Phys/Geol/APS 6610
Earth and Planetary Physics I
The Inhomogeneous String
The earth is neither homogeneous nor one-dimensional or two-dimensional, both key assumptions up to this point. We'll rst break with the rst of these assumptions and consider
waves of an inhomogeneous string, where density and/or Young's modulus vary with position
along the string. The transition to higher dimensions will be made in a later subsection. The
inhomogeneous string problem cannot be solved for a general perturbation in density or Young's
modulus ((x) or (x)), but there are a number of approximate solutions commonly used in
seismology that hold for variously constrained forms of these functions. We will consider three
of these: the JWKB method, the Eikonal equation, and rst order perturbation theory. The
rst two of these methods dene geometrical ray theory, a high frequency approximation that
holds in regions of smooth spatial variation of wave velocity.
The development of ray methods begins by referring back to the inhomogeneous string
equation that we derived earlier from Lagrange's equation for a continuous medium:
2
(1)
(x) @ y(x; t) = @ (x) @y(x; t)
@t2
@x
@x
2
(x; t)
= (x) @ y@x(x;2 t) + @@x(x) @y@x
(2)
where is Young's Modulus or tension for longitudinal or transverse vibrations, respectively.
Generally the next step invokes the following assumption.
Ray Assumption One: The spatial gradient of Young's modulus (0 =) is suciently small,
i.e. the medium is suciently smooth, that the rightmost term in equation (2) can be
ignored.
We are then left with the equation
2
2
(x) @ y@t(x;2 t) = (x) @ y@x(x;2 t) :
(3)
A more precise condition for dropping this term is that the change in the velocity gradient over
a single wavelength is small compared to the change in velocity itself. This will be discussed
further below.
Letting c2 (x) = (x)=(x), the spatial and temporal equations still separate (y(x; t) =
X (x)T (t)) and yield:
d2 T (t) + !2T (t) = 0;
0
dt2
d2 X (x) + !02 X (x) = 0:
dx2
c2 (x)
(4)
(5)
The temporal equation is exactly the same as for the homogeneous string. This is the rst observable consequence of Ray Assumption One and implies that frequency, unlike displacement,
is stationary relative to small changes in seismic structure: under this approximation frequency
1
is not changed by perturbing Young's modulus and/or density. This fact will be discussed
further in the section on seismic variational principles below. Equation (5) is dierent than the
spatial equation for the homogeneous string in that wave velocity is now an unspecied function
of x. As a consequence, we cannot simply set X (x) = A exp(iSx), where S is a constant. Ray
methods diverge on how equation (5) is treated, as we shall now see.
A. Ray Method One: JWKB 2
Under the assumption that c2!(0x) is a slowly varying function of x, we assume that
X (x) = A(x)eiS(x) :
(6)
Note that S plays the role of kx in the homogeneous string problem. We will discuss what
`slowly varying' means shortly. Upon substituting equation (6) into equation (5) we receive
A00 + 2iS 0 A0 + iS 00A , S 2 A + !c20 A = 0
(7)
2S 0 A0 + S 00 A = 0
(8)
2
0
The real and imaginary parts must separately equal zero, thus:
Imag Part
A00 + !c A , S 2A = 0:
2
0
2
Real Part
0
(9)
Another consequence of Ray Assumption One is that the spatial gradients of both S and A
will be small in comparison with the functions themselves. In particular, if we make a second
assumption, that we're dealing only with high-frequency waves:
00
Ray Assumption Two
A =A << !02 =c2 :
(10)
(11)
The real part equation then can be rewritten as
S 2 (x) = c2!(0x) ;
S 0 (x) = c!(x0) ;
Real Part
0
2
(12)
(13)
which can be integrated to yield
dx0 = ! T (x);
0
c(x0 )
Z x
dx0 :
T (x) =
c(x0 )
S (x) = !0
Z
x
(14)
(15)
The quantity T (x) is called the travel time and is range-dependent, and the quantity 1=c(x) is
commonly called the ray slowness.
The imaginary-part equation can be solved without modication. We simply note that:
d A2 S 0 = (2S 0 A0 + S 00 A)A = 0;
dx
2
(16)
where, by equation (13), we nd
A2 (x) = S 0C(x) = C c!(x) ;
0
(17)
and C is an arbitrary constant.
Finally, then, the approximate solution to equation (5) is, after substituting from equations
(14) and (17):
q
X (x) = constant c(x)ei!0 T (x) :
(18)
Combining the solutions to equations (4) and (5) yields
y(x; t) = X (x)T (t) D(x)ei!0 (tT (x)) :
(19)
Please don't confuse T (t), the solution to the temporal equation, with T (x), the travel time
here.
Solutions of the type given by equation (19) lead to ray theory as discussed in the next
subsection on the Eikonal Equation and which will be discussed in much greater detail later in
the course. It is important to note that equation (19) still has a D'Alembert-type form, where
R
the phase function t x dx0 =c(x0 ) gives the travel time of the wave through the medium from
the source at x = 0.
If one end of the string is xed, the exponential in equation (18) should be replaced with
a sine. If the other end of the string is xed at x = L, the eigenvalues !0 can be determined
from equation (18) by the boundary condition X (L) = 0 which will only hold if the argument
of the sin is an integral multiple of :
!0 = Tn
(L) :
(20)
This is the string analogue of the Bohr-Sommerfeld quantization condition of pre-1925 quantum
mechanics.
Ray Assumption Two is not a common form of this assumption. We can derive the more
common expression of the underlying assumption of ray theory by, instead of breaking equation
(7) into real and imaginary parts, we break it into parts dependent on and independent of A.
If we do so, we get the A-dependent equation:
iS 00 , S 02 + !c20 = 0:
2
(21)
We can get the next higher order estimate of S (x) than that given by equation (14) by substituting from equation (13) to nd S 00 = !o c0 =c2 (x) and
2
0
2 0 !
!
!
c
0c
0
0
S (x) = c2 (x) i c2 (x) = c2 (x) 1 i !
(22)
0
where c0 = dc(x)=dx. We see that the lower-order solutions for S (x) and A(x) given by equations
(14) and (17) have assumed that c0 =!0 << 1 or c0 =c << !0 =c or
0
0 cc << 0 !c0 2;
(23)
02
3
where we have multiplied both sides of the equation through by the average wavelength 0 =
2c=!0 . This condition can be rewritten as
Ray Assumption Two0
c0 0 << 2c;
(24)
which in words is translated as the change in wave speed over a single wavelength must be
much less than the wave speed itself. Thus, we have seen that ray theory holds for media
with properties that vary slowly (Ray Assumptions 1 and 2') and is also a high frequency
approximation (Ray Assumption 2).
The JWKB method is named after a British applied mathematician who did most of his
work in seismology, Sir Harold Jereys, and two German physicists and one French physicist
who studied quantum mechanical scattering: Wentzel, Kramers, and Brillouin. The British
call it JWKB and most US physicists leave Jereys out entirely and call the method WKB,
although Jereys was the rst to completely solve this problem. In fact, the main ideas date
back to Green and Liouville in the 19th century and Rayleigh used them in his treatise on wave
propagation in a stratied medium. Most seismologists call it WKBJ, but let's go with the
Brits in loyalty to Jereys as a seismologist. In fact, the JWKB method is considerably more
complicated than our discussion here because we have assumed that the solution is oscillatory
which is not necessarily the case if there exists turning rays as in the earth. For further discussion
see Aki and Richards, Ch. 9.
B. Ray Method Two: The Eikonal Equation
The Eikonal equation can be derived from the real part equation, equation (9), with the
variable change
S (x) = !0 (x)=c0 ;
(25)
where c0 is meant to be the average wave velocity in the string. With this substitution we can
rewrite equation (9) as
2
00
2
(26)
2 (x) , c2c(0x) = !c02 AA((xx)) :
0
By Ray Assumption Two, the right hand side can be approximated by zero and we get the
Eikonal equation
0
2 (x) = c2c(0x) :
2
0
(27)
Recall that S (x) = (x)!0 =c0 was just kx for the homogeneous string equation, which is
just the phase of the spatial eigenfunction. The spatial variability in the phase function S (x)
is completely summarized by (x). In multiple dimensions we will see that the wavenumber k
forms a vector k that points normal to the wavefront, or along a ray. The Eikonal equation is,
therefore, a PDE that relates rays to the seismic velocity distribution.
It appears that the Eikonal equation (eq. (27)) is complicated and that it is not easier to
deal with than the wave equation. However, we will see later on that very simple equations can
be obtained from it for rays which form the foundation for body wave seismology.
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C. First-Order Perturbation Theory
First-order perturbation theory is a very common technique used in seismology to compute
the eect of a small perturbation in some material property on the frequencies and displacements
of the earth. Let's consider rst the general outlines of the nondegenerate theory and then
specify the problem for the inhomogeneous string.
Consider again the dierential operator L. We wish to compute the eigenvalues n and the
eigenfunctions yn of L:
Ly n = n y n :
(28)
Suppose that the operator L is nearly equal to another operator L(0) :
L L + L;
(29)
(0)
(0)
whose eigenvalues, (0)
n , and eigenfunctions, yn , we already know:
L yn = n yn :
(0) (0)
(30)
(0) (0)
The perturbation operator L is small in some sense. Assume that the zeroth-order eigenfunctions compose a complete, orthonormal set and that the rst order perturbation in the operator
causes rst-order perturbations in the eigenvalues and the eigenfunctions:
(1)
n = (0)
n + n ;
X
(0)
yn = yn(0) + a(1)
mn ym :
(31)
(32)
m
Substitute the trial forms for n and yn into the original eigenvalue problem of equation (28)
and separate zeroth and rst order terms:
Zeroth Order
First Order
L yn = n yn ;
X
Lyn +
amn m ym = n yn + n
(0) (0)
(0) (0)
(0)
(1)
(0) (0)
(1) (0)
m
(0)
X
m
amn ym :
(1)
(0)
(33)
(34)
Multiplying both sides of equation (34) by yn(0) and integrating over all allowable values of x
we get the perturbation to the eigenvalue:
D
E
(0)
(0)
(1)
n = yn jLyn ;
(35)
where the Dirac braket notation was dened in our discussion of Sturm-Liouville problems and
Green functions. Thus, the corrected eigenvalue is
D
E
(0)
(0)
2
n = (0)
n + yn jLyn + O((L) );
(36)
where O((L)2 ) is read `terms of order delta L-squared'. To obtain the expansion coecients
for the perturbed eigenfunctions multiply both sides of equation (34) by yp(0) (p 6= n), integrate
over x as before, and solve for a(1)
pn :
D
yp(0) jLyn(0)
(1)
apn = (0) (0)
n , p
E
(p 6= n);
5
(37)
(0)
where we have assumed that (0)
n 6= p (nondegenerate perturbation theory). The corrected
eigenfunction is then:
D
yn = yn(0) +
E
ym(0) jLyn(0) (0)
ym :
(0)
(0)
n , m
X
m6=n
(38)
D
E
From equation
(37) we see that the perturbation L may be considered small if yp(0) jLyn(0) <<
(0)
(0) n , p .
You can see in equation (37) that the diagonal of the matrix a(1)
pn is not determined (p = n).
However, if the eigenfunctions yn are orthonormal then using equation (32):
hymjyni = mn + anm + anm + : : : ;
(1)
(39)
(1)
(1)
and if m = n then hym jyn i = 1 which implies that Re(a(1)
nn ) = 0 but Im(ann ) is completely
unconstrained and can be chosen to be zero without any loss of generality, thus:
a(1)
nn = 0:
(40)
This is the reason why in equation (38) the sum is over all m except when m = n. In this case
the denominator goes to zero, but so does the numerator and the term does not contribute to
the sum.
In summary, the recipe for the use of nondegenerate perturbation theory is:
Step 1. Solve the unperturbed problem (eq. (30)) for the eigenvalues n and eigenfunctions
(0)
yn(0) .
Step 2. Determine the perturbation operator L.
D
E
D
E
Step 3. Do the integrals yn jLyn and yp jLyn .
Step 4. Compute the corrected eigenvalue and eigenfunction by using equations (36) and
(0)
(0)
(0)
(0)
(38).
D. Application of First-Order Perturbation Theory.
Now, let's apply this theory to the inhomogeneous string problem (eq. (5)):
2
X (x) + !2X (x) = 0
c2 (x) d dx
2
L(X ) = ,!2X = X;
(41)
(42)
with xed boundaries (X (0) = X (L) = 0) by following these four steps. Note that we have kept
c2 on the left-hand side of the equation, this means that the eigenvalues are squared-frequencies
rather than squared-wavenumbers.
Step 1. Solution of the Unperturbed Problem
We already have solved the unperturbed problem where c(x) = c0 for the the eigenvalues,
(0)
n , and eigenfunctions, Xn(0) (x), of the homogeneous string. The notation X (x) is kind of
6
klunky with all those x's running around, so let's change notation and let yn(x) = Xn (x). The
zeroth-order eigenfunctions and eigenvalues are:
r
yn (x) =
(0)
2 sin nx ;
L
L
2
n
(0)
2
= ,n = c0 L ;
!n(0)2
(43)
(44)
where the eigenfunctions have been normalized (< yn(0) jym(0) >= mn ) and c0 is dened directly
below.
Step 2. Determine the Perturbation Operator L
From equations (41) and (42), if we write c(x) = c0 + c(x) and discard terms of O((c)2 ):
d
L = c (x) dx
2
2
(45)
2
d
= (c0 + c(x))2 dx
2
2
(46)
d + 2c c(x) d + O((c)2 )
= c20 dx
0
2
dx2
= L0 + L + L2 :
2
2
(47)
(48)
Thus,
d2 :
L = 2c0 c(x) dx
2
(49)
2
(0)2
n
!
n
(0)
Lyn (x) = ,2c0 L c(x)yn (x) = ,2 c c(x)yn(0) (x)
0
(50)
Step 3. Compute Matrix Elements
Applying L to the unperturbed eigenfunction yields:
(0)
so that
(0)2 Z L
(0)2 Z L
!
4!n
n
2
(0)2
yn jLyn
= ,2
(51)
c0 0 c(x)yn (x)dx = , Lc0 0 c(x) sin (nx=L)dx:
(0)2 Z L
(0)2 Z L
D
E
ym(0)jLyn(0) = ,2 !cn
c(x)yn(0) (x)ym(0) (x)dx = , 4!Lcn
c(x) sin(nx=L) sin(mx=L)(52)
dx:
0
0
0
0
Step 4. Compute Perturbed Frequency and Eigenfunction
The perturbed eigenvalues and eigenfunctions can be determined by inserting equations (51)
and (52) into equations (36) and (38).
D
(0)
(0)
E
D
E
(0)
(0)
n = (0)
n + yn jLyn ;
2
4!n(0)2 Z L c(x) sin2 (nx=L)dx;
!n2 = !n(0)2 + !n(1)2 = ,n = c20 n
+
L
Lc0 0
(53)
(54)
where we have substituted from equation (44) and
D
yn (x)
=
E
ym(0) jLyn(0) (0)
(0)
(1)
(0)
ym
yn + yn = yn +
(0)
(0)
m=n n , m
X
6
7
(55)
r
=
"R
r
#
nx + 4!n(0)2 2 X 0L c(x ) sin(nx =L) sin(mx =L)dx sin(mx=L(56)
sin
);
, m 2 , n 2
L
L
Lc30 L m=n
L , L
"R L
#
r
r
nx 4n2
c(x ) sin(nx =L) sin(mx =L)dx
2
2 X
0
sin(mx=L): (57)
L sin L + Lc0 L m=n
m2 , n2
2
0
0
0
0
6
0
=
0
0
0
6
where in equation (56) we have changed the sign of the perturbation term by swapping the
order of the m and n terms in the denominator and have factored a c20 out into the front from
that denominator and have then used equation (44) to get the nal equation.
It is common to proceed further in this derivation by expanding the wave speed distribution
in the zeroth-order eigenfunctions, in this case sines. This reduces the integral in equation
(57) to a form that can be calculated analytically. We will do this as an example below. It is
important to note that although the zeroth order eigenfunctions form a complete set for any
displacement of the string, they do not form a complete set for any wave speed perturbation.
For example, although all displacements are zero at the end points, the wave speed perturbation does not have to be. Therefore, commonly wave speed perturbations are expanded in the
eigenfunctions (sines) and their rst derivatives (cosines). We will see the result of this below,
but the terms of the expansion in cosines integrate to zero so they can be neglected. Nevertheless, this is an important general point, it is not always the case that the eigenfunctions form
a complete set for a dierent quantity { wave speed here.
Equation (57) demonstrates the phenomenon of coupling. The new eigenfunctions are linear
combinations of the old eigenfunctions. Consequently, if you would estimate the energy density
as we in an earlier set of notes, cross-terms no longer cancel perfectly and you would nd that the
energy of every perturbed mode of oscillation depends on the energy of most of the unperturbed
modes of oscillation. By mode, we mean a displacement pattern (or eigenfunction) that has
a single frequency. By unperturbed mode, we mean a mode of oscillation of the homogeneous
string { that is, a mode whose eigenfunction is a single sine. By perturbed mode, we mean a
mode of the inhomogeneous string, which is a sum of many sines, in fact it is a sum over all the
unperturbed modes. Modes are usually denoted by their index (or quantum number) n. We
say that the mode n is coupled to the mode m if the expansion coecient in equation (57) is
non-zero for those values of n and m. This will depend on whether the integral in this equation
is non-zero. We will discuss below what are called selection rules which provide information
about whether the integral is zero and, therefore, about which modes couple.
In summary, the perturbed squared-frequencies and eigenfunctions are given by equations
(54) and (57), respectively. The use of these equations requires the integral of a product of sines
and the velocity perturbation. Let's now consider two examples of velocity perturbations: (1) a
delta function perturbation and (2) a general Fourier series representation of the perturbation.
Example 1: Delta Function Velocity Perturbation
Let c(x) = c(x , x0 ). In this case,
!n
(1)2
yn
(1)
=
"
#
4c sin2 (nx0 =L) !(0)2 ;
n
Lc
r
n2 c 2
= 4Lc
L
0
0
X m6=n
sin(nx0 =L) sin(mx0 =L) sin(mx=L):
m2 , n2
8
(58)
(59)
Example 2: Fourier Series Velocity Perturbation
Expand c(x) in a Fourier Sine series:
c(x) =
X
p
cp sin(px=L):
(60)
This is not a general perturbation in wave speed. A general perturbation would include a
cosine term. But when substituted into equation (54), the cosine part integrates to zero which
is easy to see by consideration of eveness and oddness. Physically, this is consistent with the
observation that a sound speed perturbation at a displacement node has no impact on the
oscillation pattern or frequency. The wave does not know it's there. So we simply ignore it
here. Evaluation of equations (54) and (57) will then require that we compute the following
integral:
Z
L
0
0
00
,
n
x
n
x
nx
sin L sin L sin L dx nn0n00 :
(61)
The right hand side introduces some new notation. Let's call this a 3 , n symbol. With this
notation we can then rewrite equations (54) and (57):
#
"
X
4
(1)2
(62)
!n = Lc cp (pnn) !n(0)2 ;
0
r
2
p
"
#
mx :
c
(
pnm
)
sin
(63)
L m6=n m2 , n2 p p
L
0
The evaluation of both of these equations reduces to computing the 3 , n symbol. Doing so we
yn
(1)
4n
= Lc
2
X
1
X
nd that
if m + n + p is odd(64)
(pnm) = (m , n , p)(m + n,,4Lmnp
p)(m , n + p)(m + n + p)
= 0
if m + n + p is even(65)
:
and
2
(pnn) = p(,p24Ln
if p is odd
(66)
, 4n2 )
= 0
if p is even:
(67)
Thus, the eigenfunction and frequency perturbations are only non-zero for certain values of
p; m, and n. These conditions are called Selection Rules. We will state two of them as well as
an observation resulting from the Selection Rules.
Selection Rule 1: Contributions to the perturbed eigenfunctions will only result if p + m + n
is odd.
Selection Rule 2: Only odd p contributes to perturb the eigenfrequencies.
Observation 1: Wave speed perturbations that would be represented as cosines rather than
sines, have no impact on either the eigenfunctions or eigenfrequencies.
Observation 2: Even-even or odd-odd coupling (m + n even) occurs only through odd degree
structure (p odd), even-odd coupling (m + n odd) occurs only through even structure (p
even).
9