Chapter 7 Propagation of waves inside a layer

Chapter 7 Propagation of waves inside a layer In this chapter we investigate the main features of waves caused by a spherical source, and we focus on the formation of normal modes in the frequency and time domain.

7.1

Acoustic potential of an elementary source located inside a layer

Derivation of the potential Consider the more complicated problem of an elementary spherical source, located inside a layer overlying a half-space at depth d from its free surface, Fig. 7.1. The layer thickness is H, and Ci, pi and C2, P2 are the density and acoustic velocity in the layer and lower medium, respectively. In this section we derive formulae describing the waves propagating within the layer and the half-space. It is obvious that a spherical wave, traveling away from the source, experiences reflections from layer boundaries and, correspondingly, gives rise to the multiple reflections called the multiples in seismology. Let us assume at the beginning that the source generates a sinusoidal wave and then consider the transient waves. Taking into account the axial symmetry of the problem it is natural to choose the cylindrical system of coordinates with the z-axis directed downward. The coordinate origin O is located at the free boundary. As usual, we formulate the boundary value problem in terms of the complex amplitude of potential U, which is related to the pressure V and displacement S as V = uj'^pU

and

5 = gradW

(7.1)

The potential should satisfy the following conditions: (a) At regular points, that is inside the layer and in the lower medium, it obeys the Helmholtz equation 451

CHAPTER

452

7. PROPAGATION

OF WAVES INSIDE A

Free surface

LAYER

r

O H

S Cp p, <^v P2

f z Figure 7.1: Elementary source inside a layer.

(7.2)

V^W +fc^ZY= 0, where if

0
if

z> H


k= { k2 = C2

(b) At the free surface the pressure vanishes, and therefore U = {)

if

(7.3)

z =0

(c) At the boundary z = H between the layer and the half-space, the pressure and the vertical component of the displacement are continuous functions. This means that

pi Ui = p2 U2

and

dUi dz

dU2 dz

if

z = H

(7.4)

Here Ui and U2 are the potentials in the layer and the half-space, respectively, (d) The potential of the direct wave generated by the source can be represented as

AkiR Uo = Cwhere C is the known constant and

R

(7.5)

7.1. ACOUSTIC

POTENTIAL

OF AN ELEMENTARY

SOURCE

453

R = ^ r 2 -f (^ - d)2

(7.6)

is the distance from the source to the observation point. It is natural that the potential in the layer consists of two parts Ui=Uo-^Us

(7.7)

Here Us is the potential of the secondary waves which has to be finite everywhere. Approaching the source, the total potential Ui tends to UQ: Ui^Uo

if

i? -> 0,

(7.8)

and the latter is the condition near the source. (e) Since the spherical wave excited by the source has finite amplitude, the energy of the total wave field, which includes all reflected, transmitted, and head waves, is also finite. Therefore, the wave amplitude should decrease with the distance from the source. This leads to the condition at infinity ZY->0

if

R^oo

(7.9)

Thus, we have formulated the boundary value problem. As follows from the theorem of uniqueness and from the physical point of view there is only one wave field that satisfies all five conditions, eqs. 7.2 - 7.4 and 7.8 - 7.9. First we find the solution of the Helmholtz equation. Applying the method of separation of variables and using the results of the previous chapter, the solution of this equation can be written as oo

W(r, ^, ^ ) ^ / (^m e^^^ + Bm e ~ ^ ^ ^ ) Jo(mr) dm ,

(7.10)

0

where

mi = \Jm? — k\ ^

7712 = ym? — A:!

and

Im m„ < 0 ,

n = 1, 2

(7.11)

The coefficients A^ and Bm, which are independent of r and z, are the unknowns. As we already know, the function U obeys eq. 7.2 regardless of the values of A^ and Bjn- In other words, the Helmholtz equation has an infinite number of solutions, and we have to choose such Am and Bm that U satisfies all other conditions of the boundary value problem. With this purpose in mind, we again make use of the Sommerfeld integral

CHAPTER

454

oikiR

7. PROPAGATION

OF WAVES INSIDE A

LAYER

CO

=

R

f^e-'^^\^-'^\jo{mr)dm

J mi

(7.12)

0

Then the expressions for the complex amphtude of the potential in the layer and the lower medium, satisfying conditions near the source and at infinity, are oo

dm 0

^

(7.13) U2 =

C f

B2me~^^^Mmr)dm

Next we find coefficients Aim, ^im and B2m such that conditions at both interfaces are satisfied. Substitution of eqs. 7.13 into eqs. 7.3-7.4 gives the following set of linear equations: m mi

-mi d

+ Aim + Bim

=

mi

-me

mi (//— d) , -__ A

jniH

_

p

^—miH

(7.14)

= 0,

_

B^,„e-'^^"

^, R

_—m2^

where (7.15)

6 = ^

P2 Eliminating B2m from the last two equations, we obtain rni'^b-l) Vmi

/

e - ^ i iH-d)

+ (bm^ + mi) A„„ e ^ i ^ + (tm^ - rm) B,m e'"^^^

= 0

or TTll

where i?12 =

6m2 — mi bm.2 -I- mi

(7.16)

7,1. ACOUSTIC

POTENTIAL

OF AN ELEMENTARY

SOURCE

455

Thus, in place of the system 7.14, we have m

-m\d

mi (7.17) m

mi Whence m

Birn =

e

i2e-2^i(^-^) -mid l^ -~i ? -^12

(7.18)

l-i2i2e-2^i^

mi and

mi

1 — /ti2e ^ ^ 1 ^

(7.19)

The last two equations of the set 7.14 yield 2 6 e - ^ i ^ fm e^^^ + mi ^ i ^ ) ^2me~ „—7712//

\

/

67712 + ^^1

Substitution of Bim into the latter gives ^^mid

_^ mi £?i^ =

m

^mi d _ g - m i d

l-/?i2e-2'»i(«-rf) 1 - i?,o e-2'"i«

m _ l-i?i2e-2"^i

(^e^id _

e-^i^^

Hence 2 6m e""^!^ e'^^// ^gmid _ g-mld^ -^2m —

(6m2 + m i ) ( l - i ? i 2 e - 2 ^ i ^ )

(7.20)

Thus, we have solved the boundary value problem and expressed the potentials in each medium in terms of the integrals. Correspondingly, in general, computation of the wave fields is reduced to numerical integration. Consider several special cases. (a) If pi = p2 and Ci = C2, we arrive at the expressions for a homogeneous half-space with the free surface. As follows from eqs. 7.18-7.20, in this case

456

CHAPTER 7. PROPAGATION OF WAVES INSIDE A LAYER

Bim =

e"^^^

and

Aim = 0,

mi since Ru = 0, a n d

^2m —

(b) If t h e b o u n d a r y z = H

^

/^mid _

^-mid\

I rui ^

is also free

h—

)>oc,

1712 = m

and

Ru = 1

P2

Consequently, m

1

mi

f)2m\d

1 - e-2^1^ '

(7.21) m

^ 1 _e-2mi(if-d)

mi

1 - e-2mi//

(c) In contrast, if t h e lower m e d i u m is rigid, t h e n Ru = —I a n d A

_

_ J ^ p - m i (2 if+rf) mi

^ ^ l_^g-2mi//

(7.22) 5im

=

m -—e--^^ mi

1 _|_ g - 2 m i ( / / - r f )

l + e-2^1^

(d) Suppose t h a t t h e source approaches t h e free surface, t h a t is d —> 0. Therefore,

Rim

-^

5

Aim

-^ 0 ,

B2m -^ 0

mi

and Wi ^ 0 ,

W2 - ^ 0

This means t h a t t h e t o t a l wave field disappears due t o t h e b o u n d a r y condition a t t h e free surface (eq. 7.3).

7,2, EXPANSION

7.2

OF INTEGRANDS

IN SERIES

457

Expansion of integrands in series

As before, along with numerical integration, it is useful to derive asymptotic expressions for the wave field which are valid within a certain range of frequencies and observation distances. With this purpose in mind consider two methods, and the first one is based on the expansion of the integrands in eqs. 7.18-7.20 in a series. As is well known, (7.23)

l - i 2 i^1212 e - 2 ^ i-

n=o

As |jRi2| < 1, this series converges, except in the case when 1-^121 = 1

and

Re mi = 0 ,

and it happens if the lower boundary is either free or rigid. Substituting eq. 7.23 into eqs. 7.18-7.20 and making use of the first equation of the set 7.13, we can represent the potential lAi as pikiR

Ui

=

C

_oo ^

^ ^

/*

777

+ ^ —Ln n=0 { "*!

e™'^ Jo(mr) dm (7.24)

- V / — M„ e-""'' Mmr) ±:^ J m-i

dm

where L„=

e-"'idj^n+l^-2m,{n+l)H(^-^_^2m,d>^

(7.25)

The latter shows that there is only one term with the zero power of Ru. other terms, we have pikiR

Uio = C

R

Ignoring all

oo

mi J rui 0

or Ak\R

U,n = C where

R

jLk\Ri

Ri

I

(7.26)

CHAPTER

458

7. PROPAGATION

OF WAVES INSIDE A

LAYER

Free surface ^Ve^>^.^^W.^yf|Mfe^v^--4Mfeb'^r^^:"-r^;^Lf^:i''!:^'J'^y^'g

^

Figure 7.2: Reflection from free surface.

Ri = ^/^^TJ^Tdy The first term in the right-hand side of eq. 7.26 describes the direct wave while the second one characterizes the reflection from the free surface. It is clear that the reflected wave is also spherical. Although there are two integrals in eq. 7.24 for each given n ^ 0 that contain the term Ri2^ they cannot be associated with any particular reflected wave. However, there is one exception, and in order to investigate it consider the integral of the following type: CX)

In =

/ ^ ^ i ' 2 e - ' " ' ^ " J o ( m r ) dm (7.27) J mi \om.2 -\- nriij

which is related to potentials Ui and W2. The parameters / and z^ in eq. 7.27 are related to n, z, H and d. Assuming that the observations are performed relatively far away from the source, kiR> 1, we can apply the method of the stationary phase in the same manner as it was previously done. In particular, we replace the Bessel function Jo{m,r) with its asymptotic, J^^rnr)

« ^ [ ^ ^

V Trmr

cos ( m r - - ) =

V

4/

J — i —

Ui^nr-n/A)

V 27rmr L

^

g-i(mr-./4)l

-•

Consequently, the stationary point mon is deflned from the equality d (p/d m = 0. Here

7.2. EXPANSION

OF INTEGRANDS

IN SERIES

459

(p{m) = —mr — riiZn + 75

Ui = \Jk\ — rn?

Whence mon =

I = -^ = ki sin On yjr'^ H- Zl ^n

(7.28)

Note that the other exponent, describing the Bessel function, does not have a stationary point within the interval of integration. Correspondingly, the ratio bm2-mi ^12 = 7 ; 01712 + mi

at the stationary point becomes b yjkl sin^ On — kl — yjkl sin^ On - kl R12

=

—/

/

b yjkl sin^ On-kl-\-

yjkf sin^ On - kf

b yk2 — kl sin^ On — ki cos On b yjkl — kl sin^ On 4- ki cos On or Zi]

Rn =

h F-

\

Z2 COS On

"!,„

^M

(7.29) h Z2 COS On

The latter is known as the Rayleigh coefficient . Assuming that the vicinity of the stationary point gives the main contribution to the integral, in place of eq. 7.27, we have 00

In « i?i2(mon) / — e-^^ "^"1 Jo(mr) dm J mi 0

if

kir > 1

(7.30)

Then, the use of the Sommerfeld integral gives r>ik\Rn In ^ R[2{m^nf-—^

Kn

where

Ak\Rn = (3n^—^-

tin

,

(7.31)

460

CHAPTER

7. PROPAGATION

OF WAVES INSIDE A

LAYER

Free surface

Figure 7.3: Rays reflected once from the lower boundary of the layer.

Substitution of latter into eq. 7.24 clearly shows that in the wave zone the total wave field is represented by superposition of spherical waves which experience multiple reflections at both layer boundaries and decay as 1/r, provided that r ^ z^- The center of phase surfaces of those waves is located on the z-axis (r = 0) and it moves away from the layer when n increases. Respectively, the apparent incidence angle 0^ becomes smaller. It is essential that the rays of each spherical wave obey Snell's law. Consider the waves arriving at point p after a single reflection from the layer bottom. Therefore, their amplitudes are proportional to R\2- As we already know, there are four such waves and, in accordance with eqs. 7.25 and 7.30, the centers of their spherical fronts are located at points of the z-axis

z=

±{2H-d)

and

z = ±{2H-hd)

Correspondingly, those waves can be described with the help of rays shown in Figures 7.27.3. In the same manner we can plot the ray trajectories that illustrate multiple reflections from the lower boundary. Inasmuch as those rays have different incidence angles 9^, their apparent velocities differ from each other. Thus, the use of the concept of the stationary point and the Sommerfeld integral allows us to treat the total wave field as an infinite sum of spherical waves, provided that r > Ai. In contrast, at relatively small distances, r < Ai diffusive harmonics may play a significant role, and the wave behavior becomes more complicated.

7.3

Integration along branch cuts and around poles

Our goal is to derive asymptotic expressions for the potential in the wave zone, that are simpler than the exact formulae 7.13. Expansion of the function Ui in an infinite sum

7.3. INTEGRATION

ALONG BRANCH

CUTS AND AROUND POLES

461

of spherical waves, however, is not always convenient. For this reason, we make use of the method based on deformation of the path of integration in complex plane m. As in section 6.3, we proceed from Cauchy's theorem, according to which an integral of an analytic function along a closed path is determined by its singularities located inside the area surrounded by this path. Before we apply this approach, it is useful to represent the coefficients Ai^, Bim and B2m, eqs. 7.18-7.20 in a different form. Let us start from 52, eq. 7.20. Since 1 1 - i?i2 e-2^1^

hm2-\-mi (6m2 + mi) - (6m2 - mi) e-^^^i^ 6m2 -h m i

6m2 (1 - e-2^1^) + mi (1 + e-^^^^) ' we obtain ^

2 6 m e ^ 2 ^ sinhmic? 6m2 s m h m i i i + m i cosh m i / /

/«^^x

It follows from eq. 7.19 that mi^

=

1 - i^i2e-2^i^

^ e - -

mi "^

m mi

rui

d (6m2 + mi) - (bm2 - mQ e-^"^^(^-^) _ m_ ^_^^, (6m2-f mi) — (6m2 — mi) e~2"^i^ mi

™,. &m, [ l - e - ^ " ' ^ ( ^ - ' ' ) ] + m , [l + e-^--(^-'^)] 6m2 (1 - e-2mi//) _^^^ (]^ _l_ g-2mi//)

^ _^, ^^

Thus

"^

m 6m2 sinhmi(i7 — d) + mi coshmi(ii/— 0?) mi 6m2 s i n h m i i / + mi c o s h m i i /

m mi

.^^^

From eq. 7.18 we obtain

"^

m 6m2 sinhmi(//'— d) + mi cosh?ni(i/— c?) mi 6m2 sinhmii? + mi c o s h m i / /

Consider first the potential inside the layer. Introducing the notation

/7 QQ\

CHAPTER

462

7. PROPAGATION

OF WAVES INSIDE A

LAYER

(7.35)

D = bm2 sinhrriiH -f mi coshrriiH and making use of eq. 7.13, we have oo

Ui

(7.36)

= C / — ( e - ' " ' l ^ - ' * l - e - ' " i ' * e " ' i ^ ) Jo(mr)dm

OC

+ 2C

m 6m2 sinhmi(i/— d) + mi coshmi(/f — d) . sinh mi2: Jo(mr) dm D

/ mi

At the beginning suppose that z < d. Then \z — d\ = d — z and g - m i (d-2:) _ g - m i d ^miz _ Q

Therefore

Ui = 2C f

m b 1712 sinh mi(H — d)-h mi cosh mi(H — d) ... _. , , sinh mi2: Joimr) dm mi D (7.37) if

z
In the opposite case when z > d, we have ^-m,\z-d\_^-rmd^m,z

^

^-m,{z-d)

_ ^m, {z-d)

_

-2

smhmi{z

-

d)

Thus oo

Ui = 2C f m

6m2 sinhmi(i/— G?) . ., — smhmiz

mi

H

mi coshmi(/f — c?) . ., i / ix — smhmiz — smhmi [z — d) Jo(mr) dm

The expression in brackets can be represented as [...] = — {6m2 [sinh mi (/f — d) sinhmi^: — sinhmi/f sinh mi (2: — d)] (7.38) + mi [ cosh m.i{H — d) sinh miz — cosh mi H sinh mi (z — d) ] }

7.3. INTEGRATION

ALONG BRANCH

CUTS AND AROUND POLES

463

Using well-known expressions for sinh(x it y) and cosh(x it y) the right-hand side of eq. 7.38 can be greatly simplified: bm2 [' "] = bm2 sinhmirf sinhmi {H — z) Also mi [... ] = mi sinhmid cosh mi {H — z) Therefore oo

Ui=2C J

m b 1712 sinhmiiH

— z)-\-mi — D

mi

cosh mi(H — z) . , , T / \T sinh mi a Joimr) dm if

z >d

(7.39)

After simple but rather cumbersome transformations, we obtain CX)

U\ =2C

m,Fi(m^) Jo(mr) dm,, 0

(7.40) CXD

U2 = 2C / mF2(m^) Jo(mr) c?m , 0 where Fi{m?) and F2(m^) are the even functions of m given by sinh 771 iz [6m2 sinh m i ( i / — c?) -h mi cosh mi {H — d)] mi if

Fi{m^) =

D

z
(7.41) sinh mid [6m2 sinh m i ( i / — z) -|- mi cosh mi ( i / — z)] mi if z> d

and, eq. 7.32, F2(m2) = - e"^'-^-") sinh mid Now we are ready to apply the Cauchy's theorem , so we again use the equahty 2 Jo(mr) = H^^\mr)

+

H^o\mr)

(7.42)

CHAPTER

464

7. PROPAGATION

OF WAVES INSIDE A

LAYER

Rem - •

•

•

•

•

•

Figure 7.4: Singularities on the real m-axis.

Substitution of the latter into eq. 7.40 yields OO

Ui = C

CO

/ m F i ( m 2 ) H^^\mr)

dm + jmFi{m^)

0

H^^\mr)

dm

0

or W i = C ( / i + /2),

(7.43)

where OO

Ii=

fmFi

(m^) H^Q^^ {mr) dm, (7.44)

l2=

fmFi(m^)

H^^^(mr) dm

0

In order to choose the new integration paths properly and apply Cauchy's theorem, it is necessary to consider the singularities of the function Fi{m?). First, there are branch points defined by the equalities mi = 0

and

m2 = 0

(7.45)

and the latter give four such points located on the real m-axis: (7.46) In accordance with eq. 7.18, the function Fi{m?) may have poles, and their position is found from the condition £) = 0, that is. bm2-mi^_2^^^^^ 6m2 -hmi

(7.47)

Suppose that C2 > Ci or fci > ^2 and consider the axis Re ?n, Fig. 7.4. If m > /ci, coefficient i?i2 and exponent e"^^^ are real quantities less than unity. This means that i?i2e-2-^^
7.3. INTEGRATION

ALONG BRANCH

CUTS AND AROUND POLES

465

and therefore the poles are absent when m > ki. Next, assume that m < k2. Correspondingly, ^ 1 = —i ykl — m?

and

m2 — —i yk^ — m?

are purely imaginary, but i?i2 still remains real. Since eyi^(-2iH^kl-mA = 1 and

|i?i2|
the condition 7.47 is not satisfied, that is the poles are also absent. Now consider the interval between the branch points k2 < m < ki In this case, the condition for the existence of poles becomes •

hm2 + i\mi\

2i\m,

\H

_ .

0 7712 — 2 I m i I

(7.48)

where m2 = \m2\

and

| mi | = \Jk\ — w?

It is clear that regardless of the value of m, the magnitude of the complex number in the left-hand side of eq. 7.48 is equal to unity. Therefore, if its argument is zero, the poles are located between the branch points. The same analysis shows that in the opposite case, when C2 < Ci, the poles are also absent if m > ^2 and m < ki. For the interval ki < m < k2 , the left-hand side of eq. 7.47 is i6|m2|-|mi|^_2|^^^l ^ ^ z 6 I m2 I + I mi I Correspondingly, there are no poles along the real axis of m. At the same time, the study of eq. 7.47 shows that its roots, that is the poles, are complex. Next we analyze only the case C2 > Ci when the real poles are located between the branch points. Layer has lower velocity, C2 > Ci It is convenient to assume, as was done in section 6.3, that all singularities are slightly moved away from the real m-axis and, then, after integration, they are placed back to their original positions. Making use of this approach, we change the integration path, eq. 7.40, 0 < 771 < OO

466

CHAPTER

7. PROPAGATION

OF WAVES INSIDE A

LAYER

and first consider the integral Li = i m Fi{m^) H^^\mr)

dm

along the closed line surrounding the forth quadrant, Fig. 7.5a. Since the singularities are absent inside this area, Li vanishes, and we have jmFi

(m^) H^^^ {mr) dm = 0

or CX)

JmFi

(m^) H^o\rnr) dm + J m Fiim") H^o\mr)

dm (7.49)

0

H- I

m Fi{m^) H^Q\mr) dm = 0

-ioo

Taking into account that the path element Co© has an infinitely large radius, and function HQ \m,r) -^ 0 when m,r —> oc, the second integral in eq. 7.49 is zero. This gives oo

0

J m Fi{m^) H^^\mr) dm = 0

J

m Fi{m^) H^^\mr)

dm

—ioo

From the second equation of the set 7.44, we have —ioo

72=

I mFi{m^)H^^\mr)dm

(7.50)

0

Thus, the integration along the real axis is replaced with that along the negative part of the imaginary axis. Next, consider the integral L2= jmFi(m^)

H'^^ {mr) dm,

where C is also a closed path that surrounds the first quadrant in such a way that the singularities are situated outside it, Fig. 7.5b. Then, as follows from Cauchy's theorem, L2= i m Fiim'^) H^Q^\mr) dm = 0 (7.51) c Let us note that, for convenience, only two poles are shown in Fig. 7.5b. The path C, which is more complicated than the previous one. Fig. 7.5a, includes several elements, namely.

7.3. INTEGRATION ALONG BRANCH CUTS AND AROUND

POLES

467

t Im m

b/

Re m

b;

V-

K Poles

k,

Rem

Figure 7.5: Integration paths. (a) the real axis of m 0 < m < CXD, (b) branch Unes 6i",

bi

and

62",

6^,

(c) circles around the poles, (d) the portion of the circle Ccx^ with infinitely large radius R —> 00, (e) the imaginary axis of m. Because of the exponential decay of function HQ ^ (mr) for large values of its argument, the integral along the segment CQO is zero. Therefore, in place of eq. 7.51 we can write 00

fmFi{m^)

0

H^o^\mr) ^m + ^ 1 + Z 1 + [ mFi{m^)

0

^

^

H^o\mr) dm = 0

(7.52)

ioo

Here Yl is the sum of integrals along the branch cuts, while J2 is the sum of integrals 6

p

around the poles. In accordance with the first equation of set 7.44, we have

/i = - E - E b

P

fmF,{m')Hi'\mr) dm

(7.53)

468

CHAPTER 7. PROPAGATION

OF WAVES INSIDE A LAYER

Due to the equality

the integral /2, eq. 7.50, can be represented as —icx)

I2 =

-

J

mFi{w?)H^^\-mr)dm

0 ioo

=

-

f mFi{m^) H^^\mr) dm

(7.54)

0 0

=

I mFi{m^)

H^^''(mr) dm

ioo

Thus, /i + /2 = - E - E 6

(7-55)

P

and, correspondingly, the expression for the potential inside the layer, eq. 7.43, becomes WI = - C ( E + E )

(7-56)

It is essential that integration along the imaginary axis is eliminated and the function Ui is expressed in terms of integrals around the poles and along the branch cuts. As follows from the theory of functions of a complex variable (Appendix A), N

- ^

= 27ri X^Res^-.

p

(7.57)

j=i

where the right-hand side is the sum of residues of the function

Finally, we have Ui=Uip-^Uib

(7.58)

Here N

Uip = 27ri^

Resj j=l

and

ZY^^ = - ^ b

(7.59)

7.4. NORMAL

7.4

MODES

469

N o r m a l modes

Numerical integration of eqs. 7.40 allows us to find the wave field at any distance r from the source. This task is relatively simple when separation between the source and the observation point is suSiciently small, and this case is of great practical importance for exploration seismology. With the increase of distance r, however, evaluation of infinite integrals (eq. 7.40) becomes more difiicult because of rapid oscillations of the Bessel function Jo(mr). For example, such a situation arises when we investigate propagation of waves in water layer, provided that the distance r is much greater than its thickness. Consequently, along with numerical integration, it is useful to derive asymptotic formulae describing these waves. Besides, this procedure allows us to understand wave behavior better. With this purpose in mind, we make use of eqs. 7.59 and, first, examine waves inside layers, which are related to the poles Z^ip-27riC^Res,

(7.60)

where Res is the residue of the integrand, eqs. 7.44, mFi{m'^)H^o^\mr) Such waves are usually called normal modes. Suppose that the z-coordinate of the observation point satisfies the condition d < z < H, but later this restriction will be removed. Then, in accordance with eq. 7.41, we have 2.

sinh mid hm2 sinh mi {H — z) -\-rui cosh mi {H — z) rui D

and D = bm2 sinh m i i / + mi cosh rriiH. Taking into account that sinh(—ix) = —i sinx

and

cosh(—zx) = cosx,

the function Fi(m^) can be represented as 2\ sin mid 6 m2 sin mi (if — 2) + mi cos mi (i/— z) FAm ) = —z ; :—z—— z z—77 , mi om2 smmiH -h mi cosruiH where

r^ a^\ ('•61)

CHAPTER

470

7, PROPAGATION

mi = yki — w?

and

OF WAVES INSIDE A

LAYER

m2 = ym2 — fci

Correspondingly, the integrand in eq. 7.44 is mFi(m^)H^^\mr)

(7.62)

(t>2{m)

Here 01 (^)

=

m sin fhid [bm2 sin fhi (H — z) -\-mi cos fhi {H — z)] HQ -z—

(mr), (7.63)

02 (m)

=

brn2 sinmiH

-\-mi

cosrhiH

We assume throughout most of this section that velocity C2 in the lower medium exceeds that in the layer: C2 > Ci

As was shown earlier, the real poles, x^, of the integrand are located between the branch points k2 < Xn < ki and their positions are defined by the condition 02(^n) = 0

or

nil

tan m i / / =

(7.64)

bm2

Inasmuch as the poles are simple (Appendix A), their residue is determined as Res

01 (m)

01 (m)

02 ( ^ )

02 ( ^ ) '

(7.65)

where '{m) =

d(j){m) dm

The last equation of set 7.63 gives d(j)2

—— = dm

bXn

.

^

jj

smmi// m,2

hm2XnH

z rui

^

rj

^n

-

rj ,

u

cosrriiH —:r- cos m i / / + x^/z mi

'

-

u

smmiH (7.66)

mi

H (fhi sinrhiH — hm.2 cosrhiH) — cosrhiH -]

sinrhiH m2

7.4. NORMAL MODES

471

Substitution of eq. 7.64 into eq. 7.66 yields d 02 dm

_

Xn \ u (' ~ LI , ^1 cosmiH\ z—77— rhi \ H [mi smm^iH H L \ tanmi/i / — cosrhiH — b^ sinrhiH tanrhiH

or —— = — — mi i/— sin m i / f cos mi if — 6^ sin^ mi i? tan mi i / dm, TTii sinrhiH ^ -•

(7.67)

Consider the function (^i(m). The sum in brackets in eq. 7.63, can be simphfied the following way: [...]

=

cos mi z (6 m2 sin mi if-I-mi cos mi//") + s i n m i z ( m i sinrfiiH — bm.2 cosrhiH)

Making use of eq. 7.64, the latter becomes ^ / ^ r, ^ cos^mi//\ [... J = sm mi2: I mi s m m i / f + mi —7sinrhiH J

mi sinmiz sinrhiH

Thus, 0i(a:n) = —^—:—77- smmio? H^ {Xnr) smmiH

(7.68)

Finally, from eqs. 7.60, 7.65, 7.67, and 7.68, we obtain 2 7ri . - ^ m i i / s i n m i z s i n m i d (1) Wip = -jf C 2 ^ — WQ \xnr) and Mn = rhiH — sinffiiH

cosfhiH — 6^tanmiif sin^mi/f,

where

rhi = yfkf- xl We note the following.

(7.69)

472

CHAPTER

7. PROPAGATION

OF WAVES INSIDE A

LAYER

(a) To avoid confusion with mi and m2, the notation Xn for poles is used. (b) We have arrived at eq. 7.69 assuming that z > d. However, similar derivations for the second equation of the set 7.41 show that eq. 7.69 remains valid for all points inside the layer except the source location. Consider some features of the function ZYip, eq. 7.69. When the argument XnV is small, XnT ^ 1, as is usually the case in the near zone, none of the terms of the sum displays wave behavior. Rather they have the meaning of the diffusive harmonics. In contrast, if Xnr > 1, these terms can be interpreted as waves. In fact, taking into account the asymptotic expression for the Hankel function

eq. 7.69 becomes

Wip = ^

y ^ E Gn e^ (-^^^/^^,

(7.70)

where ^n =

rn smrn— siuTn — y / : r^TT ^~2 \" y/Xn [rn — s m Tn COS r „ — 0"^ t a n r „ s m r „ )

(7-71)

and Vn = rhiH

(7.72)

In accordance with the definition C/ip-Re(Wipe-^^^), we have Uip = —J—

Yl^n

cos{ujt-Xnr

- -)

(7.73)

We see that Xn has a meaning of a wavenumber, and Cpn = — is a phase velocity of the nth propagating mode with frequency a;.

(7.74)

7.4. NORMAL

MODES

473

Figure 7.6: The left-hand side (sohd), the right-hand side (dashed) of eq. 7.77 and its roots (gray dots). Finding poles To find positions of poles, we need to solve eq. 7.64. Let us introduce a new parameter p characterizing the relation between the wavelength A and the thickness of the layer H p = kifiH = 27r/j.— = —^—^ > 0 Ai ci

(7.75)

where / is a frequency and

^^Fl'" < 1

(7.76)

02/

Using eqs. 7.72 and 7.76, we transform eq. 7.64 into (7.77) and search for roots r„ of this equation for any given value of p. The left-hand side of this equation is independent of p and is a periodic function of r^. In contrast, the right-hand side depends on the parameter p. The intersections of curves describing these two functions define the roots of eq. 7.77, as shown in Fig. 7.6. We ignore the trivial root ro = 0, which does not correspond to traveling waves. Then, depending on the value of p, there are no roots, just one root, or several roots of eq. 7.77. It is clear that with the increase of parameter p, the number of roots also increases. Let us discuss this question in more detail. Analysis of eq. 7.77 for different values of p shows the following. (a) The roots obeying eq. 7.77 are absent within the interval 0 < p < 7r/2.

474

CHAPTER

7. PROPAGATION

n

2K

OF WAVES INSIDE A

3n

LAYER

p

Figure 7.7: Roots r„ of eq. 7.77 as functions of p. (b) The first real root ri appears when p = -K 12. As is seen from eq. 7.77, it is equal to ri = 7r/2 if p = 7r/2. (c) There is still one real root within the range 7r/2 < p < 37r/2, which becomes greater when p grows. (d) The second real root r2 arises when p = 37r/2. It is also equal to r2 = 37r/2. (e) There are two real roots within the interval, and they both gradually increase when the parameter p becomes larger. Continuing this consideration, we obtain an infinite number of intervals of p which contain the roots of eq. 7.77. The boundaries between them are defined from the equality p = rn = (2n + l ) | , (7.78) and these pairs of p and r^ satisfy eq. 7.77. It is also useful to notice that each root approaches its limit as p increases. For instance, ri^>7r,

r2—^27r,

TS^STT,

...

if

p -^ oo,

that is -
—
—
and, in general. TT

( n - 1 ) - < Tn < 7rn

(7.79)

The behavior of the first three roots is shown in Fig. 7.7. We see that the n-th pole exists

7.4. NORMAL MODES

475 SP/^1.,__JJZ/^1

^p/^J.

Figure 7.8: Phase velocity dispersion curves for first three normal modes. in the range Pn < P < oo or 7r(2n -f- l ) / 2 < ^ < oc, which corresponds to the range of frequencies fn < f <.C)o, where f = uj/27r and (2n + 1) ci /n =

*"FW) 2 '

(7.80)

fn is called a cut-off frequency. Dispersion The analytical solution of eq. 7.77 for an arbitrary value of ct; is absent. This means that Xn as well as the phase velocity UJ

LJ

^pn

(7.81)

^H-n) should be found numerically. The relation 7.81 demonstrates that the phase velocity of each normal mode is frequency-dependent. Such dependence is called dispersion (Part I), and curves Cpn(^) are called phase velocity dispersion curves. Prom analysis of behavior of roots Xn it is clear that Cpn(^) is changing in the interval C2 > Cpn ^ Ci when u changes in the range Un <^ < oo (Fig. 7.8). The case 6 = 0 It is interesting to consider a special case when the lower medium is rigid and, therefore, 6 = 0. Then eq. 7.77 is greatly simplified and becomes cos r^i = 0

(7.82)

476

CHAPTER

7. PROPAGATION

OF WAVES INSIDE A

LAYER

Consequently, the roots are rn = 7r(n--y

(7.83)

where n = 1, 2 , . . . is an integer. Unhke the general case, 6 7*^ 0, the roots of eq. 7.77 are independent of the frequency and medium parameters. Therefore, the poles x„, which characterize the behavior of normal modes, are defined as x„ = ^^klH^-Tr^(n-iy

(7.84)

From eqs. 7.83 and 7.71, the expression for the phase velocity of the nth mode is ujl^

(7.85)

^kim-iT^{n-\) For the cut-off frequency of the same mode we have /n = ^ ( n - ^ )

(7.86)

Dependence of normal mode amplitude on depth The expression 7.71 determines the change of the amplitude of a normal mode with the depth of the observation point and the source: Gn ^smrn—smrn

—

We see that normal modes behave as standing waves along the 2:-axis, and with an increase of their order n, the number of nodes also increases, Fig. 7.9. The function Gn, characterizing the potential in the wave zone, is independent of the distance r from the source and, correspondingly, the relationship between amplitudes of different modes is defined by the frequency and the order n. Waves in t h e lower medium Let us briefly discuss the behavior of waves, which associate with normal modes, in the lower medium. As follows from eq. 7.42 and 7.60, the potential U2p is U2p = '2 7ri Y^Res mF2{m'^) H^^\mr), where F2(m2) = A e ^ 2 ( ^ - 2 : ) sinhmi^/

(7.87)

7.5. ASYMPTOTIC

BEHAVIOR

OF WAVES RELATED

TO POTENTIAL

UB

477

Pressure [relative units]

n=0

A2=l

«=2

«=3

Figure 7.9: Standing waves. and z >H >d Since Re m2 > 0, each term in eq. 7.87 describes an evanescent wave propagating in the horizontal direction with phase velocity Cpn and exponentially decaying along the 2:-axis for k2r > 1. The case Ci > C2 Until now we assumed that C2 > Ci. In the opposite case, the wave behavior is different. As the internal reflection is absent, all waves inside the layer quickly attenuate as the distance from the source increases. The same result follows from eq. 7.64 because its roots become complex. Note that if C2 <^ Ci, the reflection coefficient is close to unity and, correspondingly, attenuation is relatively small.

7.5

A s y m p t o t i c behavior of waves related t o potential Ub

In the previous section, we studied function Wp, which describes the waves related to the poles of the integrand of eq. 7.56,

F ( m i , 7712)

sinhmid bm2 smhmi{H — z) -h mi cosh m i ( i / — z) bm2 sinhmiH -\-mi cosh mi if mi

(7.88)

CHAPTER 7. PROPAGATION OF WAVES INSIDE A LAYER

478

Since the function F ( m i , 7712) is even with respect to mi, the integrals along the lines of branch point m = ki cancel each other, and we have k2

Uib = -C

I

mHQ{mr)

[ F ( m i , 7712) - F{mi, -7712)] dm

(7.89)

k2-\-ioo

As follows from eq. 7.88, the difference of functions A F = F ( m i , 7712) — F(77ii, —77^2) is equal to

AF

sinh77710? [67712 sinh77ii(if — z) -\-mi cosh.m.i{H — z) 7?7i [ 6772-2 sinh77iii/+ 77ii cos]im>iH

(7.90) 77ii coshmi(//" — z) — 6772-2 smhm,i{H — z) 77ii coshm^iH — 67712 sinh77ii7/ To simplify this expression, we perform some elementary algebra:

[677i2 sinh77ii(i/ — 2;) + 77ii coshm.i{H — z)] x [77ii coshm^iH — bm.2 sinh77iiH] = 677117712 sinh 7711 (7/— z) cosh 7711//+ 771^ coshm.i{H — z) cosh m i i /

(7.91)

6 m.2 sinh m.i{H — z) sinh m i i / — 6 mi 772-2 sinh m i H cosh m.i{H — z)

Also, [ 6 m2 sinh 772-1 i / -h mi cosh m.\H] x [mi cosh mi (7/ — 2:) — 6 m2 sinh mi ( i / — z) ] = 6mi m2 sinh m i / f cosh m.i{H — z) -\- m\ cosh 7?7iiif cosh m,i{H — z) 6 m2 sinh m i H sinh mi{H — z) — 6 mi m2 sinh m.i{H — z) cosh m i i /

Subtraction of eq. 7.92 from eq. 7.91 gives

(7.92)

7.5. ASYMPTOTIC

BEHAVIOR OF WAVES RELATED TO POTENTIAL UB 479

2 b mi 1712 smh.mi{H — z) cosh mi if — 2 6mim2 s i n h m i / / cosh mi (/f — z) = —2bmi 7X12 [ s i n h m i i / c o s h m i ( / / — z) — coshrriiH smhmi{H = —2bmim2

sinhmiz

— z)] (7.93)

Certainly it is a great simpHfication. Thus, we have 2 b 1712 sinhmid sinhmiz AF= — ml cosh^ vfiiH — b'^ ml sinh^ miH and ^,^ Uit = 2bC

k2

f / ./

m2 smhmirf s m h m i z ^^n, , , -^ -2 „ .2 2 • u2 -mH^'\mr)dm mf cosh mfiiH — b'^m.Q^ smh m i / /

/^^.x (7.94)

Asymptotic expression for tixt, As before, we examine the case when k2r > 1. This means that the Hankel function can be replaced by its asymptotic expression

V Trmr In other words, it is assumed that the initial part of integration gives the main contribution due to the exponential decay of the integrand along the branch cut. To evaluate this integral, eq. 7.94, let us introduce a new variable m = ki(—-j-ix\

(7.95)

dm = ikidx

(7.96)

Whence

Near the branch point, we approximately have m ^ k2

and

mi = y/c^ — kj = ikifi,

where M

1 - ^

4

(7.97)

480

CHAPTER

7. PROPAGATION

OF WAVES INSIDE A

LAYER

^ hJ2ix^ = kiJ2x^e'''^^

(7.98)

The radical m2 becomes m2

= J(^-^ix\\l-kl

We also have sinh?Tiid sinhmi2: = —sinkifid

sinki/jLZ ,

and the denominator of function AF is equal to ml cosh^ ruiH -b'^ m\ sinh^ rriiH =

(7.99)

= -kl 11^ cos^ k^^iH ^2kfb^x^

e^"/2 sin^

Making use of eqs. 7.95-7.99, the function A F can be written as 2kib,

2x — e*^/^ sinki/j.d V ^- C2

_ AF =

sinkifxz ,

(7.100)

where ai = -kl 11^ cos^ ki^H,

a2 = 2klb'^-e^^/^

sin^ ki^xH

(7.101)

C2

Also m dm = ik^ — dx C2

and

Substitution of eqs. 7.100-7.102 into eq. 7.94 gives AibCkn

Uib «

k^^^

r— ^ Wnr

• 1

sinkifid sinki/ize^^'^^I,

(7.103)

where /=

f^^^^^—^dx J

ai H- ao X

(7.104)

7.5. ASYMPTOTIC

BEHAVIOR

OF WAVES RELATED

TO POTENTIAL

UB

481

Bearing in mind that the value of / is defined mainly by integration over small x, it can be approximately represented as 1

°°

— a\ J

Vxi

As is well known, this integral is tabular, and it is equal to

r^__V^__ 2ai(A;ir)3/2 Correspondingly, in place of eq. 7.103 we obtain ^

^ 2ibCk2 kl r^

sinkifid sinki/iz ^^^^^ (JP- COS^ kijiH

The latter shows that at sufficiently large distances from the source the potential Uih characterizes a wave that propagates inside the layer with the phase velocity equal to C2 and the amplitude decaying with the distance as 1/r^. This wave is the result of constructive interference of head waves and multiple reflections with the angle of incidence, equal to the critical angle 6 = sin~^(ci/c2). Note that this wave behaves as the standing wave along the z-coordinate. It is obvious that we could obtain the same expression for Wi6, eq. 7.105, if it is assumed in the beginning that smki/2H

=0

or

ki/j,H = 7rn

(7.106)

This condition can be rewritten as — /j,H = 7rn

or

fn =

7===T

(7.107)

Since the critical angle 0c is defined as sm 6c = — 5 C2

we also have fn = ^ T ^ ^

(7.108)

Thus, when the frequencies obey this condition, we observe a relatively strong decrease of the wave field Uit with distance r. Now suppose that parameter ai vanishes, that is

482

CHAPTER

coski/jiH

7. PROPAGATION

=0

or

OF WAVES INSIDE A

k^^iH = ^^^

^^ ^

LAYER

(7.109)

Correspondingly, eq. 7.104 becomes 1

'y?e-^i^^

/ = — /

a2 J 0

•=— dx

Jx ^

Letting y = y/x, we have 1 c?x

dx

This yields 2

f

_^, ^,,2 ,

/ = — fe-^'^'y'dy a2 J

or

^

/

1

a2\kir

Substitution of the latter into eq. 7.103 gives

Otib^-T

2 sinkifid sin kifiz J k^j^ :-j-j e ^ , or sm^ kijiH

(7.110)

that is, the wave again propagates with the velocity C2 but its amplitude decreases with the distance slower that in the previous case, a2 = 0. Such behavior is observed if frequencies satisfy eq. 7.109:

As was shown earlier this set of frequencies coincides with the cut-off frequencies for the normal modes, and therefore constructive interference occurs. Comparison of waves related to both potentials Uip and Un) shows that normal modes play a dominant role in the wave zone.

7.6

Transient waves inside a layer

Now we discuss propagation of transient waves in the layer, and, as in the previous sections, the main attention is paid to the wave zone. As it is already known, the field of normal modes prevails in this case, that is Up ^ Wfe. Correspondingly, the complex amphtude of the potential is U[v.^)=Up(jp,u) (7.112)

7.6, TRANSIENT

WAVES INSIDE A LAYER

483

Potential of transient waves In accordance with eq. 7.70, the function Up, caused by a source of sinusoidal waves with the unit amphtude, C = 1, is Z Y = l J I l f ; G „ e ^ ( ^ " ' ' + ^/4),

(7.113)

where Gn = -T^T^ ; „ ;r„ _cos : r^ —r.1b'^ tan r^. „sin^ 2 ^ ^r. ^ ) ' Ic^ (r„ —„sin

(7-114)

and Tn are the roots of the dispersion equation Tn

tanr^j —

.

(7.115)

^..

(p2 - rl)

Here 6= — ,

p = ki^H,

and

M = 1^\

^2

(7.116)

Finally, the wavenumber of the nth normal mode is

Xn =

~yjkim-rl,

(7.117)

where r^ is a known function of the wave frequency, layer thickness H, and velocities Ci and C2. Suppose that the potential, caused by motion of the primary source, behaves as 0,

^ <0

! (7.118) Uo{i), ^> 0 Applying the Fourier transform and taking into account eq. 7.118, we have for the spectrum of the source oo

U^[p, u) = JUoip,

t)e^'^Ut

(7.119)

0

The latter allows us to assume that the source simultaneously generates an infinite number of sinusoidal waves with infinitely small amplitudes and different phases, and each of them may give rise to a normal mode. Then, making use of the principle of superposition (Fourier transform) we obtain the potential of total field in the wave zone.

484

CHAPTER 7. PROPAGATION

OF WAVES INSIDE A LAYER

1 °° U{p,t) = — f Uo{uj)U{cj)e-'^^*dw,

(7.120)

—oo

where U and UQ are the known functions given by eqs. 7.113 and 7.119, respectively. Substitution of eqs. 7.113 into eq. 7.120 and change the order of integration and summation gives U{p, t) = j^J—E

J GnUoe-'^^'-^-^-^/'Uou

(7.121)

or Uip, t) = Yl UniP, t),

(7.122)

71=1

where UniP,t) = ~

^ jG„Uoe-i(^t-^nr-n/4)j^

(7.123)

— OO

By analogy with the frequency domain the transient wave is now represented as a sum of normal modes. Since the functions Gn(ct;), UQ{U) and Xn{
1 [Y

— \ —|GnWo| exp(-#n)c?^, /7 Y Trr

where the phase V^n = ijot — XnV ~ 7r/4. The condition of constructive interference of these sinusoidal waves at given distance r and at time t is the closeness or stationarity of their phases, that is.

As it was discussed in Part I, the function Cgn = duj/dx^ is called a group velocity. This function defines the waveform of a transient (non-stationary) normal mode. In this light, note that we already considered several kinds of velocities. One of them is the physical velocity defined by the medium bulk modulus M and its density p. In this model.

7.6. TRANSIENT

WAVES INSIDE A

LAYER

and

Cl

485

C2

The second is the phase velocity Cp which characterizes the propagation of phase surfaces of any single sinusoidal wave, for instance, a normal mode. The phase velocity may depend on frequency and usually differs from the physical velocity. Also, we often deal with the apparent velocity c^, which describes the movement of phase surface of a sinusoidal wave along a given direction. It is related to phase velocity Cp as Ca =

COS a

where a is the angle between this direction and the normal to the phase surface. Now we deal with a different kind of velocity which defines the speed of propagation of wave groups (Part I). To find the relation between the group and phase velocities we use the formulae ^vn

Then, duj

duj

^''" ~ d:

^pn

U

1

^pn (^)

(7.124)

u> dc.pn Cpn duJ

It is possible to find an expression for the group velocity which does not contain dcpn/du using the formula for the derivative of an implicit function known from calculus:

dx

dF{x,y) dx dF{x,y) dy

In our case x = u>,y = Cp„ and

F{uj, Cpn) = tan (UJH^C^

^ - c'^^

"^pn

+-A ^pn

^pn

After performing some cumbersome algebra, we arrive at the expression for c^n, (7.125)

^gn

l + (c2cr2-l) 1 +

Cj

C2

bic^'-c^f/'ujH

Knowing Cpn for a given mode, it is possible to calculate Cgn using eq. 7.125. The result

486

CHAPTER

7. PROPAGATION

JL JR

JA

OF WAVES INSIDE A

LAYER

J

Figure 7.10: Group velocity as a function of frequency. of such evaluation for the first normal mode, characterizing the dependence of the group velocity, c^^, on frequency / , is shown in Fig. 7.10. As we see, the wave group with the cut-off frequency JL propagates with velocity C2 of the lower medium. This is the highest velocity of wave groups. Then with an increase of frequency the group velocity decreases and reaches its minimum at the frequency JA- With further increase of / , the group velocity begins to grow and asymptotically approaches layer velocity C\. Certainly this is an important information because it tells us about arrival times of different wave groups. As an illustration, consider the normal transient mode of the first order inside the layer, when k\r > 1. Note that the left and right parts of curve Cgn{f) in Fig. 7.10 are called the ground and water waves, respectively. It follows from the behavior Cgn(f) that the ground wave with the lowest frequency arrives first, and its group velocity is equal to C2. With an increase of time the dominant frequency of wave groups gradually increases, which takes place when / is less than /R: /L ^ f < IR- Within this frequency range the group velocity changes in the limits Ci < Cgn < C2. Both water wave and ground wave behave as amplitude modulated and frequency modulated signals. Then there is always an instant when both the ground and water waves arrive simultaneously at the observation point, and their group velocities are equal to ci. The ground wave is characterized by a relatively low frequency /R, while the water wave has a much higher frequency. Thus, starting from this moment, the transient mode is formed by wave groups with two distinct frequencies. In other words, we observe superposition of the ground and water waves. With the further increase of time the group velocities of these waves become smaller and their dominant frequencies approach each other. The second portion of the transient wave is terminated when the wave group starts to move with the velocity Cgno, equal to its minimum value with dominant frequency /A- This third and final stage of the transient wave is named the Airy phase. An example showing all three stages of the first normal mode is given in Fig. 7.11.

7.7. EVAL UATION OF NORMAL

60

70

80

MODES

90

100

487

110

120

130

140

150

Cft/H

Figure 7.11: Behavior of the first transient mode.

7.7

Evaluation of normal modes

In the previous section we qualitatively described behavior of transient modes and, in particular, distinguished three different parts of the wave train, namely, the ground wave, the water wave, and the Airy phase. However, the dependence of their magnitudes on time and distance still remains unknown. In order to find those relationships, we can, in principle, perform numerical integration of each term in the right-hand side of eq. 7.121. At the same time it is possible to arrive at sufficiently accurate values of the wave magnitude by applying approximate methods. Let us assume that the function Uo{t) is 0,

t <0 (7.126)

Uo{t) = t >0

where /? is the constant characterizing the rate of change of the source pressure. In fact, from the relationship

we have P - - p / ? M e"^* if ^ > 0. As follows from the inverse Fourier transform, the spectrum of function Uo{t) is Uoiou) = A J 0

e-^'e'""'dt

or

Uo{uj) =

(3 — iuj

Therefore, the transient normal mode of the nth order (eq. 7.121) can be written as

488

CHAPTER

7. PROPAGATION

OF WAVES INSIDE A

LAYER

It is useful to make some transformations of the integrand in this expression. To the accuracy of a constant, the function F(cu) ^ ^ntlllA p — iu

e-' (^" ^ - ^ / 4 )

(7.128)

represents the spectrum of U{t). As follows from the properties of the Fourier transform, the real and imaginary parts of its spectrum are even and odd function of a;, respectively. Re F{uj) = Re F{-cj)

and

Im F{uj) = - I m

F{-uj)

Therefore, OD

OO

/ F(w)e-^'^*da; = 2Re j F{u)e-'''^^

(kj

(7.129)

0

-OO

Taking into account that Gn is real and 1 —

1 ^— =

• •

A _ia; = exp I tan —

we have r 77/ N -^uJt^ Re [F{uj)e '^^\

Gn{uj,r,t) ( TT _^uj = - ^ ^ = = ^ cos (^o;t - x„ r - - - tan ' -

Whence Un{t) = ^ \ — I n

(7.130)

Here '" = 1 ^ $ ^

COS (u;^ - x „ r - ^ - t a n - fj

3^

(7.131)

0

The integrand represents a product of two functions. One of them Gn{uj, r, t)

(7.132)

varies with uj relatively slowly. The same is true for the term tan ^{uj/(3). In contrast, the function TT

_-, a ;

cos [ujt — XnT —4— — t a n (3—

7.7. EVALUATION

OF NORMAL MODES

489

rapidly changes even when the variations of a; are small, because the wave is considered at large distances from the source, XnV ^ 1. It is natural to treat the integrand as a sinusoidal wave whose phase depends on frequency. Applying the terminology of interference, we can say that if the phase changes rapidly in the vicinity of a certain frequency, this interval of integration does not make a noticeable contribution, due to destructive interference, and therefore it can be neglected. However, near such frequencies, where the phase variation is small, the constructive interference takes place and those ranges of a; determine the value of the integral. Respectively, our goal is to find such stationary points and evaluate the integrals around them. As we know, this is the essence of the method of stationary phase, and it is used in studying the ground and water stages of transient waves. In accordance with eq. 7.131, the phase ^n is

(Pn((^) =ujt-Xnr-^-

tan"^ ^

(7.133)

Suppose that CUQ is the stationary point. Taking into account that the interval of integration around UQ is small, it is proper to express the phase at its points in terms of ^n(^o) and corresponding derivatives. This procedure greatly simplifies evaluation of U{t). Expanding ^n{^) in the Taylor series, we obtain ^n(^) = ^n{uJo) + - r - ^ + -T-T V2 + Tduj^ T ^ ^6 + ' ' ' ' iUJ

(7.134)

where I UJ — C^o I

u = u — (JJQ

and


Here all the derivatives are evaluated at point cj = ojoMaking use of eq. 7.133 and neglecting the change of the last term, we find "-V^n _

du

.

- rx'

d'^iPn

„

d^ifn

du^

^'

di

= -rx':,

...

(7.135)

Correspondingly eq. 7.134 becomes (fn{^) = an-\-ani'

-\-bniy'^ -i-Cniy^ -\-... ,

(7.136)

where an=iOot-

Xn{uJo) T - - - tan ^ - ^

(7.137)

and x"

x'"

490

CHAPTER 7. PROPAGATION

I

txXyi

ff

OF WAVES INSIDE A LAYER

Qj Xfi

III

(I

Jjfi

By definition, in the vicinity of the stationary point a change of phase is minimal, and therefore the second term in the series, a^ i^, which usually causes the main variation in (Pn{(^)i vanishes. For this reason, in place of eq. 7.136, we have ^n{uj) « ^n(a;o) - - y ' ^ ' - ^

^'

(7-138)

and T

t-rx'^

=Q

or

c^nC^o) = T

(7.139)

The latter is the known expression for the group velocity and it is of a great importance. In fact, eq. 7.139 allows us to determine the velocity of a group formed at a given instant t and at a certain distance r from the source. Since the dispersion curve Cgn{oj) is assumed to be known, the dominant frequency UJQ (that is, the stationary point) corresponding to the group velocity c^n(^o) can be easily found. The ratio ^ " ( ^ ' ^' ^)

(7 140)

can, of course, be calculated, as well as the first term of the series, that is (/?n(^o)The second-order derivative xj^ is represented as dxn \ diJ \ du )

d du

I 1 \ \ Can I

1 dc,gn c^^ duj

(7.141)

Thus, knowing the group velocity as a function of the frequency a;, we can also calculate the second-order derivative of the wavenumber and, correspondingly, the second term in the Taylor series, eq. 7.138. The last term containing x'^ is evaluated in a similar manner and makes it possible to approximate the phase (pn{(^) around the stationary point. Next, consider two cases:

x:: ^ 0

and x;: - o

First, let us discuss the meaning of these conditions. With this purpose in mind, it is necessary to refer again to dispersion curve Cgn{u;) in Fig. 7.10. At the low-frequency part (jj < UA, the group velocity decreases with an increase of the frequency, and therefore, du

7.7. EVALUATION

OF NORMAL

MODES

491

As follows from eq. 7.141, xj( > 0 within this range. In contrast, at the high-frequencypart of the curve, where ou > UA, the group velocity grows with the frequency and, correspondingly. dc.gn

> 0

doj

and

x" < 0

Thus, the first case above includes all the frequencies except one, where the group, velocity reaches its minimum, and da gn du

0

x'l = 0

or

This is the second case which describes the Airy stage of the transient wave. Before we consider the two cases, it is useful to simplify eq. 7.131. In view of the fact that the function, given by eq. 7.140, can be treated as a constant in the vicinity of CJQ, it can be taken out of the integral. This yields

/„ =

Gn{uJo, "^1 r, t)t) f ( 7^ -1 ^ \ -1 - ' do; ==- / cos \ujt — XnV — — — tan — 1 \ / ^ - 0

0

or, using Euler's formula _ Gn(a;o, r, t)

(7.142)

V ^ + ^a where oc

!„ = / exp

tan

i I Ujt — XrjV

dcu

(7.143)

0

We also assume that tan - 1

tan

Thus, we have the following expression for Un{t):

t/„W = ^,/i:^.L(^£,I^KeX„ ^

(7.144)

^-^^ yJP^+Lul

C a s e one: cc^ ^ 0 Since the difference of frequencies i/ = a; — CJQ is small the third term in series 7.138 can be ignored, and we have (fn{^)

rx^

^ ^n(^o) - ^

(^ - ^o)^

(7.145)

CHAPTER

492

7. PROPAGATION

OF WAVES INSIDE A

LAYER

Substituting the latter into eq. 7.143, we obtain oo

(7.146)

duj

«2^n('^)('^~'^o)^

As we already know, only the vicinity of the stationary point gives the main contribution to the integral. For this reason, the integration hmits can be changed, and in place of eq. 7.146, we can write J„ = e - ^ ' ^ " M

|exp

dou

^2^nM(^-^0)'

(7.147)

— OO

Since this integral is tabular oo

/ exp ^ 2 ^ n H ( ^ - ^ o ) ^

duj

2 7r

-,—TT e r X''

^±iE 4

we have ^

/ ^^ r \x''

p - ^ ^ n M ^ ie=^ ^ f4

(7.148)

,

where the plus sign is taken if xJJ < 0 and the minus when x^ > 0. Finally, making use of eqs. 7.144 and 7.148 we obtain ^ri{t) = -[TX

^

/

VTrr

.

U

I // I coslujt-XnV-

^/32 + ^2 V^ <

V

- ± -

4

-tan

—-

4

/?

or rr /.x _ £ ^

Gn(u;o, r, t)

cos I u; i — 2;„ r — tan ^ -5- 1

if

a;J^ < 0

^ r - ^ | x ; j | ( / ? 2 + a;g) (7.149) Ur.it) = —

^ " ( " « ' ^ ' * ) = sin I cj ^ — Xn r — tan ^ -r- 1

^'•yKIC^M^

if

x'' > 0

These equations describe transient modes corresponding to the ground and water waves. They allow us to calculate the waveform at any distance from the source, as long as the observation point is located in the wave zone. Thus, the procedure of obtaining the wave field consists of the following steps: (a) The group velocity Cgn = r/t is found for any given values of r and t.

7.7. EVALUATION

OF NORMAL

MODES

493

|i/WW0

20

40

60

80

100

C^t/H

Figure 7.12: Transient mode at different distances from the source. (b) The stationary point CJQ is calculated from the dispersion curve Cgn{
dcj

= 0

and

= 0,

(7.150)

which indicate that the group velocity varies relatively slowly near frequency CJQ, and, therefore, constructive interference is observed. Assuming, as before, that only a small range of frequencies close to LUQ makes the most significant contribution, we use Taylor

494

CHAPTER

7. PROPAGATION

OF WAVES INSIDE A

LAYER

Ai((J

Figure 7.13: Airy function. series 7.136 again: ^n{^)

= an + aniy -\-Cnl'^

^

...

(7.151)

The term proportional to z/^ is discarded, since the conditions 7.150 are met at UJQ. The coefficients o;^, a^, and c^ are given by eq. 7.137. The second term of the series, a^v = \t

I u

in general differs from zero except those pairs of r and t, when the equality t = r/cgno takes place. Substitution eq. 7.151 into eq. 7.143 gives CO

In =

exp [ -i (^an -i-aniy -\-Cn i/^jJ duj, —oo

because the value of the integral is defined by the vicinity of UJQ. Bearing in mind that ly = (JU — (Jo

and

du = du ,

we have exp [ -i (^an -\-aniy-\-Cn ly^)] du

In= —oo

In accordance with eq. 7.142,

(7.152)

7.7. EVALUATION OF NORMAL MODES

.

J _

495

. 0 0

2G„{uJo,r,t) r ^ ° ' ""' ^^ /"cos(a„ + a„i/ + c„i/^)di/

(7.153)

The integral in eq. 7.153 can be presented as oo

/ COS (an -\- any -\- Cn y^)du

(7.154)

0 oo

oo

= COS an I COS [cLn ^ + Cn fj

dv — sin an I sin [an y -\- Cn yj

0

du

0

Since u is small, the second integral can be ignored and we have oo

oo

/ COS [ocn + ani^ -\- Cn y) du = COS an I COS (a„ y -\- Cn yj dy 0

(7.155)

0

The last integral is expressed in terms of the Airy function Ai{C,) (Chapter 3) in the following way: oo

j COS {anV + Cn u^) dv = T ^ ^ ^ W MO

,

(7-156)

where

c = 3^/2 '

(7.157)

>^ \Cn

The behavior of the Airy function is different for negative and positive an = t — r/cgn (Fig. 7.13). In fact, for positive a^ Ai{() gradually decreases with increase of its argument ( and decays almost as an exponent when C is large. For negative a^ Ai{C) passes through the maximum and then slowly decreases. Finally, from eqs. 7.130 and 7.153-7.156, we obtain U (t)

=

^ ^ - 4 V2¥Gn(c^o,r,0

AijC) (7.158)

UQI-

Xn(a;o) r - - - tan ^ — I

The amplitude of the Airy phase decreases with distance r as r~^/^, that is, shghtly slower than amplitudes of water and ground waves. This means that far away from the

496

CHAPTER

7. PROPAGATION

OF WAVES INSIDE A

LAYER

source the Airy phase may be a dominant part of the signal carried out by a given mode. This phase represents the amphtude modulated signal, as the frequency of oscillations UQ remains constant.