavesThe body waves propagate within a body of rock The faster of these body waves is called Primary wave (P-wave) or longitudinal wave or compressional wave and the slower one is called Secondary wave (S-wave) or shear wave

P-wave The P-wave motion same as that of sound wave in air alternately pushes (compresses) and pulls (dilates) the rock (Fig1) The motion of the particles is always in the direction of propagation The P-wave just like sound wave travels through both solid rock such as granite and liquid material such as volcanic magma or water It may be mentioned that because of sound like nature when P-wave emerges from deep in the Earth to the surface a fraction of it is transmitted into atmosphere as sound waves Such sounds if frequency is greater than 15 cycles per second are audible to animals or human beings These are known as earthquake sound

Fig 1 Diagram illustrating ground motion for body waves(a) P-wave (b) S-wave (Bolt 1999)

The relation between compressional or P-wave velocity (Vp) and the elastic constants E (Youngrsquos modulus) σ (Poissonrsquos ratio) K (bulk modulus) μ (rigidity modulus) λ (Lamersquos constant) and density ρ is given as follows

Although Lamersquos constants are convenient other elastic constants are also used From Hookersquos law we can obtain the following relations

1048587 10485871048587 1048587

1048587

Thus Vp can be expressed as

The above equations show that the P-wave velocity in ahomogeneous solid is a function only of elastic constants and density One might expect that the elastic constants would be relatively insensitive to pressure whereas density would increase with pressure This would mean that the velocity should decrease with depth of burial in the Earth In fact this is contrary to actual observations An explanation for such paradoxical observation is that with increase in density the elastic constants increase much more which cause higher velocity with higher density The variation of velocity with depth is reasonably systematic as we go to greater depth

Polarization of P-wave when propagating in a homogeneous and isotropic medium is linear In inhomogeneous Earth higher frequency waves are however affected they show irregular particle motion

S-wave It is known that the S-wave or the shear wave shears the rock sideways at right angle to the direction of propagation (Fig1) As shear deformation cannot be sustained in liquid shear waves cannot propagate through liquid materials at all The outer portion of Earthrsquos core is assumed to be liquid because it does not transmit shear waves from earthquakes The particle motion of the S-wave is perpendicular (transverse) to the propagation In Fig1 the particle motion of the S-wave is up and down in vertical plane it is named SV wave However S-wave may also oscillate in horizontal plane which is called SH wave

The relation between S-wave velocity Vs the elastic constants and density is given as

An alternative expression is

Motion of the medium in P-waves being longitudinal there is no polarization of a P-wave it is linear but S-waves being transverse are polarized A horizontally travelling S-wave if so polarized that the particle motion is all vertical then it is called an SV wave and if particle motion is all horizontal it is called SH wave The velocity ratio VPVS is

Either expression tells that the P-wave velocity is always greater than the S-wave velocity The ratio is always greater than 1 first because K and μ are always positive second because σ cannot be

greater than 12 in an ideal solid ( Poisson solid for which λ = μ) For most consolidated rocks VpVs ranges between 15 and 20

2 Surface WavesThe second general type of earthquake wave is called surface wave because its motion is restricted to near the ground surface Such waves correspond to ripples of water that travel across a lake The wave motion is located at the outside surface itself and as the depth below this surface increases wave displacement becomes less and less Surface waves in earthquakes can be divided into two types Love waves and Rayleigh waves The Love waves are denoted as LQ (or G) and the Rayleigh waves as LR (or R) While Rayleigh waves exist at any free surface Love waves require some kind of wave guide formed by velocity gradient Both conditions are fulfilled in the real Earth

Love Wave (LQ)The British mathematician AEH Love demonstrated that if an SH ray strikes a reflecting horizon near surface at post critical angle all the energy is lsquotrappedrsquo within the wave guide (Love 1911) These waves propagate by multiple reflections between the top and bottom surfaces of the low speed layer near the surface The waves

are called Love waves and denoted as LQ or G Its motion is same as that of the SH-waves that have no vertical displacement

It moves the ground from side to side in a horizontal plane parallel to Earthrsquos surface but at right angle to the direction of propagation (Fig 2) so the wave motion is horizontal and transverse

Fig 2 Diagram illustrating ground motion for surface waves(b) Rayleigh wave and (d) Love wave (Bolt 1999)

The Love wave velocity (VL) is equal to that of shear waves in the upper layer (VS1) for very short wave lengths and to the velocity of shear waves in the lower layer (Vs2) for very long wave-lengths ie velocity VS1 lt VLlt VS2

The effects of Love waves are result of the horizontal shaking which produces damage to the foundation of structures Love waves do not propagate through water it affects surface water only It causes the sides of the lakes and ocean bays to move backwards and forwards pushing the water sideways like the sides of a vibrating tank

Rayleigh Wave (LR)Rayleigh (1885) demonstrated that the surface boundary condition can be satisfied leading to the existence of a lsquocoupledrsquo and lsquotrappedrsquo P-SV wave travelling along the surface such as the Earth-air interface with a velocity lower than shear velocity and with an amplitude decaying exponentially away from the surface This second type of surface wave is known as Rayleigh wave

The free surface boundary equations yield the following equation from which velocity( c) of Rayleigh waves may be determined

c6β6 ndash 8 c4 β4 + c2 ( 24 β2 - 16 α2) ndash 16 ( 1- β2 α2) = 0

If we substitute the values c = β and c = 0 in the lhs of the above equation we obtain unity and - 16 ( 1- β2 α2) respectively this last expression is negative since β lt α Hence the Rayleigh equation has a real root of c lying between 0 and β When Poissonrsquos relation holds this equation yields three real roots of c2 β2 namely 4 ( 2+ 2 radic 3 ) and ( 2 ndash 2 radic3) The first two of these values are both greater than 3 and thus make both r and s real so that there could be no corresponding surface wave solutions The third value leads to the results

c = 092 β

Which shows that the speed of Rayleigh waves in a homogeneous isotropic perfectly elastic half space is 092 of the speed of S body waves in the medium

In general the surface waves with periods 3 to 60s are denoted R or LR Like rolling ocean waves the Rayleigh waves develop the particle motion both vertically and horizontally in a vertical plane pointed in the direction of wave propagation (Fig 2) Since Rayleigh waves generate from coupled P and SV waves the particle motion is always in vertical plane and due to phase shift between P and SV the particle motion is elliptical and retrograde (counter clockwise) with respect to the direction of propagation The amplitude of the motion decreases exponentially with depth below the surface

For short wave-lengths VRg corresponds to ~ 092 VS of the material comprising the surface layer For very long wave-lengths the VRL corresponds to ~ 092 VS of the substratum material since effect of the surface layer is negligible when most of the waves travel in the zone below it

As seen from the above equations S-wave is slower than P wave and Rayleigh wave is slower than Love wave Thus as the waves radiate outwards from an earthquake source the different types of waves separate out from one another in a predictable pattern An illustration of the pattern seen at a distant place is exemplified in Figs 3 amp 4

Fig 3 Seismogram showing P PP S LQ and LR phases (Kuthanek 1990

3 SEISMIC WAVES AND GROUND SHAKINGThe body waves (P and S-waves) when move through the layers of rock in the crust are reflected andor refracted at the interfaces between the rock types or layers

When P and S-waves reach the surface of the ground most of their energy is reflected back into the crust Thus the surface is affected simultaneously by upward and downward moving waves After a few shakes a combination of two kinds of waves is felt in ground shaking

A considerable amplification of shaking occurs near the surface This surface amplification enhances the shaking at the surface of the Earth On the other hand earthquake shaking below ground surface say in the mine is much less

Again combination of two kinds of waves in shaking is not felt at sea The only motion felt on ship is from the P-waves because S-waves cannot travel through water beneath the ship

A similar effect occurs as sand layers liquefy in earthquake shaking which is appropriately known as liquefaction

There is progressive decrease in the amount of S-wave energy that is able to propagate through liquefied layers ultimately only P-wave can pass through it The above description is not adequate to explain the heavy shaking due to a large earthquake The effect of surface waves (Love wave and Rayleigh wave) and various kinds of mixed seismic waves including converted and reflected seismic phases at the rock interfaces complicate the matter and type of ground shaking is further muddled together The horizontal and transverse motion of the Love waves and elliptical and retrograde motion of the Rayleigh waves cause severe damage to the foundations of engineering structures and buildings The ground shaking is also much affected by soil conditions and topography For example in weathered surface

rocks in alluvium and water filled soil the amplification of seismic waves may increase or decrease as the waves reach the surface from the more rigid basement rocks

Also at the top or bottom of a ridge shaking may intensify depending on the direction from which waves are coming and whether the wavelengths are short or long The site amplifications play an important role in microzonation study (eg Field and Jacob 1995 Hartzell 1992) that identifies sites vulnerable for more damage by seismic waves

4 SEISMIC PHASES AT THE ROCK BOUNDARIES

Snellrsquos lawIt is easily deduced that

sin i sin r2 = V1 V2

where r2 is the angle of refraction V1 and V2 are the velocity of the upper and lower layer respectively This formula can be extended to the case of reflection or refraction of a wave of different type eg reflected or refracted S from an incident P leading to a generalised form of Snellrsquos law

V sin i = constant (p) where V stands for either VP or VS on either side of the boundary and i is the angle between the corresponding ray (incident reflected or refracted) and the normal on the same side and p is

called seismic parameter or ray parameter The ray parameter is constant for the entire travel path of a ray The consequence of a ray traversing material of changing velocity V is a change in incidence angle i with respect to a reference plane As the ray enters material of increasing velocity the ray is deflected toward the horizontal Conversely as it enters material of decreasing velocity it is deflected towards the vertical If the ray is traversing vertically then p = 0 and the ray will have no deflection as velocity changes

The body waves (the P and S-waves) are reflected or refracted at the interfaces between rock types In addition to reflection or refraction of one type the seismic waves are also converted to other types As illustrated in Fig5 P-wave travels upwards and strikes the bottom of a layer of different rock type part of its energy will pass through the upper layer as P-wave and part as converted S-wave which is known as P to S conversion (or PS phase) and part of energy will be reflected back downwards as P and S waves Similar reflection refraction and conversion may occur with S-wave All these converted phases are useful for velocity and geological structure study

Fig 5 Seismic waves generated by an incident P-wave at the boundary between two rocks

Huygensrsquo PrincipleThe new direction of a ray-path can be inferred from Huygensrsquo principle It states that an incident ray say on a rock boundary may be treated as new source about which new hemispherical wave fronts expand on each side of the boundary (Fig6) Since each of these elementary wave fronts corresponds to only an infinitesimal amount of energy a physically realistic wave front consists of a surface to which an infinite number of them are tangent Figure 6 illustrates an incoming plane wave It strikes the rock boundary at point A at time t1 and becomes active as an infinitesimal source By the time the incident wave front reaches B at time t2 wave fronts from A have spread hemispherically into both media It may be noted that while

the radius of the hemisphere is V1 (t2 ndash t1) in the upper layer it is V2 (t2 ndash t1) in the lower medium The figure further suggests that proportionately smaller hemispheres exist about all points between A and B The slopping planes which are tangential to these physically represent real wave fronts The new wave directions reflected or refracted are normal to these planes

Fig 6 Reflected and refracted wave fronts at a medium boundary by Huygenrsquos principle

The upper part of the diagram (Fig 6) demonstrates the law of reflection ie angle of incidence i is equal to angle of reflection r1

The lower part of the diagram demonstrates the law of refraction or Snellrsquos law

Special Cases1 Case of incident SH waves

When the incident wave is of SH type then the reflected and refracted waves can only be SH type

When tanf(prime) is imaginary it follows that |CC0 | = 1 Hence in this case there is total reflection of the incident wave but with a change of phase given byδ [C=C0 exp(-iδ)] For the angle of emergence cos (inverse)(vvprime) which gives fprime a zero value there is total reflection with zero change of phase

If μprime = 0ie if medium Mprime is fluid or vaccum there is complete reflection without change of type or phase

2 P wave incident against a free plane boundary For normal incidence we have e=f=π2 and for grazing

incidence e = 0 There exits a reflected disturbance of SV type for all angles of incidence except zero and π2

For e = 128 and 30 degree there are no reflected waves of P type

For 2deg lt e lt 63 deg at least half of the reflected energy is in the SV type

3 SV wave incident against a free plane boundary There are no reflected or refracted SH waves If e = 128 deg or 30 deg ie if f = 557 deg or 60 deg

there is little reflection in the SV type e is imaginary if 0 le f le cos -1(1 radic3) = 547 deg For

this range of values of f there is then complete reflection in the SV type with a change of phase in general e

Polarization

At appreciable distances from an earthquake or explosion source S seismic waves can be treated as plane waves in which the particle of the medium vibrate in planes π normal to the direction of wave advance As a result of prior reflections or refractions of the waves at one or more boundaries the particle vibrations may be confined to straight lines in the plane π and the waves are then raid to be polarised (plane polarised) similarly to the case of optical waves The polarised waves are referred to as SH when the particles vibrate horizontally and as SV when they vibrate in vertical planes

Case of Plane Waves

Sufficiently distant from the source of an initially confined disturbance the waves may be regarded as plane This approximation is relevant to many seismological problems for the distance of a station recording the local seismic displacements is often great compared with the dimensions of the initially disturbed region In this case called the ldquofar ndash fieldrdquo the displacements associated with the P and S waves are in effect longitudinal and transverse respectively

The theory of plane waves may be setup independently of the scalar and vector wave equations by making a trial substitution of the form

ui = Ai exp ik (ljxj ndash ct)

where lj2=1 in equation the equation of motion in terms of

displacements On eliminating the three Ai it is found that the above form is a possible solution of equations of motion if and only if the square of the speed c obeys a cubic equation whose roots are α2 β2 β2 where α(VP) β(VS) are given by

The speeds α β are found as before to be associated with longitudinal and transverse waves respectively it follows that the types of waves described in the last paragraph are the only possible types of body plane waves

It emerges also that the two types P and S are independent of each other and further that the latter may be plane polarised In seismology when an S wave is polarised so that all particles of the substance move horizontally during its passage it is denoted SH when the particles all move in vertical planes containing the direction of propagation the wave is denoted SV

Dispersion

Dependence of wave velocity on wave number and hence on the wavelength and period implies

that the shape of the disturbance will in general continually change as time goes on since each simple harmonic constituent will now travel with a wave velocity special to itself

If the initial disturbance is confined to a finite range of values of x and the medium is unlimited it follows that as time goes on there will be a continual spreading out of the disturbance into trains of waves This phenomenon is called dispersion

Scattering

Deflections of a portion of wave energy occur when elastic waves encounter an obstacle or a region in which the elastic properties of the medium

differ from values outside the region (small scale heterogeneities) As a result a seismic wave incident on an anomalous region in the earth will produce in addition to an undisturbed plane wave an interfering scattered wave that will spread out from the obstacle in all directions

Small scale heterogeneities cause scattering that partitions the high-frequency wave field into a sequence of arrivals that are often called coda waves Scattering can also decrease the amplitude of a seismic phase by shifting energy from the direct arrival back into the coda

Seismic Diffraction

Diffraction is defined as the transmission of energy by non-geometric ray paths In optics the classic example of diffraction is light ldquoleakingrdquo around the edge of an opaque screen In seismology diffraction occurs whenever the radius of curvature of a reflecting interface is less than a few wavelengths of the propagating wave Whenever

the wave strikes an edge or end of a reflecting surface this edgecorner of the boundary acts like a secondary source (Huygensrsquo principle) and radiates energy forward in all directions The waves produced are called diffracted waves The phenomenon is called seismic diffraction

Attenuation Thus far we have been concerned with the elastic properties of the Earth in our discussion of wave propagation In an idealized purely elastic Earth geometric spreading and the reflection and transmission of energy at boundaries control the amplitude of a seismic pulse Once excited these waves would persist indefinitely The real Earth is not perfectly elastic and propagating waves attenuate with time due to various energy-loss mechanisms The successive conversion of potential energy (particle velocity) to kinetic energy (particle velocity) as a wave propagates is not perfectly reversible and other work is done such as movement along mineral dislocation or shear heating at grain boundaries that taps the wave energy We usually describe these processes collectively as internal friction and we ldquomodelrdquo the internal-friction effects with phenomenological

descriptions because the microscopic processes are complex