133
318
THS

PRQPAGAHQN C)? A fiGUNE PEJLSE EM 13¢
MEfiEUM WETH [a CGMPLZX ELASTSC

.‘iGBULfiS

Thesis for fine Degree 91‘ Ph. D.
MlCHIGAN STATE CGLLEGE
Saiah izzai‘ Tahsén

1953

    

    

Thlsiltoeenlfgthat the
thesis entitled ‘ _
”Dqum... 4 a 3......\ P4“. :
”.2. a. My...“ uldk ..
Q... Wmmsm Mum

$4“ Ieaad TL...

r

has been accepted towards fulfillment *
of the requirements for

 

3

Major professor

Date “N. \O

Salah Izzat Tahsin
Candidate for the degree of

Doctor of Philosophy

Final examination: 3-5 p.m., Friday, November 20, 1953, Room
314 P.M. Building

Dissertation: Propagation of a Sound Pulse in a Medium with
A Complex Modulus

Outline of Studies:
Major Subject: Physics
Minor Subject: Mathematics

Biographical Items:
Born, Feb}. 10, 1917, Baghdad, Iraq
Undergraduate Studies, Am. U. of Beirut, Beirut, Lebanon,
1938-1941.
Graduate Studies, Michigan State College, 1949-1953.

Experience: Teacher of physics in secondary schools of Iraq,
1941-1948.

Member of the Society of Sigma Xi, Sigma Pi Sigma, Pi Mu Epsilon.

PROPAGATION OF A SOUND PULSE IN A MEDIUM WITH

A COMPLEX ELASTIC MODULUS

BY

SALAH IZ ZAT TAHSIN

A THESIS

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

DOC TOR OF PHILOSOPHY

Department of Physic s

1953

ACKNOWLEDGMENT

I wish to express my gratitude to Professor R. D. Spence for
suggesting the problem and for his continuous help in every phase of

this work.

PROPAGATION OF A SOUND PULSE IN A MEDIUM WITH

A COMPLEX ELASTIC MODULUS

by

Salah Izzat Tahsin

AN ABSTRACT

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

DOC TOR OF PHILOSOPHY

Department of Physic 5

Year 1953

 

App roved :2 D 3%“,01

SALAH IZZAT TAHSIN ABSTRACT

In the first part of this thesis, the physical causality prin-
ciple along with the assumption that a real (not complex) cause
gives rise to a real effect, is used to derive the following proper-
ties of the complex propagation constant k(u) _ .

1. It is analytic in the lower half of the complex u-plane.

Z. Lim kiu)=% as u_.(n ,

3. Its real and imaginary parts satisfy the Kronig-Kramers
relations.

4. It. possesses symmetry properties; i.e.,

RT) = -k(-fi) .
where the bar represents the complex conjugate.

The second part of this thesis discusses the propagation of a
plane wave pulse in an infinite, homogeneous, and isotropic medium
whose elastic modulus is assumed to be represented by a simple
form consistent with the relaxation theory of the elastic moduli.

Since the Fourier integral which occurs in this problem cannot
be evaluated exactly, two approximate methods are employed to find
the shape of the pulse. The precursors of the pulse are shown to

be exponential in form, negative for w°<<g and positive for

SALAH IZZAT TAHSIN ABSTRACT

(A)¢,>>‘/-,-t!I , where (a) is the amplitude of the relaxing part of the

modulus and (T) is the relaxation time.

TABLE OF CONTENTS

INTRODUCTION ..............................

PART I. FUNCTION THEORETICAL CONDITIONS
ON THE PROPAGATION CONSTANT OF A SOUND
WAVE .................. . ..................

PART II. THE PROPAGATION OF A SOUND PULSE
IN A RELAXING MEDIUM .......................

Evaluation of Integral (20) .....................
Case 'I ................................

Case II ................................
Mapping the complex plane .................
Location of the saddle points .................

The integration path ........... ’ ............
Discussion of the solutions (29) and (39) .........

The pulse ............................ .
CONCLUSION .......... ' ......................

12

14

17

21

22

Z4

Z9

35

36

43

45

C i

54.

IN TR ODU C TION

This thesis is divided into two parts. In Part I we shall
find some properties of the propagation constant applying the
causality principle and the assumption that a real (not complex)
cause gives use to a real effect. In Part II we shall discuss the
prOpagation of a plane sound wave in a dispersive medium, charac-
terized by a complex propagation constant.

Before we start the work on this thesis, we would like to
give a brief account of what has been done in connection with the
principles we are going to use.

The physical causality principle, namely that the effect cannot
precede the cause, has been, in various equivalent forms, the reason
in establishing some properties of many physical constants. Upon
the suggestion of Kro'nig (1), Shfitzer and Tiomno (2) analyzed the
relation between causality and the scattering matrix 5. They found
that, for the causality principle to apply, the analytic function S(k)
must have its singularities either in the lower half plane or on the
imaginary axis. The same principle is believed to be the deep
cause of the properties of Wigner's function (3) R (the reciprocal

logarithmic derivative of the wave function). Van Kampen (4) worked

on the same relation and found more properties of the S-matrix

for the scattering of the electromagnetic field by a fixed center and
for nonrelativistic particles. E. Hiedemann and R. D. Spence (5),

by applying the causality principle and using function theory methods,
obtained the Kronig-Kramers relations (6) between the real and
imaginary parts of the complex elastic modulus K. They further -
found that the singularities of K must lie in the upper half of the
complex frequency plane.

Sommerfeld (7) and Brillouin (8) discussed the propagation
of light in dispersive media and established the presence of ”pre—
cursors” or forerunners to the main pulse. They showed that the
first precursor travels in any optical medium with the velocity of
light in vacuum. The propagation function for light waves used by
Brillouin leads to pure resonance phenomena. In our work in Part
II we Will use a propagation constant for sound waves that lead to
pure relaxation phenomena. The mathematical difference between

the two phenomena will be discussed at the end of Part I.

PART I

FUNCTION THEORETICAL CONDITIONS ON THE PROPAGATION
CONSTANT OF A SOUND WAVE

In this portion we define the propagation constant and

present a discussion of its most general properties.
Let a pulse be applied at the plane x = O in an infinite,

homogeneous and isotropic medium. The amplitude of the pulse in

this plane is given by

f 0 (<0
(1)

f

where {(0,1‘) is a real quantity. The frequency spectrum of the

{(0.19 5 %0

pulse is then.

<p(u.) = .._’__ {(0,6) e dfi (2)

Where (it) represents the complex frequency a.) + ill which is intro-
duced to insure convergence of the Fourier representations used in
this thesis and to facilitate a discussion of the problem in terms
of the usual methods of the theory of the complex variable.

By application of Cauchy's theorem and Jordan's lemma it

Can be shown that (1) implies that in the lower half plane LP(“) is

i . "=.§"3II—'—

analytic and tends uniformly to zero as u an . The statement
that {(0,(') is real leads to the requirement that (pot) a (Ppfi)
where the bar denotes the complex conjugate.

The amplitude of the pulse which arrives at the plane X¢o
can be found by superposing all the plane waves which belong to
the various frequencies in the original pulse. These plane waves

{at-luau]

are of the form 9 where kuu is the propagation constant

appropriate to the frequency (a) . Thus the amplitude of the pulse

in the plane X¢o is given by

«boo—id

(Iaf- ‘(UJXJ
f(x,é/ = 50m; e 61¢ (3)

~co-th

If we set

99‘“) = P’mwso’m
then

too-I'd +0044
f(0,t) = [wzalmaf~lf?u).dévaljdu +5 (fiftyAJnuf-i-Ylulanulj d“ (4)
-aoac'cx -coq"
The response must be real since the pulse is real which demands

the vanishing of the second integral in the last expression, thus

 

4-09-5d
filfhjm 441‘ +507“) £01 at] da- = 0 (5)

-w—o'd
This in turn requires that
(any) = W’(-- «1.11)
307(4),“ = -. 90"(—w,2/)

This is the condition for a real pulse at x = 0.

Now let

. zL . u
90m; .2“ a sum 2 Van +cym)

where
SUI“) = (Pkujuvat‘ - Viz“! dénaf
9/7“) = (,Q'(u),4é4a/+ 99'7“] mat
and gym,“ a 5117-“), 11) , (Jody) = .. 5V”(-u),1/)
Thus equation (3) becomes
+ao-c‘ot -
~¢°£(“IX
f(x,t) .-. 5w“) 9 ‘ du
—ao-¢.'d
(6)
1-m-CK +OO-‘O.d

gnaw-ah +y"(u)4¢hix]du +i [Cl/"(441w 4x - Wu)». idea

«I’m-Dc.“ "ND-'5.“

Let Am) a ("(4) -t' 1‘7“) (7)

Substituting (7) in (6) we Obtain

+00-“

f(x,t)=/[y’cmu.'x1wo/tk it) + y‘hsawuu mn’fl'k)

— V910}: (I. ’x} Md "xi 1- V (ca-1M 'x) MM'X/Jcfl,
+09-£d
+.~ ”Cessna/(A's; - weavré'wM/é "x2

«vb-clot

+5Vlicay(fl’x)601{(£&) - Viao'nli'x) M(‘”X)I (2’64

Requiring a real response as before leads to the following

p rope rtie 5

FM, 22) - _ /r'(—w,z/)

Anna, 2/)

k ”(— ‘4): 7))

However, if we expand ‘(UJ in a power series

. __ a .- g- , -. ll 8
Am) .. 7km; - C[law ¢k‘(u)] ()
where
k7“) = I A .21. ---
+ a. + a. +
k"(u)=_é:. 2.5.. 3.5.: .---
a + a3 + a, +-
then

K’(w,7/) == Kl(-“)/V)

‘K"(w,2/) =_K"(-w,u)

Hence K(”-)= K(-E) .
At a certain plane Xoal-O there is aT¢0 such that the

response is zero for (17', then

+ 00 -c'a(
~¢.k(u)x° [at

(pm) 6‘ e du [- < T (9)

II
0

40-64:
If we make the substitution
the last equation becomes
+OD-éa!

-"iugxc —¢'a//4/ +c'aT
(pm) e e da = 0

Mao-('0‘
Now we set Xo=cT. where (C) is an undetermined constant,

to get the integral in the form

«:01
+00 («[I- 29‘de «kt/H .
90m; 9 . e c!“ = o (10)
-m-(d

The condition that this integral vanish requires that

' -5.‘ ]T
t I (a)
(a)(’0(a) e“f u is analytic in the lower half—plane.

(«[I- fikatllT -zulH
(b) (.f(a) e e a 3 o (11)

where C is an infinite semicircle in the lower half plane.

From (b), on the application of Jordan's lemma, we find that

(up - 724(th T

 

integral (10) is satisfied if ('0(u)e is analytic within C
and tends to zero uniformly as Ul—v- 00in C . But we already stated
that

got“) 7‘ Q” P 0

in the same region. Therefore

(«[1- ékflle
e ; finite constant
a —-> go

and («[I— §£(u)]7' a. -—)eo‘> c' D

 

where D is also a finite constant. This leads us to the result

 

+41... .17..
m) “a” r c c7. "’2“ (12)

which shows that the undetermined constant (c) is the velocity of

propagation as the frequency tends to infinity. By virtue of (8),

 

(12) shows also that kfll) 4, 1 (12a)
ll—Pw

(up - f‘IUIJT 4' [7(a) g' [“(Wayl 4- iflfldfi'l]
e

Let Q = e

' . -/3. . . "
e and +4? ma = 3 +ul

I
Upon the application of the Cauchy—Riemann conditions on A

0!
and A , and after some simple algebra, we obtain

25-2.4 ' a~._a
9w'3v and 57 3g ’

which proves the analyticity of FY“). We can also write
£007.: aT-Ffl‘)
to Show that (mu is analytic, since the sum of analytic functions is
analytic.
Making use of relation (8) and following the method of E.

Hiedemann and R. D. Spence (5), we obtain the following:

 

(“L‘- “21)

5b) )hfF/w“ KW) “ du. (13a)

5:91). a 41—91. .___ aloe (13b)

«2 (aha/'0

moo) .. 1 -.. _I_/___k"(“) an. (14)
77' u
K’au 2 Ken) 4- #jk'lu) ..._~£_.. du. (15)
ul_wl

The above four forms of the Kronig-Kramer relations can be
supplemented by the following four forms if we assume that K(“)

possesses no singularities on the real frequency axis other than the

o rigin

10

 

 

, co
K1212 = £32. + 3.1” m._al_ 51/“ (16a)
w w 77' a (60-40”
.59.! : _ .2. P 4’7“). “ a/ 16
w W U (u‘-w‘) “ ( b)
w
Kurd-K10); _ .2. j K’” 021. (17)
77' a

klw) =K(oo)+_2_P {mu—“— d“ (18)
. 7" (a

‘-W")

where (P) stands for the Cauchy principal value.

Equations (13a), (13b), (14), and (15), as well as the last
four relations, make clear the reciprocal dependence of the real
and imaginary parts of (RI).

The singularities of K(u) have been shown to lie in the upper
half plane. The effect of such singularities on the propagation con-
stant may conveniently be discussed on the basis of whether they
lie on or off the positive imaginary axis.

The singularities which lie off the imaginary axis physically
represent resonance phenomena such as occurs in diSpersion of

light waves in optical media and which are almost never present

11
in sonic media. The singularities which occur on the positive
imaginary axis represent relaxation processes and lead to the type
of dispersion found in many sonic problems and in electromagnetic
waves of long wavelength. Throughout the remainder of this thesis
we shall assume that all the singularities of Kt“) lie on the positive

imaginary axis.

PART II

THE PROPAGATION OF A SOUND PULSE
IN A RELAXING MEDIUM

In this portion of the thesis we consider the propagation of
a plane wave pulse in a medium characterized by a complex prop-

agation constant which arises from a complex elastic modulus.

We assume, as in Part I, an infinite, homogeneous and iso—

tropic medium. The pulse applied at the plane X20 is given by

{(6) :0 ('(0 ,é>A

f“) =44" (4.1"

0

IA

tell

where “AH? .51- and A = m ..

p is the period and m is an integer.

Mp,

The Fourier transform of {(Z') is

A -;
-4“
(9(a) =/Mw,[' e c!!-
a

A (TEL-‘0‘! -‘:(wo+“)t
=:/_§ a2 dé
O

 

 

 

(19)
26
zrrc'm -££:—:"~u vzfl'im -£’_"_‘£Zu
___!_ e e -I + e e ’ -:_
2( (“do ‘“)

an). +u)

. mm“ 13
"‘ «I. /
rpm) .-.- w. e "
(I'M-6d}

 

The amplitude of the pulse observed at a point X=I50 at a
time (t) is given by

. “a ('[al‘ - hm]
font) ”iii/(Pm) e 0’“
~00

+oo

-——— _«_/._ - e do: (201

177' (flu-6d,?-
~00

: L+Iz

( if“! aha) x] ([uu - A) - km) 1:]
e 1

To find an expression for ‘02) we assume that the relaxing

modulus of elasticity may be represented by

 

M(«) = Mano) (1— liar)

Where (a) is a constant characteristic of the medium and('Z'/
is the relaxation time. This is the simplest form of the complex
modulus which can be employed to exhibit the effect of relaxation
on the propagation of the pulse. More general representations
would include the contributions from a spectrum (discrete and/or
continuous) of relaxation times. In actual practice it is often found

that the previous expression for the modulus suffices over a fairly

wide frequency range.

14
As was shown previously, (C) is the velocity of propagation

as (0.) tends to infinity, thus we write

c 4.47.221. and q; .,. /M(a)
D "”1:

where D is the density of the medium and (1!) is the complex
velocity associated with the frequency (u) .

By definition the complex propagation is

 

h<u2=£‘.=_‘L.————— =_“.-— 43—5—55 21
1r C /,_ a“ c. (I-a)+£ut’ ( 1
l-Hiut'

 

This form of 4(a) satisfies the restrictions which were de-—

veloped in the first portion of this thesis. From (8) and (21) we get

 

/ +£u‘c
u _ _._
K( ) '" (I-a)+£u1.'

 

which is analytic in the lower half-plane. And

a —-—~ oo (22)

It possesses two branch points on the positive imaginary
axis. Furthermore, Kl“) = K(—¢7) .

Evaluation of Integral (29;

Making use of (22), the exponent in 1. ,equation (20), becomes
(«U—‘55) as u -+ao , it has a negative real part,

in the lower half-plane when [-<

nlx

in the upper half—plane when f > €—

Similar results apply for iu(5-— {3) , 5: l ._ A

15
x
When t < E-
Both I, and I2 vanish along a path deep below the real axis.
When .5. E L‘.
c < < c + A
In this case the integration path has to be moved up to the
upper half-plane. In so doing, the path encounters the singularities
at u at“), and the branch cut and branch points, if any, of the func-
tion 4(a) . These, as was assumed earlier, lie on the positive
imaginary axis. Figures 1 and Z Show the paths for I, and II.
The contribution of those parts of the path around the singularities

two are easily found by Cauchy's theorem. For this case we obtain

—‘“(“)0) " £- - h
I, = e :19» wai— Hum] +;'— Li’zL. e 0‘ JQa’u (23)

11:0
When f>—é-+A

In this case I, still has the value (23), and It becomes

(:(tlg -- ‘5)
e

—£?WO)X
I,2 -.- — e MPJ -l:'(w.)x] - {if/fli— 6!“ (24)
C

, (flu-w."

Thus in this case the main pulse cancels. Only the integrals
in (2.3) and (24) remain. The integral in (23) will be discussed fully

later. The integral in (24), however, starts when the main pulse

 

 

 

 

 

 

 

 

 

e e —>w
-UJo +wa
.5.
K C
are >—

 

 

Figure 1. The integration path for I, , eq. (20).

 

 

 

 

 

 

 

 

 

V
t>€~+ A
v
V [j .%_. V
, '1- a
C L “r
._ t.\ A.\ w
v U T”
“we +wo
Riga-A
\ a: as

 

 

Figure 2. The integration path for 12 , eq. (20).

l7
vanishes and is of the same form as the integral in (23) except for
the sign. Since we are mainly interested in the manner of build-up
of the pulse, we shall, in the following, devote our attention to the
integral in (23), namely

' f-fi
LIMEL ed“ ’0 a’u (25)
C

277 a‘-%‘
1

.0.

The contour C! can be deformed into other paths, such as
the one shown in Figure 3, after reorienting the branch cut.

However, the evaluation will not be carried out along C’ or
the latter alternative path. Instead two approximate methods will be

used. One for «2,, << % and the other for Cd. >> 7?: .

Case I. (do << 4?:

We first make the following changes in our variable

€=¢r=P+C7 9

and set

ai—lzx. $41— .9 -(k—C-i—n = 55:51”— k; ,

an ’3 —§ --§- 1:55— .— ) o
d K .é at 'c': (1%)“: /

Now consider integral (25) in terms of the new variable.

where [-*=[- ..

nix

 

’Q

 

 

 

 

 

 

 

 

 

 

 

18
. I- O
: 4*
U (I) T k
F K .
\E J) ”‘0’ + '-
N
P (A)
Figure 3. Alternate path for integral (25).
_ I}
u a P

 

 

 

 

 

 

 

7

Figure 4. Integration path for integral (26) and its equivalent.

19

P “$6 - K‘a/éx)
[(2949: -25? 21.32., e (26)
Y

where ‘7' is the path shown in Figure 4, along with two more equiva-

lent paths. Foré )>l , we can write

,

‘C-(PZ)

 

 

 

 

act c‘r‘
Let now
. .s. a ’ .. aw)
6 2c? and “ th'(a 9' 7
then
. 6 ‘ aer'
—5x ‘(T'f "' )
_ P. 1 5
[ow - - n. e .9. 6 0'5 . f3. «)5! (27)
Y .
We write
g); «at _ (“(5. J; .. :5. a:
r g 1‘ xx 3 1
Let
v —(-0- (a-
__§_E%.=le and égfia-ie (28)
then ét' .. “"T = 2" Fax (.010-

 

20

Now onU')
“F
E, :: Re
multiplying by TL 25—),- , we obtain
' . 7r
.. r--—
$77 "r7". Pe‘he“ " from<28>
_L._t_‘_ R = R = r/“x
r xx I or 7‘..—
l I" u'a-
a d _ ._ .______.
n E! ' axr‘

Function (27) now becomes

-ex €42.”de Cavr+r)
‘C’ «tx 2—77'0

-ex _ 4
zé—lfi—e .(‘L(2L/£“X) (29)

For small values of (a) we can write

_ l -9. ‘ ~ _____..Q
or ' zct’i(a ‘f'a _ let"

then
-——(5—-x
{01.0.13 /-3—(€-/) e “r "1(1'3-5)‘ [—(9-0) (30)
where 9:55: -

For very small (9-!) , (1,03) :1 ——£—y , so that in this case

we can write

 

/(x,H-.- _w,-g— e “r -(6’-/) (31)

21
To get an idea of the orders of magnitude, let us take
.5' —6 5'
a=0.1 , C=/.5K10 (rm/sec. , T: 10 sec. , LU, = /0 C-/sec.
Then for (0-!) of the order of 10-2, f(x,{) is of the order of

~3 . . .
la . This is a very small magnitude compared to that of the

pulse.
The interval of validity of expression (30) is obtained from

the condition in (27), i.e.
F. —. _. 95:5
Po <<tg‘ whare ’él "' R- t tax
From which we get the condition '
6—] << ___£z___
. 2%1'Z'I

To make the expression valid for a large interval, we demand

that
Id." 2" << a
And hence the limitation on (’0) which was set for this case,
0
i.e. (do << % .
. Md
Case 11 (4)0 >7 _

T

For this case we need to use the saddle point method. This
method is used because of the difficulty the function hat) presents
in a conventional evaluation. We have already defined the propaga—

tion function as

 

k(u) =.‘£.k(u) :3. /*i“: See (8) and (21).
C C (1*“)+Lu.r

22

Thus 4(a) has two branch points, one of the type of a pole
at u 35-51—23 and an ordinary branch point at “=54. .

The first of these points presents an essential singularity of
the integrand which renders the evaluation impractical. An approx-
imate nethod (the saddle point method) will be discussed in the
following.

The complex u-plane will be surveyed as to the negative do-
mains of the real part of the exponent for varying(€). Saddle
points of the exponent will be located and shown to shift with time.
The ‘path of integration will be made to run in the negative domains,
passing through saddle points whenever it goes from one negative
domain to the neighboring one. The integral will have negligible
values along most of the path. It yields an appreciable value only
when it goes through a saddle point, where the real part is not too

negative. The shifting with time of the saddle points forces the path

to intersect the poles and thus give rise to the main pole.

Mapping the complex plane.

 

The integral to be evaluated is integral (25). We rewrite

the integral in terms of the new variable

23

.1. We§[g-K(€)]

I P. (I
-{7}. 27:2? 8 J45, (32)
cl
Let now
we; = .' efa-Kzan .- X(,o.7) +z Y(,a.7)
where

 

X(,0.7)= —- 79 +7Em(wgfi) +Pf-ETAJ.‘ (ii—fl) (33a)

Y‘W= P9 -ff-%i:““(i§-f‘) WE” (fig-‘52) <3”)

 

 

r, afiu- (7-1)‘ , r; a/l;+[7-(I-dl]‘
if}: 5:14.151. . , (fl: andI-Ig-a}

It was shown earlier that K(§) 2:?» 1, so that for large &, we
can write
ME.) = if. (6-!) = rpm-I) - 7119-!)

Then Xz-ll(9-I) behaves as follows at infinity
For 9<l is negative infinity below the real axis.

is positive infinity above the real axis.
For 9:] is zero.
For 5)] is positive infinity below the real axis.

is negative infinity above the real axis.

24
The domains of X(’o,7]) in the finite plane are shown in Fig-
ures 5 to 8. In the shaded region X is positive; in the unshaded
region it is negative. These figures were obtained by plotting X(/0,77)

as given in (33a).

Location of the saddle points.

 

Since the function
7N6 z X09171) +£ Y(,o.7])
is analytic except at 5:20 and .5: ((I—a) ,X60.7/ and Yfp; 7)
cannot possess finite maxima and minima, they can only possess

saddle points, at which

Xgflgflgg—yaa )
‘gp 97 9F 7
orwhere
Bar“
22:“

Carrying out the differentiation and equating to zero we get
the quartic,
't . 3 —a z. . :3 b bu- 5') 3 ‘(I-b
e, -.(sb+1)g —[3b(1+6)+1L—H;‘ lg+¢[1.(+)-————jg +[b’b19'JI (34)
Where b = I— a
The solutions of this equation represent the loci of the saddle

points for a certain (a) and a varying (3) . Since the algebraic

solution of this equation for arbitrary (a) and (9) is unwieldy, it

1’ .5

Figure 5. Topographic map of the function X(P,77) in the complex
(fa-plane. Shaded area is positive.

\\\\\\

9<1

 

Td

 

I
|<9<fl:a—

 

 

\V \\\\\\‘

Figure 6.

26

n

 

7/1,?

 

///

Figure 7.

 

Figure 8.

27

was solved numerically for a number of values of (a) and (9}

sufficient to obtain a general picture of the location of the saddle

points.

The exact solution of the quartic for a:a.1 and varrying(€)

is shown in Figure 9.

Thus two sets of saddle points are obtained, one pair starts

at 55:11» when 9: I and proceeds towards the finite plane with

increasing (9) The other pair is in the finite plane and on the

im inar axis at 9:1 . The two points of the latter pair a roach
ag Y PP

each other with increasing (9) until they meet in the upper half—
plane and fork out as shown.

’ It is possible to obtain an approximate representation of the
locus of the points on the imaginary axis for values of (9) very

near unity. For such values of (5) , the saddle points lie very

near the origin where Ea <1 This makes it possible to drop the

te rms with the third and fourth powers of (5,). Thus we obtain

the following quadratic which proves to be a very good representa-

fi-On of the locus:

[3b(+b)+:b(;; 1‘qu "‘("”’*”"E§%39]’7 [13.60 +17; o (35)

 

'r) 2 8
l /—\\

_,,,o

 

 

Figure 9. Loci of saddle points for varrying(6). Arrows show
direction of shift with increasing (9)

 

 

 

Figure 10. Location of saddle points on the imaginary axis for
varying (9) . The solid curve is obtained from eq.
34 and the broken curve from eq. 35 and both for a=0.l.

29
Figure 10 shows how good the approximate equation is, es-
pecially for small values of (6), which we are most interested in.
We can obtain another good approximate expression for the

locus of the other pair of saddle points, if we divied (34) by (35)
and drop the remainder after obtaining a quadratic. We will prove

later, however, that the contribution of these saddle points will be

canceled by the contribution of other parts of the path.
The integration path.

Integral (32) is to be evaluated for 6)] and ’00 >> £25:

Figure 11 is.the same as Figure 6 with the addition of less
negative ridges in the negative domains and the integration path.
The integration path is (—oo)fadcd'6{'(+ao) , made to pass through
Sfiddle points whenever possible.

For €<l , the path runs completely below the real axis
and gives no contribution. At 9:/ or l'z—é- , the path remains in
the lower finite plane except at infinity. However, the path has to

Shift completely to the upper plane at 9: VJ— , see Figure 7.
[-0

 

The part of the path going around the poles‘tP moves up before
0

9;. i . In so doing, the path intersects the poles and an
[-61

 

30

I
\
§ 0
\
I
I

/ I ‘\
f/——---—————~—1d dL——————~————
/ I '

 

 

 

 

Figure 11. The true integration path for eq. (32). a , b , and C are
saddle points.

 

Figure 12. The approximate integration path for eq. (32), before
(broken) and after (solid) the arrival of the pulse.

31

oscillating term is obtained, which builds up very quickly to an ap-
preciable value. We will discuss this point with more detail in a
later section.

The total contribution of the path to the integral is obtained
by finding the contribution of those parts of the path that are in the
immediate neighborhood of the saddle points. However, as will be
shown Shortly, the contributions of the saddle points(a)and(b) ,
Figure 11, are canceled by the contribution of that part of the path
crossing the ridge at the far end of the plane. This leaves the con-
tribution from the vicinity of the point (‘6) to be calculated.

QTO prove the above statement, we draw the shortcuts (df) and
(d7! ), Figure 11. These two shortcuts are equivalent to (faa’)
and (alblfl) respectively. Let G be the contribution of (a’f),L the
Contribution of the saddle point (a), and L‘ the contribution of the
part crossing the ridge between (1") and (a) . it follows that

G ,_._ L + L,

We now proceed to show that G is negligible, thus proving
that L ..’~.=-.LI and hence the contribution of the saddle point (a) is
ca—nceled.

Alongdd) and(dl‘Fl),&, is very large, so that we can rewrite

integral (32), making use of relation (12) as follows:

32

71 x x
.—..£,o(0-U .. 7- mo-u
’Zér/é‘efl- e c d5 A<g<o
«00

The second exponent is a constant, Since(7) is constant
along (id). The integral thus has an upper bound of 3’- , which is
very small except when the path approaches the pole. In this case
form (32) of the integral must be considered. However, by this
time, the contribution of the path in the neighborhood of the pole
becomes too prominent, giving rise to the main pulse, and the
contribution of the rest of the path becomes negligible. Thus, be-
fore the pulse builds up, the contribution G) of(fd) is negligible and
we are justified in [considering the relation L z—LL .as valid.
Furthermore, Q can still be kept negligible after the pulse builds
up by taking the path along a large circle around (3) . Figure 12
Shows the new path before and after it crosses the poles at if.

Next we proceed to find the contribution of the path in the
immediate neighborhood of the saddle point at (C), Figure 11.

We start by expanding the function

”((5) :2 Cafe "" K(58

in a. Taylor series around the saddle point as obtaining

we) -_-. 7/05..) --z'-|V”<'ci,))(e, .55)‘ (36)

33

whe re

(mag) =

. 3
ll-(l-(l) +a(4~3a)c£, - 35': '- 35 is
20"“)‘5’6651-72 (1 ‘ngsy/z

the first derivative is zero at the saddle points.

 

 

Integral (32) now becomes
1‘ . - . a, . - l.
I P. fiy(5,) fijvfidflé 5,)d
_... 7—— e .e 5., (37)
z7r g 70‘
At the saddle points, on the negative imaginary axis, the path

is parallel to the real axis. If now we limit our integration path to

a very small portion in the neighborhood of the saddle point, we can

write
5, : E +i. 75 where 1° 7: = as,
SO that d6, = d6.

Along such a small path we may consider ___F a constant,

E.

which is especially true for I0 >) E, as we are considering in

this case. Taking all constant terms out of the integration sign we

 

get
i’lflé ) +6 x a 1
Ct ‘ -227.er (£3)! 6
.. Be , 1 e de (38)
me. 437
-€
1/(55) is always negative and 7/15,) is of the order of ID“
for a =0.l . If now we choose X= Icm° : C=/o53’oacm'/5€¢o and

__1 u
T: ’0 sec. or smaller. The order of magnitude of fil’V(51)|is

34

about (3) or larger. This value of the exponent makes most of the
contribution come from a very small interval about the saddle point.
Thus, extending the limits of integration to infinity does not alter

the result appreciably. In so doing we obtain the result

-Ex?)v<e;|
P» e an 2’ (39)
“Ha; - A") xlv'ugn)

For this expression to be valid we should keep ESE-é? as
nearly a constant as possible. This can only be done by requiring
that

/2’ >> )a.)‘

An expression for the maximum value of (£351 is obtained by
setting (€‘-/) equal to zero in equation (35). The result is sim—
plified considerably if we note that (a) is usually very small com-

pared to unity. After some simplifications we obtain the following

expression for as Max.

lé’maxJ = Cl + J;

Now fi> Cl , so that we can write
,9 » fa“ or w, » Lg:

WhiCh is the condition of validity we stated for this case.

35

Qiscussion of the solutions (29) and (3_l.

Expression (31) shows a linear dependence of f(x,t) on (6)
for very small (0-/) . However, for large ((9-0 expression (29)
decreases exponentially and rapidly with increasing (5) . This ex—
ponential decrease of the function continues until the function loses

its validity according to the condition

_ .9:—
9 /<< 1’0,

obtained earlier. To find the order of magnitude of the function for
values of (€-/) restricted by the above condition, we assume 61-70.],
It: -.- o./ , 1': la-‘sec. , x =1 Icm. and Ca- I.SXI05cm./3¢c. . These
values require that €-/<< 5 . Let us choose €~/=a.05' , then
from (29) we get font) of the order of [0-3 which is very small
compared to the amplitude of the pulse. We will show later that

it is not possible to determine exactly the time of build-up of an
observable pulse. However, to be safe in applying expression (29),
W8 should assume a very small (9-!) in accord with the above con-
dition. This in turn limits the amplitude of the function to very
Small values until the arrival of the pulse. It can easily be seen
from expression (30) and the condition on(€-/) that, within the limits

0f Validity, an increase in(/:) decreases the absolute value of the

func tion.

36

Expression (39), on the other hand, builds up exponentially
from a finite nonzero value at 9==/ . It reaches a maximum at a
certain (9) which is a function off?) alone. The amplitude, in-
cluding that .of the maximum,is a function of both (,9) and ( T) ,
it changes directly with (T) and inversely with {/3 ) .

The difference in sign between the two expressions (29) and
(39) is expected because of the factor ETC-:33 which changes sign

a

as '5' assumes values smaller and larger than (’00). Figurel3
shows three plots of (39) versus (a) for f: -_-_/0 and three differ-:
ent values of (T).
- The contributions of both expressions (29) and (39) are usually

referred to as the "precursors" or ”forerunners” to the observable

pulse, which will be discussed next.

lhe pulse.

 

We have shown in the last section that the amplitude of the
precursor is very small compared to the pertaining pulse. We also
Stated that, the moment the integration path crosses a pole, the
amplitude builds up to a noticeable value.

Theoretically, the pulse arrives at the plane tho with a

Velocity (C) , the maximum velocity. This is certainly true of those

“fix,”

7'

- l0
0.: «»

 

ii f(Xal’
0.051» 1': l0"7

 

®
11

 

Aff(XaI)
T: 10-.

0.011

 

‘-
‘7

9=I 14

Figure 13. Plots of expression (39) versus
(9) for three values of('t') aides/0.

k ‘9

 

 

38
Fourier components of the pulse with the highest frequencies.
These, however, do not constitute the bulk of the components con-
stituting the pulse. Most of the components have lower frequencies
and hence lower velocities compared to (C) . The whole set of
components can be subdivided into groups of different average
frequencies which move with different average velocities. The
"group velocity” 72"? is defined by the relation

.L-i’ :Td—A, ) 40
113"“:‘M’ JP (,0 ()

where (6(a)) is a function of the real average frequencymu and is

the real part of km».

The group of frequency components centered on («2.) arrives

.9.
15?
Thus (6?) is the ratio of the maximum velocity to the group velocity.

at the plane X4=0 at t: 3; , and since 6:35- , at 9:19} =
Figure 14 is a plot of((€/) as a function off/o). This was
obtained from (21) according to the definition (40). The figure shows
that((%) approaches unity as the frequency tends to infinity. In other
Words, as the frequency increases, the group velocity approaches
the maximum velocity.
It is not possible to find a reasonable definition for the pulse
Velocity in a medium for reasons that will become clear later.

However, it is evident from the past discussion that the pulse

13.11:..Lh 7L

 

1“

v: “.02-

39

[.3 .0: .4

1.2'

 

 

I-O ’ 4 — a- a
0.5 LG LG P

 

V

Figure 14. 5’ as a function of(/D)(solid curves). Broken

curve
represents (6) as a function of gaff.

 

 

 

 

4O
velocity must not exceed the group velocity. We now make the as-
sumption that the pulse velocity is only slightly smaller than the
group velocity. This is especially true for high and low frequen-
cies; it is not so true for frequencies lying in the region of
anomaly.

The broken line in Figure 14 represents the limit of validity
of both (29) and (39). Expression (29) is valid in a region to the
right of the point of intersection of 'the broken line with the (9;
curve for a certain value of (a) . Expression (39), on the other
hand, is valid in a region to the left of the points of intersection.
The figure yields the following useful information. First, the
interval of validity of (29) increases with increasing (a) . Second,
9} is smaller for (39) than it is for (29). Taking notice of our
assumption in the preceding paragraph, we conclude that the signal
arrives earlier when (39) is used than it does when (29) is used.
Usually the constant (a) is of the order of [0—2, which means that
6; is very small as can be seen from Figure 14. Thus our choice,
in a previous section, of 6 : [.05 t0 find the order 0f magnitude

of (29), was rather generous. Nevei'theless, (29) was found to be

Very small compared to the pulse.

 

 

 

 

41
Figures 15 and 16 are schematic representations of what

happens at the plane X4=0 , using expressions (29) and (39) respec-

J‘. .5. (15,:

tively. Until l' = ‘2" nothing happens, at I": v =5 3
I ”I-

pulse velocity), the pulse builds up very rapidly to its full ampli-
tude.

Nothing was said, so far, about the decay of the pulse. In
equation (20) we had two integrals I, and 12. These two integrals
are identical except for their sign and their lower time limits. A
was picked such that 12 starts in phase with I, , but the difference
in sign puts them completely out of phase and leads to the destruc‘
tion of the pulse. Thus I2 yields the same contributions as I, ,
only the former leads to a trailing phenomenon instead of the
precursor. This tail phenomenon, not being of any special value,
is not shown in Figures 15 and 16. From Figure 14 we can easily
establish that, for the same(Cu), the pulse arrives earlier the larger
(T) is. Figure 13, on the other hand, shows earlier maxima for
larger T3. Thus we find that the rise of (39) to its maximum value

Occurs at the same range of (%)for the pulse arrival. In Figure 16

We let the pulse build up from the maximum value of (39).

 

 

 

 

42

I fix.”

 

 

 

+1

 

 

(Ix <—[

 

 

 

o|x <—

 

Figure 15. The pulse build—up for w,<<1.r@: , eq. (29).

 

 

 

 

 

 

 

 

OIX 4‘—
<1" <—

 

(a;

Figure 16. The pulse build-up for a), >> T 9

eq. (39).

 

 

 

 

 

 

CON CL USION

A pulse traveling in an infinite, homogeneous and isotropic
medium, characterized by a complex elastic medium, moves with
a maximum velocity (C) . This is the velocity of the Fourier com-
ponent of the pulse with the highest frequency. Other components
move with lower velocities, each a function of the pertaining fre—
quency.

Because of the above-stated dispersive property of all media,
a pulse arrives distorted at the plane X¢o . Of particular interest
to the Observer is the manner in which the pulse builds up to an
observable amplitude. Experimental determination of sound velocity
in different media is a case where a knowledge of the process of
building up of a pulse might prove to be of considerable importance.

We have found that a sonic pulse, in a medium having a re-
laxation time (T) builds up exponentially at first and later assumes

its Oscillatory property.

A detector of infinite sensitivity, at a distanceOUfrom the

Source of the pulse, registers the arrival of the pulse at f: .25..

As the sensitivity of the detector decreases, the detection of the

 

 

 

 

 

 

 

44
pulse arrival is delayed. This is one of many effects that make the
experimental determination of sound velocity subject to controversies.
Such a determination leads to values dependent on the following prop-
erties:

l. the dispersive properties of the medium,

2. the Fourier composition of the pulse,

3. the sensitivity of the detecting device.

Further treatment is needed to discuss the behavior of a
pulse for the case Paw/awhich is not cOvered by the approximate
methods developed in this thesis. However, it appears that, for
most of the values of (a) met in practice, the error introduced, in
the velocity determination, by the presenceof the precursor will not
be too large.

Finally, it must be pointed out that the general methods de—
veloped here may be applied to the propagation of an electromagnetic
pulse in a medium in which the dispersion arises from a relaxation
process. In this case, however, the values of (a) will not,in gen-
eral,be small and the effects of the precursor may be much larger

than in the sonic case.

 

 

 

 

 

 

BIBLIOG RAPHY

Kronig, R. deL. Physica. 12, 543 (1946).

Shiitzer, W., and J. Tiomno. Phys. Rev. 83, 249 (1951).

Wigner, E. P., and L. Eisenbud. Phys. Rev. 72, 29 (1947).

Van Kampen, N. G. Phys. Rev. 89, 1072 (1953) and Phys.
Rev. 91, 1267 (1953).

Hiedemann, E., and R. D. Spence. Z. Physik. 133, 109 (1952).

Kramers, H. A. Atti. cong. intern. fisici, Como, 1927, Vol. 2,
p. 545.

Kronig, R. DeL. J. Opt. Soc. Am. 12, 547 (1926).

Sommerfeld, A. Ann.-Physik. 44, 177 (1914).

Brillouin, L. Ann. Physik. 44, 203 (1914).

 

. . . ~ . . .. a. . .
L .. . a . . . H .. . J . I. an? .i. twin-Elf

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