g 377 Wave Energy

8/19/2019 g 377 Wave Energy

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Chapter 3: Wave propagation fundamentals: From energy point of view, energy

partitioning at interfaces

Before pursuing further on discussing specific topics in seismic exploration to a

variety of applications, it is critical to understand the basic wave propagation phenomena.

In this chapter, we introduce the fundamental solutions for the wave equation we derivedin Chapter 2, and discuss the fundamental propagation phenomena.

First, we discuss the energy conservation in terms of plane wave, cylindrical wave,

and spherical wave, corresponding to 1D, 2D, and 3D propagations. This phenomenon isknown as geometric spreading.

Then, we discuss absorption, another main reason for having seismic energy

dissipation.Finally, by discussing the Snell’s law, the reflection, refraction and seismic phase

conversion at a material contrast interface, as well as the energy distribution, we discuss

the approximation by using the ray theory, or geometric wave propagation.

3.1 Wave front and the Huygen’s Principle

It is clear that the wave field at a particular moment is an extension and continuity of thefield in the last moment. By using the concept of wavefront , which is defined as the

surface on which the wave motion is in the same phase, this concept is used to describe

the Huygens’ Principle. Huygens’ principle states that every point on a wavefront can beregarded as a new source of waves. In Figure 3.1, the points P1, P2, P3, P4, and Pi, on the

wavefront AB can be regarded as the news sources. After an infinitesimal time of dt , the

waves from each of the new source propagated to a distance of vdt , and forms a newwavefront A’B’. Be aware of that between A and B there are infinite number of points so

the newly formed wave front is a continuous phenomenon.

A

A’

B

B’

P1

Pi

P4

P3

P2

..................

vdt

Figure 3.1 Illustration of the Huygens’ principle.


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From Figure 3.1, it is easy to see that the shape of the wave front depends on medium’s

velocity. For example, the wave generated by a point source propagates in a medium withconstant velocity, the wavefront is in a shape of a spherical shell.

3.2 Wave energy

A simplified case for the wave equation discussed in Chapter 2 is the plane wave

propagating in one direction, say the x-direction. In this case, the wave equation can be

written as

2

2

22

2 1

t

u

v x

u

∂

∂=

∂

∂ (3.1)

One solution for a plane wave propagating in an unbounded, uniform medium can be

expressed as

)sin()cos()](exp[ 000 kxt iukxt ukxt iuu +++=+= (3.2)

This plane wave can be viewed as the wave generated by a plane source occupying theentire yz-plane (x=0) to generate waves propagating in x-direction. For simplicity, let’s

only consider the real part of Eqn (3.2). In this equation, u0 is the amplitude, ω is theangular frequency; k is called the wave number. We will show the relationship of k with

respect to angular frequency ω, by demonstrating Eqn (3.2) does satisfy the 1-D waveequation (3.1). Taking the secondary derivative of u with respect to space, here the x-

coordinate, is

)cos(02

2

2

kxt uk x

u +−=∂∂ ω

and putting the second derivative of u with respect to time on the right hand side of Eqn

(3.1) gives

)cos(1

02

2

2

2

2kxt u

vt

u

v+−=

∂

∂ω

ω

comparing the last 2 equations leads to

vk =

It is clear that the wave number k in space domain defines how many revolutions in a unit

length scale, just like the angular frequency’s (ω) function in time domain, to define howmany revolutions in a unit time. They are linked through the propagation velocity v,

which is determined by the physical properties of the media the wave travels through. In


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the case discussed in this section, the wave does not experience any lose of energy.

Clearly, it is not realistic in the real world. We will discuss the energy equilibrium in thenext section.

3.2.1 Kinetic energy

At one particular position with r=r, the simplest wave displacement is not depending on

the position anymore and can be written as

t uu cos0=

the particle velocity at this point then is

0 sinu u t ω ω = −

(and what is the expression for acceleration?) the kinetic energy density is then

2 2 2 2

0

1 1sin

2 2k

m E u u

V t ρ ω = = ω

Please make sure you understand the difference of wave propagation velocity as it

passing through the media, and the particle motion velocity that only vibrating about an

equilibrium point of the particle itself. Since the function of sin(ω t) has a range of [-1, 1],

then sin2ω t varies in the range of [0 1]. This gives the maximum kinetic energy is

22

0max2

1ω ρ u E k =

3.2.2 Potential energy: Strain Energy

In elasticity, the elastic deformation (strain) is caused by applying stress, the product ofstress and strain, in analogue to work which is the product of force and displacement, in a

macroscopic sense, is in a dimension of energy. Since we are taking the case in a

microscopic sense, the energy actually is energy density, i.e., the energy in a unit volume

of the medium. The work done by stress is converted to elastic strain energy in the sameamount and stored in the medium. Elastic strain energy is a kind of potential energy due

to some kind of recovery force (gravity is a recovery force, and there is a gravity

potential).

]2[(2

1

])2[(2

1

)(2

1

2

1

ijijiiii

ijijij

zx zx yz yz xy xy zz zz yy yy xx xxijij E

ε µε ε λε

ε µε λθδ

ε σ ε σ ε σ ε σ ε σ ε σ ε σ

+=

+=

+++++==


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since iiijij ε ε δ = , then

ijijijijij

E

σ ε σ ε ε =∂

∂

=∂

∂

)(2

1

This expression means that for a given strain, if the stress applied is larger, the energystored is also large (This is due to the elastic modulus is larger).

The strain energy is a kind of the potential energy that stored in the elastic medium whenit is deformed. From physical principle, the summation of the potential energy and the

kinetic energy get to be a constant at any given moment. Let us examine the situation at

two particular moments: ω t=0, and ω t =π /2. At ω t=0, we have

00 cos ut uu == and

00sin0 =−== uuv

At ωt =π/2 we have

0cos0 == t uu

and

ω π

ω 00 2sin uuuv −=−==

This is to say that at the moment the displacement is in its maximum u0, the particle

velocity is zero, and the kinetic energy is zero, all energy has been stored as the elastic

strain energy. In contrast, when the displacement is zero, the velocity reaches its

maximum u0ω , and the kinetic energy is in its maximum, and all the elastic strain energy

has been released. The total energy at this point is

22

0max2

1ω ρ u E E E k pk ==+


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t

T Ek=0

Ep=Ekmax

Ek=EkmaxEp=0

u0

Figure 3.2, the balance of the kinetic energy and elastic potential energy at any moment.

3.2.3 Energy Intensity I

Energy intensity is the total energy flow through a unit area in a unit time, so that it is the

energy density we have learned above (times the volume, then divided by the area and

time). Imagine a cylinder, it happened have the wave energy propagation direction

coincides with the axis of the cylinder as shown in Figure 3.3 below

dA

vdt

k

Figure 3.3

so that


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vu EvdAdt

vdtdA E

dAdt

E I total

22

02

1 ρω ====

where v is the propagation velocity of the waves. It is clear that energy intensity can also be called as energy flow density. For the sake of energy conservation, we should expect

the total energy in the entire domain of the media at any moment should be a constant;the total amount of energy depends on the source has radiated. Now we can discuss thewaves with different type of sources and their relationship with energy and energy flow

density.

3.3 Geometric spreading: Spherical wave, cylindrical wave and planar wave

3.3.1 Point source –a 3D case

For the simplest case, i.e., a point source in an infinite homogeneous medium, we shouldexpect the following. Let’s imagine two wave fronts, which make 2 spherical shells with

the same origin (location of the source). The radius to the outer shell is r 2, which is

greater than that of the radius of the inner shell r 1. Thus, the surface areas of the outer andinner shells are 4πr 2

2 and 4πr 1

2, respectively. By energy conservation, the total energy

flowing through the outer shell and the inner shell at a given time should keep the sameso that we have

12

12

2

1

12

)(

2

1)(

2

1

)(

44

2

1

222

2

122

12

2

2

1

2

2

1122

12

ur

r u

ur

r u

I r r I

and

r I r I

S I S I

E E

=

=

=

=

=

=

ρω ρω

π π

to generalize this relation with the inner shell becoming a constant reference shell closer

to the source and the outer shell a generic one we have

0)(0

ur

r u =

This state that the amplitude is decaying against 1/r for the waves generated by a pintsource, since the shape of the wavefront is spherical, this is generally referred as the

geometric spreading for spherical waves.

3.3.2 Line source –a 2D case


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For an infinitely long line source, the shape of the wavefront is a cylinder; this is calledthe cylindrical wave

1

2

2 1

2 1

2 1

2 2 1 1

2 1

2 1

2 2 2 21

2

1

2

2 2

( )

1 1( )

2 2

E E

I S I S

I rL I rL

and

r I I

r

r u u

r

r u u

r

π π

ρω ρω

=

=

=

=

=

=

to generalize this relation with the inner shell becoming a constant reference shell closer

to the source and the outer shell a generic one we have

0

0 ur

r u =

This state that the amplitude is decaying against 1/√r for waves generated by a linesource, since the shape of the wavefront is cylindrical, this is generally referred as thegeometric spreading for cylindrical waves.

3.3.3 Plane source –a 1D case

If the source occupies the entire x=0 plane (as shown in the beginning of Section 3.2), the

shape of the wave front is planar, this wave is called the plane wave; there is no

amplitude decay for plane wave. In summary we can view the energy density flow ofdifferent waves as:

Point source: spherical wave 1/r decay;Line source: cylindrical wave 1/√r decay;Plane source: plane wave no decay;

In reality, the energy decays by the wave field occupying larger and larger volume andenergy in the unit volume become less and less when the wave propagating farther and

farther, this phenomenon is called geometric spreading.

The geometric spreading alone can not lead to the complete dead-off of the seismic wave

energy. The ultimate dead-off of the kinetic energy of seismic waves is due to the energy

absorption caused by the imperfection of the earth materials, i.e., the elastic energy has been completely transferred to earth mantle.


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3.4 Energy dissipation caused by absorption (intrinsic attenuation)

Absorption is the energy loss caused by the imperfection or defect of the material, in the

form of energy conversion from mechanic to thermal. This loss can be accounted for byusing the absorption coefficient α in the form as

r

0

α −= e A A

Now we introduce the concept of the quality factor Q, which is defined as the ratio of the

total elastic energy and the energy lost in one cycle, i.e.,

E

E Q

∆=

π 2

Q can be thought as: after how many cycles of vibration the elastic energy can be

dissipated, apparently larger Q means many cycles to dissipate the energy so that the

material tends to be more close to perfect or more purely elastic. In contrast, if only aftervery few cycles the energy is gone, the material is far more from elastic.

We have learned that the kinetic energy is proportional to the square of the amplitude,i.e., we have

22

2

1 A E ρω =

taking the reference point at point 1, so we have

∆=≈−=−≈

−≈

−+=

−≈

∆=

2)ln(2)1(2)(2

)(2))((2

2

1

1

2

1

21

2

1

211

2

1

2121

2

1

2

2

2

1

A A

A A

A A A

A

A A A

A

A A A A

A

A A

E

E

Q

π

Taylor expansion has been applied in the last step, since

)0.....(1

1ln >+−= x x

x

On the other hand, from the original definition of the absorption coefficient we have the

amplitudes at 2 points with only one cycle apart (one wavelength in space) can beexpressed as

)r (

02

r 01

0

0

λ α

α

+−

−

=

=

e A A

e A A

then

αλ αλ α α λ α α

λ α

α

eeeee A

e A

A

A==== ++−+−

+−

−

0000

0

0

r r )r (r

)r (

0

r

0

2

1

and


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αλ αλ == )ln()ln(2

1 e A

A

so we got that

αλ =∆

and

αλ

π π π =

∆=

∆=

2

22

E

E Q

This is the relation between the quality factor Q and the absorption coefficient α.

Finally, after consider the geometrical spreading and the absorption, a general form of the

solution of wave equation can be written as:

0 cos( )r

uu e t kr

r

α ω −= − for point source (3D propagation)

0 cos( )r u

u e t kr r

α ω −= −

−

for line source (2D propagation)

0cos( )r u u e t kr α ω −= for plane source (1D propagation)

Again, be make sure that the absorption is the mechanism responsible for complete dead-

off of seismic vibrations.

After we discuss geometric spreading and absorption, which occurs even for a uniform

medium, we need discuss energy partitioning at interfaces caused by heterogeneousmedium. This is the basis for diffraction, and scattering.

3.5 Diffraction, and its kinetic approximation: The ray theory, or geometric wave

propagation (Snell’s law)

3.5.1 Geometric wave theory, Snell’s law

The process of wave reflection may be defined as the return of all or part of a wave beamwhen it encounters the boundary between two media. The most important rule ofreflection is that the angle of incidence is equal to the angle of reflection., which is

known as the Snell’s law (actually the simplest expression of it). Where both these angles

are measured relative to an imaginary line which is normal to the boundary. Figure 3.4shows the situation of the incidence of a p-wave to a planar interface separating medium1

and medium 2.


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Figure 3.4. A plane p-wave impinging at the interface with ρ2v2 > ρ1v1.

The ray direction of the reflected P- and S-wave, the transmitted (refracted) P- and S-

wave are obeys the Snell’s law:

2

2

1

1 sinsin

vv

θ θ =

In the expression above the wave incidence and reflection/transmission are expressed bythe direction of rays. Only consider geometry, and kinetics – not consider the causes of

the deformation or motion. This is the essence of the geometric wave theory. It is an

approximation of the physical wave theory. The premise of this approximation is that thefrequency of the waveforms is assumed to be infinitely high, or the wavelength is very,

very short compared with the features it studies. No diffraction phenomenon is

considered in the treatment here, only reflections and refractions.

3.5.2 the Reflection Coefficient and transmission coefficient

Reflection is often quantified in term of the reflection coefficient ‘R’. R is defined simplyas the ratio of the reflected and incident wave amplitudes.

R = Ar / Ai

Where ‘Ai’ and ‘Ar’ are the incident and reflected wave amplitudes respectively. Thevalue of the reflection coefficient relates to the magnitude of reflection from the interface

between two media with different physical properties.

The acoustic impedance Z of the two media involved dictates the magnitude of reflection

from a boundary. You will remember from our discussion of 'acoustic parameters' that

the acoustic impedance is simply the product of the density (ρ) and the sound speed (v) ofthe media.


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Z = ρv

In acoustics, the acoustic impedance Z is measured in Rayles (1 Rayle = 1 m/s.kg/m3 = 1

kg/m2/s); but it is not that popular in seismology.

The full expression for sound reflection coefficient versus the incident angle α is:

i

i

n Z Z

n Z Z R

α

α

2

12

2

12

tan)1(1)/(

tan)1(1)/(

−−+

−−−=

where n = (v2/v1)2 and α i is the angle of incidence of the wave ray.

Matlab Exercise:

Plot the Reflection coefficient against the incident angle for

a) air-water interface;

b) water-rock (v=3000 m/s) interface.

Notice that since energy is always conserved the remaining energy that is not absorbedmust be transmitted into the second medium whereby it will undergo refraction if the

velocities in the two layers differ. If absorption is negligible (a lossless medium) the total

energy of the reflected and the transmitted waves should be equal to the original energy

of the incident wave.

3.5.2 Normal incidence

If the waves are normally incident to the boundary the reflection equation can be

simplified to:

012

12 =+

−= i for

Z Z

Z Z R α

and the transmission coefficient is

1 2

2 1

20i

Z Z T for

Z Z α = =

+

Thus, reflection is a simple function of the impedance of the two media. If the two mediahave the same impedance there will be no reflection. Since the impedance is the productof velocity and density it is possible for example to have two media with different

densities or sound speed but the same acoustic impedance.

In-class exercise: Reflection and transmission coefficient for power (energy in a unit

time) is the amplitude R and T squared. What do you expect for RR+TT=? Explain why

you get the result.


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Reflection coefficients have values that range between -1 and +1. From this range we canidentify 4 different types of reflection:

1) z2 >> z1, R => 1 (Rigid boundary), i.e. most of the acoustic energy will be reflected

without a change in phase.

2) z2 -1 (Soft or pressure release boundary), i.e. most of the wave energy is

reflected with a 180 degree phase change.

3) z1 = z2, R = 0, (No Reflection)

4) Similar acoustic impedance, -1


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The fact that different materials have different acoustic impedance and reflectioncoefficients allows us to acoustically distinguish between different targets. Side-scan

sonar for example uses this characteristic to distinguish between sand (predominantly

quartz), mud and rock. This is done by examining the difference in intensity of the

acoustic returns, often in conjunction with some sort of textural analysis.

Exercise

Compute the magnitude of the reflection and transmission coefficients at the boundary between a fresh upper layer and a saline lower layer of water in a salt wedge estuary.

Assume that the angle of incidence of the acoustic ray is 5 degrees. The characteristics of

the fresh and saline water are as follows:

Fresh water: c=1426m/s, density = 1000kg/m3

Saline water: c=1519m/s, density = 1025kg/m

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