SEISMOLOGY, Lecture 3

8/22/2019 SEISMOLOGY, Lecture 3

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PEAT8002 - SEISMOLOGY

Lecture 3: The elastic wave equation

Nick Rawlinson

Research School of Earth SciencesAustralian National University
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The Elastic Wave EquationEquation of motion

The equation of motion can be derived by considering thetotal force applied to a volume element V with surface area

S.

V

We require the sum of the surface force field, derived from

the tractions T, and the external forces to balance the

mass acceleration in the volume, i.e. F = ma.
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The force balance equation can be written as:

FSURFACE + FBODY = FTOTAL

T(n) = limA

0 FSA(n)

FS =

S

T(n)dS

FB =

V gdV

The bottom integral for FB occurs because F = mg (whereg is the acceleration due to gravity) and the density is

= m/V, so dFB = gdV.
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The total force, where f is the local acceleration field, isthen given by

FT =

V

fdV

Therefore, the complete force balance equation is:S

T(n)dS+

V

gdV =

V

fdV

Since f = 2u/t2 and T(n) = niT(xi) = n,S

ndS+

V

gdV =

V

2u

t2dV

Th El i W E i


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Now, according to the divergence theorem(transformation between volume integrals and surface

integrals),

S ndS= V dVTherefore we can write:

V

+ gdV =

V

2u

t2dV

V

+ g

2u

t2

dV = 0

Th El ti W E ti
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The integrand must be zero since this relationship holds for

any arbitrary control volume, so

+ g = 2u

t2

In the absence of body forces (not valid for very low

frequencies e.g. normal modes) and away from the source,

= 2u

t2

which is the equation of motion.

Th El ti W E ti
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The Elastic Wave EquationDerivation

We can use our expression for the equation of motion andthe relationship between stress and strain in an isotropic

elastic medium to derive the elastic wave equation.

In other words, we can substitute:

ij = ij + 2eij

into

= 2u

t2

For fully anisotropic media, we would need to substitute

the expression ij = Cijklekl into the equation of motion,where {Cijkl} represent the 21 independent elastic moduli.

Th El ti W E ti


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Substitution of the isotropic stress-strain relationship into

the equation of motion yields:

[ ]i =ijxj

=

xj[ij + 2eij]

Taking the first term on the right hand side,

xj[ij] =

xi[]

which occurs because ij = 1 only if j = i.Application of the product rule to the second term yields:

xj[2eij] = 2

eijxj

+ 2eij

xj

The Elastic Wave Equation
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From the previous lecture, we showed that strain is related

to displacement by

eij =1

2 uix

j

+ujx

i

Putting these results together produces

[ ]i =

xi[] +

xj uixj

+ujxi+ 2eij

xj

Each of the four summed terms on the RHS of this

equation can be reorganised as follows:

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Term 1:

xi[]

= ()

Term 2:

2uixjxj =

2u

xjxj=

2uxx2

+2uxy2

+2uxz2

2uyx2

+2uyy2

+2uyz2

2uzx2

+2uzy2

+2uzz2

T

= 2u



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Term 3: it is assumed that u has continuous second partial

derivatives.

xj

ujxi

=

xi

ujxj

=

xux

x +uyy +

uzz

y

uxx

+uyy

+uzz

z

uxx

+ uyy

+ uzz

x( u) =

xi( u)

= ( u) =



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Note that the last equality occurs because:

u =uxx

+uyy

+uzz

=

Term 4:eij

xj

= e

Putting all four terms together yields the elastic wave

equation:

2u

t2= () + 2u + + 2e = u

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The elastic wave equation can also be written completely

in terms of displacement:

u = (u)+[u+(u)T]+(+2)(u)u

It applies to small displacements from an equilibrium state

in an elastic medium.

The first two terms on the RHS contain gradients of the

Lam parameters. They are non-zero if the material isinhomogeneous (i.e. contains velocity gradients).

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It can be shown that the strength of and tend tozero when the wave frequency is high compared to the

scale length of the variation in and . This approximationis used in most ray theoretical methods.

Thus, if we ignore the gradient terms, we get:

u = ( + 2)( u) u

which is strictly valid for homogeneous isotropic media.

Using the above equation, we can show that elastic media

support both compressional (P) and shear (S) waves.

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The Elastic Wave EquationPotential field representation

Rather than directly solve the wave equation derived on

the previous slide, we can express the displacement fieldin terms of two other functions, a scalar (x, t) and avector (x, t), via Helmholtz theorem

u = +

In this representation, the displacement is the sum of the

gradient of a scalar potential and the curl of a vector

potential.

Although this representation of the displacement field

would at first appear to introduce complexity, it actuallyclarifies the problem because of the following two vector

identities

() = 0

( ) = 0

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Due to the form of the RHS of the wave equation, these

two vector identities separates the displacement field intotwo parts:

: No curl or rotation and gives rise to compressionalwaves.

: Zero divergence; causes no volume change and

corresponds to shear waves.

If we now substitute u = + into

u = ( + 2)( u) u

we obtain:

2

t2( + ) = (+ 2)(2) ( )

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The right hand side of the above equation comes from thefact that if we take the divergence of the displacement field

then:

u = + = 2

Likewise, if we take the curl of the displacement field, then

u = +

= ( ) 2

= 2

noting the use of the vector identity

2v = ( v) v which is valid for any vector v.

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It is also assumed without loss of generality that = 0- the vector potential has zero divergence. This can be

done since taking the curl discards any part of the vector

potential that would give rise to a non-zero divergence.

Using the same vector identity as before, the second term

on the RHS of the wave equation can be simplified as

follows:

() = 2()+(() = 2()

since the divergence of the curl is zero.

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This now allows the wave equation to be re-organised as

follows:

( + 2)2 2

t2

=

2

2

t2

One solution to the above equation can be obtained by

setting both bracketed terms to zero. This yields two wave

equations, one for each potential.

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The Elastic Wave EquationP-waves

In this case, the scalar potential satisfies:

2

t2=

( + 2)

2

which clearly has the form of a scalar wave equation i.e.

2f

t2= c22f

The speed at which this type of wave propagates is

therefore given by:

=

+ 2

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The Elastic Wave EquationS-waves

Similarly, the the vector potential satisfies:

2

t2=

2

which clearly has the form of a vector wave equation i.e.

2v

t2= c22v

The speed at which this type of wave propagates istherefore given by:

=

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The Elastic Wave EquationP- and S- wave properties

The oscillatory variable (u = ) of P-waves orcompressional waves is volumetric change, which is

directly related to pressure.

In contrast, the oscillatory variable u = of S-waves

implies a shear disturbance with no volume change( = 0).

Although the above derivation is strictly valid for high

frequency waves in isotropic media, observational studies

confirm that P-waves invariably travel faster than S-wavesand hence always correspond to the first-arrivals on a

seismogram.

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qP- and S- wave properties

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qP- and S- wave properties

Another interesting property of these waves that isconfirmed by observation is that P-waves can propagate

through a liquid but S-waves cannot. This is because

liquids do not support a shear stress ( = 0), so =

/

and = 0.

Primary seismological evidence for the existence of a liquid

outer core is based on the differing behaviour of P- and

S-waves.

Note that in realistic media (i.e. the Earth), P- and S-wave

motions do not completely decouple in the way describedabove, but in many applications, this approximation is

acceptable.
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SEISMOLOGY, Lecture 3

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