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Local intrinsic modes: Layer with nonplanar interface
WAVE MOTION NORTH-HOLLAND
LOCAL
8 (1986)
1-14
INTRINSIC
MODES:
LAYER WITH NONPLANAR
INTERFACE
J.M. ARNOLD Department
of Electrical Engineering and Electronics,
University of Glasgow,
Glasgow GlZ SQQ, United Kingdom
L.B. FELSEN Department
Received
of Electrical Engineering and Computer Science, Polytechnic Institute of New York, Farmingdale,
12 November
1984, Revised
19 February
NY I1 735, U.S.A.
1985
Laterally inhomogeneous layered media serve as models for realistic propagation environments in electromagnetics, optics, underwater acoustics, seismology and other disciplines. When the inhomogeneities extend over long distances compared to the local wavelength, general methods (like those based on integral equations) for these nonseparable boundary value problems become intractable. For weak lateral variations, the propagation process can be localized around intrinsic modal fields, which are synthesized by plane wave spectra capable of describing propagation properties uniformly in guiding, cutoff and radiating regions. These local intrinsic modes, specified by modal invariants, traverse the medium along lateral trajectories, which may refract and form caustics similar to those for ordinary ray fields in an inhomogeneous environment. In the present investigation, these concepts are applied to a single homogeneous layer separated from a homogeneous half space by an interface with arbitrary but slowly changing surface profile, as exemplified by a shallow ocean waveguide. The previously derived intrinsic modes for a two-dimensional wedge geometry (J.M. Arnold and L.B. Felsen, “Intrinsic modes in a Journal of Acoustical Society of America 76, 850-860 (1984)), serve as the prototype for the nonseparable ocean waveguide”, construction of the wave spectra, first, for the two-dimensional configuration with nonplanar interface, and then for the general three-dimensional case.
1. Introduction Layered media with relatively arbitrary interface contours and medium properties ,model such geophysical environments as the shallow ocean, the earth’s crust, the troposphere and ionosphere, etc., as well as artificial environments such as tapered dielectric waveguides used for components in integrated optics and millimeter wave guiding and radiating systems. Wave propagation in such environments-whether acoustic, seismic, electromagnetic or optical-poses a nonseparable boundary value problem of substantial complexity. For weak lateral variations over an interval equal to the local wavelength, adiabatic (local) mode theory has provided an approximate scheme which was hampered, however, by its failure in tracking a trapped local mode through cutoff into the radiating regime in layer geometries giving rise to such 0165-2125/86/%3.50
@ 1986, Elsevier
Science
Publishers
phenomena. Only recently, in the context of underwater acoustics, has this deficiency been repaired [l-5] and thereby provided new impetus for pursuing generalized local mode propagation in these complicated environments. In particular, for the canonical two-dimensional problem of a wedgeshaped homogeneous ocean above a semi-infinite homogeneous fluid bottom, a rigorous solution has been constructed that can serve as a bench-mark for comparison with results derived by other methods. This solution, an ‘intrinsic mode’ [2] that satisfies the acoustic wave equation and boundary conditions on the ocean surface and bottom boundaries (but away from the wedge apex) exactly, is expressed as an integral over spectra of plane waves. For small bottom slopes cr, and downslope from cutoff, the intrinsic mode reduces to an adiabatic mode via leading order asymptotic approximation of the spectral integral, but that
B.V. (North-Holland)
2
integral,
J. M. Arnold, L.B. Felsen / Local intrinsic modes
when kept intact,
the cutoff transition coupling are valid
between
the adiabatic
[l, 2, 31. Thus,
‘more adaptable’ from other intrinsic resenting numerical individually excitation
an intrinsic
mode
suitable
is a
valid spectral
for efficiently
intrinsic
in superposition
by a line source
mode
applied
a function
of fields
to synthesize
has been demonstated
for small bottom slopes [6-81. Comparisons of these results with those derived by other methods [4, 5, 91 have revealed general overall agreement in the behavior of the sound field upslope from the source in the water and the bottom. Where discrepancies do arise, it may be necessary to calculate the inherently exact intrinsic mode spectral integral more accurately [lo] before deciding that the shortcomings reside in the other methods since the intrinsic mode spectral integral, as noted above, solves the wedge problem (away from the apex) exactly. Its success in treating two-dimensional propagation in a homogeneous wedge-shaped medium exemplified by a wedge-shaped ocean suggests that the spectral method might provide the foundation for addressing the three-dimensional configuration characterized by weak lateral inhomogeneities in the bottom boundary shape. The justification for this conjecture resides in the recognition that wave spectra can be adapted locally to weak changes in the environment, the local environment near any point on the bottom being modeled by its tangent plane as a local wedge. To this end, it is necessary to express the plane wave spectra for the wedge, which satisfy global conditions because of the special character of the wedge geometry, in a form localized at each vertical cross section. The clue for this localization is actually contained within the global spectral construction of the intrinsic mode for the wedge since the compatibility (selfconsistency or closure) condition for the global spectra can be interpreted as a modal resonance
locally,
parallel
The constancy
at the lateral coordinate
layer with height
of the local resonance an intrinsic
(indices)
h(x’) [2].
condition
as
the spectral invariant that
of x’ furnishes
characterizes stants
rep-
fields [2]. Feasibility of
condition
x’, to a plane
mode, and it is decoupled
excited
evaluation and
for
incorporates
modes. The latter circumstance
modes
source
uniformly
modes, where these
and more generally
object than an adiabatic makes intrinsic
accounts
and furthermore
mode, with different
pertaining
to different
con-
modes.
A
spectral attack on the fully three-dimensional laterally varying problem can thus be structured around local intrinsic modes, each identified by its spectral
invariant
at any
point
x’ in the lateral
plane where the local height is h(x’). Tracking
the
spectral invariant through the range-dependent configuration then defines lateral curves, the lateral (modal) rays, along which the local intrinsic modes propagate, the details of each lateral modal ray being dependent on the bottom profile h(x’), the mode index, and the rnitial departure direction. To make these remarks more specific, let the local plane waves in the spectrum be characterized by the three wavenumbers 5, 7, K, normalized with respect to the wavenumber k in the layer, and corresponding to the directions x, y, z, respectively. These wavenumbers, the components of the unit vector k, along which the spectral plane waves propagate, satisfy the dispersion equation [2+$+lZ=l.
(1)
The spectral invariant has the form I,[x, y, h;
K(X,
y)]
(local resonance
=
HIT,
q =
condition)
integer
(2)
where I, is a known function of x, y, h and K, and different modes are distinguished by different values of 4. To maintain the invariant intact at any point (x, y) in the range dependent environment, the vertical wavenumber has to be the range dependent function K = K (x, y) determined from solution of (2). Knowing K then specifies via (1) the horizontal wavenumbers ([‘+ n2) as a function of (x, y). Imposition of initial conditions (via a second invariant I*) where the mode is launched allows local values of 5 and 7, the horizontal projections of k,, to be determined uniquely in conformity with (l), and thereby to specify the local lateral ray direction traversed by the local
3
J.M. Arnold, L.B. Felsen / Local intrinsic modes
intrinsic
mode.
[V,,S(&
Written
as
Y)12= 52(x, y)+ T2k Y) = 1-
K2(X,
y)
=
n*(X,
spectra,
subject
to the modal
structed
directly
function,
in this local
(3)
y)
dinate
y( 1) = 0 and
(1) is nothing
but
K =
tive
shown
dashed.
The
(along
positive
x)
invariant horizontal
n(x, y).
In
coordinate
form, [VS(p)12= vector coordinate,
independent
n’(p), p the eikonal
can be solved by any of the conventional Local modes propagating along lateral
being a equation methods. rays were
actually introduced previously by Keller [ 1 l] who called them ‘horizontal rays’ (see also Pierce [12] and Weinberg and Burridge [13]), but by an entirely different approach based on direct solution of the wave equation and boundary conditions through an asymptotic ansatz, and not allowing access to the bottom medium. Thus, these earlier models cannot describe the important phenomenon of energy leakage into the bottom by intrinsic mode transition from the trapped to the radiating regime through cutoff. Applying localized spectral concepts to constructing local intrinsic mode fields requires not only a knowledge of the modal ray trajectories but also of the amplitude and phase variation of the local plane wave spectra along these trajectories. Since the local plane wave spectra in three dimensions are generally specified by two independent (see (1)) spectral parameters, a double integration in the local intrinsic mode integral is required, in contrast to the single integration for the two-dimensional case r] = 0. Moreover, determination of the spectral integrands is aided substantially by a proper parametrization of the local spectra. Therefore, before attempting the general case, it is advisable to look first at the twodimensional problem, but with nonplanar bottom profile h(x’) instead of the planar profile for the prototype wedge. Local coordinates are now chosen to fit the local geometry directly, instead of the apex centered cylindrical coordinates which were employed for the wedge [ 1,2], and the local
Although
(Fig.
l(a)).
all the lateral
coordinate
x (Fig.
cutoff, eneigy leaks into the bottom ;
1(b)). Beyond
hence the propagation are complex,
in (l)),
K(X)
are con-
on the third coor-
to the range
the eikonal equation for rays in an equivalent two-dimensional medium with equivalent refracindex
frame
Because there is no dependence rays are parallel
where S is the eikonal
invariant,
angles
of the local spectra
and the ray trajectories
the
local
are therefore
direction
of
propagation
shown
by
the
is
plane
wave
spectra
arrows. can
be
reflected back toward negative x, this would occur at a range xt> x0 at which we assume that the modal energy has been transferred almost entirely into the bottom. Therefore, reflection is neglected in this case. Reflection may, however, be important even in a two-dimensional configuration if the local
a)
Crass
s&anal
view
ye-;--,
I bl
--C
I :,
Top view and horizontal
ray
trajectories
Fig. 1. Two-dimensional local intrinsic mode propagation upslope in a two-dimensional layer with weakly sloping penetrable bottom. The mode is trapped inside the layer for x < x,, where x, is the horizontal range at cutoff. The coordinate I’ is measured along the bottom boundary, Y’ is perpendicular to that boundary, and h(x’) is the local height in a perpendicular cross section. (a) Cross seational view; (b) Top view and horizontal ray trajectories. Solid: trapped regime. dashed: radiating regime.
4
J.M. Arnold, L. B. Felsen / Local intrinsic modes
intrinsic
mode is launched
to the x-axis;
at an angle with respect
this implies
field distribution
that the initial
at x = 0 has a linear
phase
ation exp(ik7y) along y, 7 being constant of the geometrical invariance. Solutions problem
are constructed
trivially
Y
modal vari-
because for this
from the strictly
two-dimensional one on replacing 5 in the latter by ([‘- 72)1’2 when solving (1) or (2) with K = K(X).
The lateral
tical curves,
rays now form a family
differing
y (Fig. 2). Three
only by displacement
of idenalong
cases can be distinguished:
0)
7
small
b)
7
lorge.
(a) The ray trajectories do not deviate strongly from the x direction, and the local intrinsic mode therefore passes through cutoff (Fig. 2(a)); this case is similar to the one in Fig. l(a) and reflections can be neglected (unless the contrast between interior and exterior media is extreme). (b) The ray trajectories deviate strongly from the x direction and turn back before mode cutoff (Fig. 2(b)). A ray caustic is located at the turning range x, where 5 = 0, and a reflected spectrum must be included; for x > x,, the intrinsic mode field is evanescent (spectral wavenumbers are complex) but may exhibit small leakage due to tunneling into the radiating regime. (c) The ray trajectories are such that cutoff occurs near the turning range (Fig. 2(c)) ; this poses a new and more difficult transition problem which requires forward and reflected spectra. When a point source at x = y = 0 provides the excitation, local modes are launched in all directions, and the ray configuration shown in Fig. 3 contains all of the situations depicted in Fig. 2. Ray caustics do now not occur at the turning range but they are the envelope of the reflected ray system. These propagation problems in the twodimensional configuration of Fig. 1(a) provide the preliminary setting for analyzing the threedimensional configuration where h = h(x, y). It should be noted that intrinsic mode construction even in the two-dimensional model is far from trivial and constitutes, in fact, a new class of hitherto unsolved propagation problems that can accommodate the leakage phenomena depicted in
c) 7 in between. Fig. 2. Lateral projections of model ray trajectories for threedimensional local intrinsic mode propagation in the twodimensional configuration of Fig. l(a). (a) 17 small. Cutoff before turning; (b) 7 large. Turning before cutoff. Caustic at x,; (c) q in between. Turning near or at cutoff (x,=x,).
\ b
5
J.M. Arnold, L.B. Felsen / Local intrinsic modes
caustic
\
Y
J ’
source
0’
I
1’
---wx
XC
Fig. 3. Lateral modal ray trajectories for local intrinsic modes excited by a point source in the two-dimensional configuration of Fig. l(a). Rays 0, Q and @ correspond qualitatively to cases in Figs. 2(a), 2(c) and 2(b), respectively. A symmetrical ray picture for y < 0 is omitted.
Figs. 2 and 3. The sequence outlined followed in the presentation below.
lim
above is
2. Formulation of the problem Again keeping in mind the shallow ocean, we consider the two-layer configuration depicted in Fig. 1, where B, represents a planar and perfectly reflecting top boundary while B2 represents an arbitrarily shaped bottom boundary, which separates the upper medium with constant refractive index n, and density d, from the semi-infinite lower medium with constant refractive index n2 and density d2. The top layer (medium 1) and bottom layer (medium 2) will be denoted by X1 and X,, respectively. An as yet unspecified source with suppressed harmonic dependence exp( - id) is located in X1, and we seek a solution of the resulting scalar field u(x) satisfying at observation points x away from the source the scalar wave equations x E X,,
(44
(V’+n;k’)u=O,
XEX,
(4b)
subject to the boundary conditions = 0,
(5a)
.(5c) (5b)
Here, k is the wavenumber in a reference medium with refractive index n = 1, and (a/au) expresses differentiation with respect to the surface normal on B2. The solution for u is to be constructed by superposition of plane wave spectra satisfying the wave equations in (4a) and (4b). The boundary condition (5a) on the perfectly reflecting planar upper surface B, can be satisfied trivially by specular reflection of an incident (upgoing) spectral component in X1. The boundary conditions (5b) and (5~) on the lower surface B2 cannot easily be satisfied globally because of the assumed arbitrary shape of B2. Consequently, we restrict variations (changes in slope) on B2 to occur over length scales which are large compared to the wavelengths in X1 and X2. Then the boundary conditions at x on B2 can be satisfied locally at x’ E B2 by Fresnel (or Rayleigh) reflection and refraction of a downgoing spectral component in the plane interface tangent at x’ to BZ; for a.planar bottom boundary with constant slope (wedge configuration), the spectra constructed in this manner actually satisfy the lower boundary conditions exactly [2]. Satisfying both boundary conditions on B, and B2 simultaneously requires self-consistency of the upgoing and downgoing spectra which, when imposed locally, then generates a solution valid asymptotically for weakly changing bottom contours. This strategy is pursued below, first, for the twodimensional and then for the three-dimensional configurations.
3. The two-dimensional
(V2+ n:k2)u = 0,
lim Lx,+~,
&Y~-B~,
xeX,+B?
I,/’
@
=
lim
I d I / ,I
0
lim u] XCX,-rBZ
case
To implement the scenario described above, we consider first the two-dimensional problem where the contour B2 is described by the y-independent height function h = h(x’), and excitation is from a uniform source parallel to the y-axis. The spectral
J. M. Arnold, L.B. Felsen / Local intrinsic modes
6
representation
of the field u(x) is taken
u(x) =
;(x’)‘P(x,x’)dl’,
as
To satisfy we impose
XEX,
(6)
J X’in where I’ is the tangential v’ is the combination
coordinate
along BZ, and
of downgoing
and upgoing
plane wave spectra that satisfies the local boundary conditions at x’ on B,,
+exp[i4(8)]
‘P(x”, x’) = 0,
condition
(10)
where x” on B, is the point normal to x’ on B,,
intersected
XI’= x’+ h(x’)v,(x’)
along the normal
. (x-x’)].
exp[ik+
(5a) on B,,
X”E B,
Ye.
by the
(11)
and h(x’) is the local height
. (x-x’)]
V(x, x’) = exp[ik_
the boundary
of B, relative
Combining
to B2
(10) and (7)
yields (7)
Here, k-(x’)
and k+(r’) are wave vectors along the
directions of the downgoing and upgoing local plane wave spectral components in X,, respectively, with the requirement that k- and k, have identical projections on the tangent plane at x’ to satisfy the specular reflection law. Thus, points x’ on B2 parametrize the local plane wave spectra. With y and fi denoting components of k al x’, parallel to 1; and V; respectively, one has (instead of the (x, z) decomposition in (1)) k,(x’)
= l$(x’)
y2(x’)+p2(x’) where
rb = I,(x’)
f vhy(x’), = k:
and
(8)
vb = vO(x’) are unit
vectors
along the directed tangent and upward normal directions on B2. Inclusion in (7) of the reflection coefficient exp[i4( e)], where ,
(9)
exp( -iyh)
n, cos 0 = n2 cos 6“, ensures that the local reflected spectra also have the correct amplitude. The weighting function ii in (6) allows for an overall amplitude distribution in the normalized spectral combination V in (7). The domain of integration D is a subdomain of B, that will be further identified.
exp(iyh)
= 0
(12)
or (cf. (2)) I, = r,(X’)h(X’)+~{~[~(X’)]+~}= 4 = integer.
47F, (13)
Equation
(13) may be recognized as the local transcondition for the eigenvalue y= y,(x’) of the local normal mode with index q at the cross section defined by x’ on B2 and the normal Q(X)). W(x, x’) in (7) then represents the normal mode field at any point 0 < v < h(x’) along the local cross section for that value of q, and is therefore denoted by Wy(x, x’). To construct the spectral amplitude function zi,(x’) in (6), corresponding to ?Pq, we require that the total field u,(x) in the cross section be locally equal to the intrinsic mode field for the wedge geometry that approximates the local environment. Guided by the wedge solution in [2], we assume verse resonance
that the spectral 0 = 13(x’) is the angle between vector k-(x’) and the tangent to B, at x’, 0” is the angle between the refracted plane wave vector k!!(x’) and the bottom tangent, and 0 and t?” are connected by Snell’s law
+ exp(i$)
amplitude
G(x’) = a(x’) exp[ifi(x’)]
function
has the form (14)
and that stationary phase evaluation of the integral resulting from substitution of (7) (with (13)) and (14) into (6) yields the adiabatic mode field downslope from the cutoff transition region. We shall use this feature to determine a and 0 for the non-planar bottom by requiring the asymptotic approximation of the spectral integral to satisfy the local boundary conditions in the cross section. The spectral amplitudes obtained in this manner are then assumed to define u,(x) at any cross
J. M. Arnold, L. B. Felsen / Local intrinsic modes
section, provided validity
even in and beyond that the integral
the cutoff transition,
in (6) is kept intact. The
of this proposition
is verified
exactly
local plane
it is convenient
wave spectra
r:,*(X:-x’)=o,
for
the special case of the wedge and therefore, by the principle of locality, is taken to apply approximately to the general case. To proceed,
at x*=x:,
to decompose
$[lb
ponents along transverse and tangential directions at x’ (we omit the modal subscript q), * (XL -xl)]
= -1
and (17) becomes
the
aO(x’)
in (7) via (8) into com-
!P(x, x’) = !Pdx, x’) exp[iplh
. (xi -x’)]x,=x;
(Ha)
=
al’
(18)
P(4).
x’=x,’
As the observation
point x moves to different
cross
sections, the point xi moves on B2, and (18) generates a global relation which can be integrated to yield
where W,(x, x’) = {exp(-iyv)+exp(i@) X exp[$&
exp(iyv)}
* (x-x’)]
0(x’)
(15b)
with v=z.J:,*(x-x’),
(15c)
and xi is the point on the bottom B2 from which the normal passes through x (i.e., the point on B2 closest to x) such that x = x; + Y(Xi) Vo(X$
(15d)
The transverse variation ?PO(x, x’) changes slowly with x’, and, when x’ = x:, is precisely the local normal mode distribution for that cross section. Also treating a(~‘) in (14) as slowly varying, one obtains for the rapidly varying phase S in the integrand
of (6),
X’ p dl’. I x(,
(19)
The integration is with respect to the tangential direction on B, from X~E B,, where XL is a fixed reference. To find the amplitude a (x’) in ( 14), the boundary condition u(x:) = 0 (xz E B,) in (5a) is enforced to the second order in the asymptotic evaluation of the spectral integral (6) since, by the modal resonance condition (13), ?PO(x, x:) in (15a) vanishes on B, (see (lo)), thereby automatically satisfying (5a) to the leading asymptotic order. Writing u(x)=i
F(x,x’) I
aS(x, x’) al, exp[iS(x,
x’)] dl’
(204 where F(x, x’) = a(x’) WO(x, x’)/[i aS(x, x’)/al’],
S(x,x’)=R(x’)+plb. which has stationary
=
(x:-x’) points
I:(I) defined
(2Ob)
(16) one obtains
by
from integration
W(x,n, x’)
24(x:) = D
++5 *(x’-x’)]=O
by parts, noting
(lo),
that for X~E B,,
al’
exp[iS(xi,
x’)] dl’ = 0
(21)
(17)
where (a/al’) denotes differentiation along the tangent to BZ. Since the stationary phase evaluation is supposed to yield the local normal mode in the cross section, consistency with (15a) and (15b) requires xi to be the stationary phase point. Thus,
Stationary
phase
u(xy) - -7r”’
evaluation
e*im’4 exp[iS(xZ,
of the integral x:)]
yields
J.M. Arnold, L.B. F&en / Local intrinsic modes
8
which can be satisfied
azqx:,x’) al’
Moreover,
only when
variation =0
atx’=x:.
(23)
recognizing
that
of the local normal
!P’,=sin[k,(h-
!P,, is the transverse mode at x’, one has
u) sin 01 exp(i$/2).
Then from (24) and (25), with a convenient This implies
from (20b) that a(~‘) must satisfy the
differential
equation + g(x:)a(x:) Xi
=0
(24) Ignoring (27)-(29)
where
g(d) =
G(x;,aG(;;v x’q Xi) [
By integration variation
x’,
(25)
a(TI)=(Isexp{-[Ig(x’)dl.] where
a, is an arbitrary
constant,
and
xb is an
arbitrary reference point. As mentioned earlier, the validity of this construction of the spectral integrand in (6) can be tested on the special case of a planar bottom Bz with small slope CL Introducing p as the distance from the wedge apex and ~3as the angle of inclination of the spectral plane waves with respect to B2, one has /l = k, cos 8, and from (19), 0 = -
I
k, cos 8 dp
=-k,pcosfl-l where parts, local (12),
cx I
k,hsin@dtZ
(26)
the last equation results from integration by changing variables to 8 and introducing the height h = pa. With the resonance condition (26) becomes 0 = -k,p
cos 0
g
exp(-i+/2)
normalizations agree precisely
. I x’
and O(a) error terms, with the results in [2].
(244
xt)/a/l’
of (24), one finds for the amplitude
at arbitrary
1
4. Oblique propagation in the two-dimensional configuration
x’=x:
qocx:, x’)
G(xf,x’) = .#qxy,
choice
for a(), a(x’) = -
aa al’
(28)
The results in to accommodate phase variation This adds to the
Section 3 are easily generalized intrinsic modal fields with linear exp(ivy) along the y-direction. modal invariant in (2) or (13)
Z,[x’, h ; y(x’)] the further
= constant
(30a)
constraint
I, = 7~= constant
(30b)
subject to the dispersion relation y2 + p’+ v2 = 1. The corresponding wave phenomena depicted in Fig. 2, hitherto unexplored even for the twodimensional wedge, form a useful preliminary before attacking the general three-dimensional case. The generalization is accomplished on replacing the propagation coefficient p(V) in all results of Section 3 by [ P2(x’) - g2]“*. The modal ray paths are then defined by (3). The obliquely propagating modal field is exp(iny), %l)(x) = {n,(x)l P+~p2-1~~~~~~
(31)
where u,(x) is the local intrinsic mode field in Section 2, with n = 0. One observation is in order: as noted in Section 1, the spectral integrand must now generally include wave spectra with components in the negative as well as positive 1’ directions to accommodate reflection due to cutoff before reaching the radiating regime. These aspects are explored in greater detail in the next section.
9
J.M. Arnold, L.B. Felsen / Local intrinsic modes
5. The three-dimensional case
5.1. The spectral integral
To pass to the general three-dimensional case without translation invariance, two invariants are again employed to determine lateral ray paths I? We note that the second invariant I2 in (30b) is a global invariant induced solely by the translation invariant geometry assumed in the previous section. In the general case without translation symmetry, (30b) must be localized; this is done by observing that near any point X’E BZ, the direction tangent to a contour h =constant is the local equivalent of the former global symmetry direction. Thus we replace (30b) by the differential invariant I,
=
2
ak*(x’)
-*
al’
v&‘)=O
(32)
where v,,(x’) is the unit vector tangent to the curve h = constant at x’ E B2 (with a consistent orientation over B,; see Fig. 4).
To continue the synthesis of the intrinsic mode, we write u(x) =
I X’E D
;(x’)P(x,
x’) dB,
(33)
where D is a subset of B2, dB, is the differential area element on B,, ?P(x, x’) is the spectral plane wave pair attached to x’ E D c B, given by
. (x-x’)]
P(x, x’) = exp[ik_(x’)
+ wW(
WI
xexp[ik+(x')
. (x-x’)]
(34)
and k,(x’) are determined by the two invariants I, and Zi in the same manner as in Section 4. We again hypothesize that c = a eiR
(35)
and introduce the decomposition ly = q0 eiPJ= p0 eiP(x’)G;-x’)
(36)
where P = P(x’) = Ilf(k+(x’) + k-(x’))]]
(37)
is the magnitude of the tangential parts of k,, and I= I(x, x’) = l(J(x’) . (xi-x’)
Fig. 4. Implementing
the invariant case.
I, for the three-dimensional
The values of I, and I; provide two conditions which fix the components of k, tangential to B2 everywhere on B,; this in turn defines a vector field /? = i(k+ + k-) on B2 which can be integrated to yield trajectories P. A similar use of invariants has been suggested by Harrison [14,15] in connection with the ‘horizontal rays’ of Weinberg and Burridge [13], to which our trajectories are closely related. We consider these trajectories in more detail later, after completing the spectral construction around them.
(38)
is the component of (xi-x’) tangential to the trajectory P which passes through x’, xi being defined as the point on B2 nearest to x. Moreover, !P,,= ?PO(x,x’) is given by ?PO(x,x’) = {exp(-iyv) +exp(i+(@)
exp(iyv))
x exp[ib . (x-x:)]
(39)
with v= ZJb. (x-x’), fy=k,.d,
(40a) atx’EB2
(40b)
and 0 = arc sin( Y/kl).
(4Oc)
J. M. Arnold, L.B. F&en / Local intrinsic modes
10
As previously, requires that 0,(0
stationary
phase of the spectral
phase in the integrand
tion
+ pr) = 0
(41)
which in turn becomes,
phase
r&= &( r’)
point.
(42)
Integrating
(42) along
p dl’
= 0(x&)+
(43)
P(X’,X,;)
where P(x’, x6) is the trajectory from xh to x’, and dl’ is the differential arc length element on P, oriented from xh to x’. To determine the amplitude term a method similar to that in Section 3 is the necessary partial integration being along trajectories, similar to (25). The a(x’) = a(xA) exp[ -/p,.,.,i),
g dlf]
in (35), a used, with performed result is
(44)
for xl E P(x’, xh),
g(x:) = lim x,-x: G(x:,
x’) =
(45) w,(x:, x’) aS(xi, x’)/al”
S(x, x’) = .n(x’)+p(x’)l(x, Ef=O
are found by integraP is determined
by
k, = kn,,
+3(x’))=O
(SOa) (5Ob)
where sin 6 = (k, . VA)/ k, = y/k,
(5la)
and (a/al’) refers to differentiation
describe
geometrical
h(x:)v,(x:)
(Sib)
properties
of the wavevectors
k, and p (see Fig. 5).
When qn, then
I, is fixed as a global invariant equal to 8 is determined for all x’ E B2. Denoting
p = k, cos 0, as before,
p = PC,
(52)
we can show from (50b) that
I$(PC,)= VP
(53)
which determines I&, and hence the components of p. The trajectories P are integral curves of /I. In fact, (53) is identical to Snell’s law
(46) x’),
(47)
for rays P in a 2-dimensional inhomogeneous medium with refractive index n dependent on position in the medium; in this case, $ is the angle between the ray and the curve n = const. at any
P
atx’=x:E
4)/k,
cos+=(p.
along the trajec-
tory P passing through x’ E I&. In these equations, x: is the point on B, given by x:=x:+
path
and I
where,
r;=v,,(X’)
a
gives
0(x’)
function
Each
I, = k,h sin 0+!4(6J)+$rr,
where I; is the unit vector tangent to the trajectory through x’. Conversely, (42) implies I= 0 for a stationary
them.
two invariants
for I = 0,
V./n = pr;, = p
trajectory
along
20
k+
(49)
and is therefore a function of xi. The point x; E B, is an initial point for trajectories and is discussed further in the next section. 5.2. The trajectories
P
The trajectories P are of fundamental importante in this theory because both the amplitude and
Fig. 5. Geometrical paths.
quantities pertaining to the lateral k_, k, and v& are coplanar.
ray
J.M. Arnold,
point in the medium. We need only identify B/k, = n to draw this conclusion. The curves B = const.
of points
and h = const. are identical,
by Arnold
P are identified surface
the paths
as those of the ‘equivalent
tive index’ method. (along
and therefore
When projected
normals
‘horizontal rays’ similar Burridge [ 111.
11
L.B. Felsen / Local intrinsic modes
refrac-
from B2 to B1
x’ at which pairs of trajectories
ate. This construction inhomogeneous clarify multiple
to those of Weinberg
mode integral. Each
medium.
trajectories point
now
required
and caustics
x’ on each
to that given
propagation
We
the modifications
to B, or B2) they become and
is very similar
[16] for ordinary
degenerin an
proceed
to
to incorporate into the intrinsic
trajectory
generates
a
Initial conditions must be specified for these trajectories in order to determine them completely.
plane wave pair in the spectrum
These conditions
a stationary phase point, then contributions from this neighborhood on a single branch of trajectories generate a single asymptotic term in the spectral integral for u ; all branches together generate the full asymptotic value of the integral by superposition of all the individual contributions. This suggests that the integral (33) should be expressed in terms of an integral over trajectories to achieve a single valued representation, rather than as a superposition of several integrals over D = B2. This is easily done by expressing the area element dBz in terms of trajectory based coordin-
may be chosen
arbitrarily;
here
we choose them so that all trajectories originate from a single point xh E B2 which we call the source point. The initial condition for IA is determined for each trajectory
by its value at the source point
x&
where 1+4,is a parameter denoting the angle which a particular trajectory makes with the curve h = const. at xb (see Fig. 6). with conventional propagation in As inhomogeneous bulk media, it is possible for several trajectories to pass from the source x& to the same spectral parameter point x’. When two such trajectories degenerate, two stationary phase points in the intrinsic mode integral coalesce, and the stationary phase analysis used in Section 5.1 is invalid. However, the spectral function C remains well defined if, for each x’ E B2, 6(x’) is regarded as a superposition of contributions from all trajectories passing through x’, each contribution being constructed along a single trajectory according to Section 3. As x’ moves over B2, these individual terms in C(x’) can be regarded as branches of a multiple valued function on BZ, with various branches connected by caustics, the loci
Fig. 6. Initial
conditions
for lateral a point.
ray paths emanating
tion domain
of u. If the integra-
D is a large enough
neighborhood
of
ates (I’, I,&), where l’represents arc length to x’ from x;l along the trajectory which leaves x& inclined at an angle &, to the curve h = const. (Fig. 7). Thus dBz = J(x’) dl’ d$,
(56)
where J(x’) is the Jacobian formation given by J(x’) = ph.
5
of the coordinate
trans-
(57)
0
and ~6 is a unit vector tangent to P at x’. Allowing
to B2 and normal
0s l’
(584
0s *,<27r,
(5gb)
- h = const.
from Fig. 7. Trajectory
based
coordinates
for lateral
rays.
J. M. Arnold, L.B. Felsen / Local intrinsic modes
12
then
covers
all points
trajectories.
On the other
the spectrum medium,
on B2 which
of a point
by analogy
complex
values
tour C which passes contiguous end points.
from +ico to n-i00 integral
For this reason
D required
to be the domain
variable
on a Sommerfeld
sectors of the complex
a way that the spectral
&,-plane converges
&, con-
through in such at both
we take the integration
in the intrinsic
spanned
with
in a 2-dimensional
one might expect the angular
to assume
domain
hand,
source
lie on (real)
values
o=Gz
(59a)
(cloe C
(59b)
where C is a Sommerfeld contour and the integrand is analytically continued on C. Consequently, we are led to consider the spectral integral
The /‘-variable can be integrated out by the method of stationary phase. Stationary phase points for this integration are obtained as follows. The phase of the integrand of (60) is, from (35) and (36), S(x,x’)=n(x’)+p(x’).(x:-X’)
(61) x’)
(62)
where I(x, x’) = z; * (XL-X’),
P = PG
(63)
and Zh is the unit vector tangent to the trajectory with coordinate I,$,at x’ E B,. For a stationary value of S with fixed & we require
(>
$+ $1+p$=o.
construction
of stationary
condition.
mode integral
by the coordinate
= n(x’)+p(X’)I(X,
Fig. 8. Geometrical
(64)
Now 80/N = p, and 1= 0, al/al’= -1 (neglecting terms due to the curvature of P, which are second order in the ordering pertaining to small boundary slopes) when the vector (XL- x’) is exactly perpendicular to 16; hence the stationary phase point,
labeled xb, is the point on P closest to x: (see Fig. 8). Stationary phase integration leaves a remaining integral over &,, which is the equivalent for this problem of the uniform oscillatory integral derived in [ 161 for the field in a neighborhood of a caustic of ordinary rays in an inhomogeneous medium (this is known to have a uniform approximation in terms of Airy functions). In certain circumstances the above procedure must be modified. First, whenever a trajectory crosses a critical depth contour h = h,, where the phase, 4, of the Rayleigh Plk, = n,lnr, coefficient exhibits a branch point because the ‘incident’ wave in the spectral pair is at the critical direction for total internal reflection; this is simply the analog in three dimensions of the ‘local mode cutoff’ phenomenon which we have discussed in detail elsewhere [ 1,2] in connection with the twodimensional wedge. In this case the stationary phase integration along the trajectories will fail, and a more uniform method, following [l], is required. If, in addition, a caustic crosses the critical
depth
contour,
a compound
transition
function is required for observation points near the crossing. A second source of trouble with the spectral construction occurs on depth contours where the depth h is an extremum and dp/dl’ changes sign as the trajectory crosses the contour (see Fig. 9). The effects encountered here are analogous to those in ordinary ray propagation in inhomogeneous media near an extremum of refractive index. In particular, tunnelling across a depth minimum can occur. Further study is required before these effects can be fully incorporated into
J.M. Arnold, L.B. Felsen / Local intrinsic modes
for the special
;F’=h-in /s,+”
bounded and
case of a single homogeneous
on one side by a perfectly
on the other
which separates half space.
Fig. 9. Lateral rays paths when depth profile has a minimum.
intrinsic
no reason
mode
framework,
to suppose
although
there
is
that this will not eventually
be possible. There is little to be gained, at present, from explicit enumeration and asymptotic extraction of all the transition functions that can occur in this type of problem (with the possible exception of the simple caustic [16] and isolated local mode cutoff [l, 21 phenomena, which are both expressible in terms of Airy functions). It is more profitable to regard the intrinsic mode itself as a globally uniform asymptotic representation of u and to proceed to direct numerical evaluation. In that case, the localizing properties of the stationary phase principle can be exploited to devise numerical quadrature schemes for evaluating the integral efficiently. These aspects of the problem are currently being explored.
6. Summary Local intrinsic modes appear promising for describing propagation in layered media with weak lateral inhomogeneities on the scale of the local wavelength.
Such modes,
constructed
in principle
by plane wave spectral synthesis, are characterized by model invariants and, by methods similar to those employed for ordinary geometrical rays in an inhomogeneous environment, can be traced along lateral trajectories through guiding, cutoff transition and radiating regions. Intrinsic modes have actually been constructed for the prototype problem of a wedge-shaped layer with penetrable boundaries [ 1,2]. As a first step toward a more general implementation of the spectral method, local intrinsic modes have been developed here
side by a nonplanar
it from an exterior
a shallow
modes
dimensional
and
for
case, with an indication
can be used
interface
which
obtained
then
to synthesize
plane
homogeneous
ocean waveguide,
have been
layer
reflecting
In this configuration,
for example, two-dimensional
the intrinsic
13
models, the local
first for the the
three-
of how they
source-excited
fields.
Generalizations to accommodate impedance boundary conditions, medium variations, multiple layers, etc., are already being considered and are expected to introduce complications in detail, but not in the fundamental concepts that have been successful in this first extension of the rigorous canonical
wedge solution.
In assessing the scope of the local intrinsic mode spectral theory, one may be guided by analogous considerations for geometrical ray theory. In an extended environment with weak inhomogeneities, asymptotic ray theory (ART) provides one of the most widely used and effective analytical tools with feasible numerical implementation. The failures of ART in caustic, shadow boundary, and other transition regions can be repaired by more general spectral objects, often called generalized or uniformized rays (see [ 171 for the wedge configuration) given in the form of spectral integrals. This complicates the numerical evaluation but still renders the method tractable. ART fields are actually the stationary phase approximations formized ray integrals. In excessively media,
explicit
concern
with caustics,
of the unicomplicated etc., can be
avoided on replacing ray fields by Gaussian beam fields [18] which smooth out the more violent transitional ray behavior. While details of this procedure still remain to be clarified [19], it offers so attractive a numerical option that it has already found wide application in seismology. Intrinsic local modes may be regarded as spectral objects that play in laterally inhomogeneous layered media the same role as rays in non-layered inhomogeneous media. The analog of ordinary rays are well-guided (trapped) adiabatic local
14
J.M. Arnold. L.B. Felsen / Local intrinsic modes
modes obtained of the intrinsic
from stationary mode integrals.
phase reduction Adiabatic modes
fail in cutoff transition
regions
also in caustic
of the lateral
There,
regions
they must
spectral
integral
tracking
along
and beyond,
be expressed form. Thus,
lateral
much like tracking of adiabatic validity ing full spectral
ordinary
trajectories.
uniformly local intrinsic
trajectories
and
proceeds
by the mode very
ray fields, with regions
alternating treatment.
with those requirWhile uniform
asymptotics may be capable of reducing the spectral integrals in certain transition regions (for example, through mode cutoff [l] and at lateral ray caustics), it is not clear that this option is preferable, in general, as regards accuracy and ease of computation, to direct numerical evaluation of the integrals. The feasibility of such evaluation has already been demonstrated for the twodimensional wedge prototype [6-81, and one may infer therefrom a similar behavior for at least some of the generalizations of the simple wedge configuration. Concerning numerical aspects, an intriguing possibility may be use of the abovementioned Gaussian beam detailed treatment of adiabatic however, even more so than the effects of the consequent explored before this approach
method to avoid mode transitions; for the ray method, smoothing must be becomes a reliable
option.
Acknowledgment
This work was supported in part by the Office of Naval Research under Contract No. N-0001479-C-0013, by the Joint Services Elecronics Program under Contract No. F-49620-82-C-0084, and by the National Science Foundation under Grant. No. EAR-8213147. One of the authors (J.M.A.) also acknowledges sponsorship by the Science and Engineering Research Council, U.K.
References [l]
J.M. Arnold and L.B. Felsen, “Rays and local modes in a wedge-shaped ocean”, J. Acoust. Sot. Amer. 73, 1105-
1119 (1983). [2] J.M. Arnold and L.B. Felsen, “Intrinsic modes in a wedgeshaped ocean”, J. Acoust. Sot. Amer. 76, 850-860 (1984). [3] J.M. Arnold and L.B. Felsen, “Intrinsic modes and coupled mode theory”, to be published in J. Acousr. Sot. Amer., 1985. [4] A.D. Pierce, “Guided mode disappearance during upslope propagation in variable depth shallow water overlying a fluid bottom”, J. Acoust. Sot. Amer. 7.7, 523-531 (1982). [5] A.D. Pierce, “Augmented adiabatic mode theory for upslope propagation from a point source in variable-depth shallow water overlying a fluid bottom”, J. Acoust. Sot. Amer. 74, 1837-1847 (1983).. [6] L.B. Felsen, “Numerically efficient spectral representations for guided ocean acoustics”, to be published in Compur. Math. Appl. modes: numerical [71 E. Topuz and L.B. Felsen, “Intrinsic implementation”, to be published in J. Acoust. Sot. Amer. “Intrinsic 181 J.M. Arnold, A. Belghoras and A. Dendane, mode theory of tapers in integrated optics”, to be published in IEE Proc. (London), part J. “Sound propagation [91 F.B. Jensen and W.A. Kuperman, in a wedge shaped ocean with a penetrable bottom”, J. Acousr. Sot. Amer. 67, 1564-1566 (1980). [lOI J.M. Arnold and L.B. Felsen, “Theory of wave propagation in a wedge shaped layer”, to be submitted to Wave motion. waves on water of nonuniform [Ill J.B. Keller, “Surface depth”, J. Fluid Mech. 4, 607 (1958). of the method of normal modes [I21 A.D. Pierce, “Extension to sound propagation in an almost stratified medium”, J. Acoust. Sot. Amer. 37, 19-27 (1965). H. Weinberg and R. Burridge, “Horizontal ray theory for ocean acoustics”, J. Acoust. Sot. Amer. 55, 63-79 (1974). C.H. Harrison, “Acoustic shadow zones in the horizontal plane”, J. Acoust. Sot. Amer. 65, 56-61 (1979). C.H. Harrison, “Ray/mode trajectories and shadows in the horizontal plane by ray invariants”, in L.B. Felsen, ed, Hybrid Formulation of Wave Fropagarion and Scaitering, Martinus Nijhoff, The Hague, Netherlands (1984). integral theory for uniform rep[I61 J.M. Arnold, “Oscillatory resentation of wave functions”, Radio Sci. 17, 1181-l 191 (1982). [I71 Y.H. Pao and F. Ziegler, “Theory of SH-waves in a wedgeshaped layer”, Geophys. J. Roy. Ast. Sot. 71,57-77 (1982). [I81 V. Cerveny, M.M. Popov and 1. Psencik, “Computation of wave fields in inhomogeneous media-Gaussian beam approach”, Geophys. J. Roy. Asrr. Sot. 71,109-128 (1982). theory of diffraction, evanes[I91 L.B. Felsen, “Geometrical cent waves, complex rays and Gaussian beams”, Geophys. J. Roy. Asrr. Sot. 79 (1984).
LOCAL
8 (1986)
1-14
INTRINSIC
MODES:
LAYER WITH NONPLANAR
INTERFACE
J.M. ARNOLD Department
of Electrical Engineering and Electronics,
University of Glasgow,
Glasgow GlZ SQQ, United Kingdom
L.B. FELSEN Department
Received
of Electrical Engineering and Computer Science, Polytechnic Institute of New York, Farmingdale,
12 November
1984, Revised
19 February
NY I1 735, U.S.A.
1985
Laterally inhomogeneous layered media serve as models for realistic propagation environments in electromagnetics, optics, underwater acoustics, seismology and other disciplines. When the inhomogeneities extend over long distances compared to the local wavelength, general methods (like those based on integral equations) for these nonseparable boundary value problems become intractable. For weak lateral variations, the propagation process can be localized around intrinsic modal fields, which are synthesized by plane wave spectra capable of describing propagation properties uniformly in guiding, cutoff and radiating regions. These local intrinsic modes, specified by modal invariants, traverse the medium along lateral trajectories, which may refract and form caustics similar to those for ordinary ray fields in an inhomogeneous environment. In the present investigation, these concepts are applied to a single homogeneous layer separated from a homogeneous half space by an interface with arbitrary but slowly changing surface profile, as exemplified by a shallow ocean waveguide. The previously derived intrinsic modes for a two-dimensional wedge geometry (J.M. Arnold and L.B. Felsen, “Intrinsic modes in a Journal of Acoustical Society of America 76, 850-860 (1984)), serve as the prototype for the nonseparable ocean waveguide”, construction of the wave spectra, first, for the two-dimensional configuration with nonplanar interface, and then for the general three-dimensional case.
1. Introduction Layered media with relatively arbitrary interface contours and medium properties ,model such geophysical environments as the shallow ocean, the earth’s crust, the troposphere and ionosphere, etc., as well as artificial environments such as tapered dielectric waveguides used for components in integrated optics and millimeter wave guiding and radiating systems. Wave propagation in such environments-whether acoustic, seismic, electromagnetic or optical-poses a nonseparable boundary value problem of substantial complexity. For weak lateral variations over an interval equal to the local wavelength, adiabatic (local) mode theory has provided an approximate scheme which was hampered, however, by its failure in tracking a trapped local mode through cutoff into the radiating regime in layer geometries giving rise to such 0165-2125/86/%3.50
@ 1986, Elsevier
Science
Publishers
phenomena. Only recently, in the context of underwater acoustics, has this deficiency been repaired [l-5] and thereby provided new impetus for pursuing generalized local mode propagation in these complicated environments. In particular, for the canonical two-dimensional problem of a wedgeshaped homogeneous ocean above a semi-infinite homogeneous fluid bottom, a rigorous solution has been constructed that can serve as a bench-mark for comparison with results derived by other methods. This solution, an ‘intrinsic mode’ [2] that satisfies the acoustic wave equation and boundary conditions on the ocean surface and bottom boundaries (but away from the wedge apex) exactly, is expressed as an integral over spectra of plane waves. For small bottom slopes cr, and downslope from cutoff, the intrinsic mode reduces to an adiabatic mode via leading order asymptotic approximation of the spectral integral, but that
B.V. (North-Holland)
2
integral,
J. M. Arnold, L.B. Felsen / Local intrinsic modes
when kept intact,
the cutoff transition coupling are valid
between
the adiabatic
[l, 2, 31. Thus,
‘more adaptable’ from other intrinsic resenting numerical individually excitation
an intrinsic
mode
suitable
is a
valid spectral
for efficiently
intrinsic
in superposition
by a line source
mode
applied
a function
of fields
to synthesize
has been demonstated
for small bottom slopes [6-81. Comparisons of these results with those derived by other methods [4, 5, 91 have revealed general overall agreement in the behavior of the sound field upslope from the source in the water and the bottom. Where discrepancies do arise, it may be necessary to calculate the inherently exact intrinsic mode spectral integral more accurately [lo] before deciding that the shortcomings reside in the other methods since the intrinsic mode spectral integral, as noted above, solves the wedge problem (away from the apex) exactly. Its success in treating two-dimensional propagation in a homogeneous wedge-shaped medium exemplified by a wedge-shaped ocean suggests that the spectral method might provide the foundation for addressing the three-dimensional configuration characterized by weak lateral inhomogeneities in the bottom boundary shape. The justification for this conjecture resides in the recognition that wave spectra can be adapted locally to weak changes in the environment, the local environment near any point on the bottom being modeled by its tangent plane as a local wedge. To this end, it is necessary to express the plane wave spectra for the wedge, which satisfy global conditions because of the special character of the wedge geometry, in a form localized at each vertical cross section. The clue for this localization is actually contained within the global spectral construction of the intrinsic mode for the wedge since the compatibility (selfconsistency or closure) condition for the global spectra can be interpreted as a modal resonance
locally,
parallel
The constancy
at the lateral coordinate
layer with height
of the local resonance an intrinsic
(indices)
h(x’) [2].
condition
as
the spectral invariant that
of x’ furnishes
characterizes stants
rep-
fields [2]. Feasibility of
condition
x’, to a plane
mode, and it is decoupled
excited
evaluation and
for
incorporates
modes. The latter circumstance
modes
source
uniformly
modes, where these
and more generally
object than an adiabatic makes intrinsic
accounts
and furthermore
mode, with different
pertaining
to different
con-
modes.
A
spectral attack on the fully three-dimensional laterally varying problem can thus be structured around local intrinsic modes, each identified by its spectral
invariant
at any
point
x’ in the lateral
plane where the local height is h(x’). Tracking
the
spectral invariant through the range-dependent configuration then defines lateral curves, the lateral (modal) rays, along which the local intrinsic modes propagate, the details of each lateral modal ray being dependent on the bottom profile h(x’), the mode index, and the rnitial departure direction. To make these remarks more specific, let the local plane waves in the spectrum be characterized by the three wavenumbers 5, 7, K, normalized with respect to the wavenumber k in the layer, and corresponding to the directions x, y, z, respectively. These wavenumbers, the components of the unit vector k, along which the spectral plane waves propagate, satisfy the dispersion equation [2+$+lZ=l.
(1)
The spectral invariant has the form I,[x, y, h;
K(X,
y)]
(local resonance
=
HIT,
q =
condition)
integer
(2)
where I, is a known function of x, y, h and K, and different modes are distinguished by different values of 4. To maintain the invariant intact at any point (x, y) in the range dependent environment, the vertical wavenumber has to be the range dependent function K = K (x, y) determined from solution of (2). Knowing K then specifies via (1) the horizontal wavenumbers ([‘+ n2) as a function of (x, y). Imposition of initial conditions (via a second invariant I*) where the mode is launched allows local values of 5 and 7, the horizontal projections of k,, to be determined uniquely in conformity with (l), and thereby to specify the local lateral ray direction traversed by the local
3
J.M. Arnold, L.B. Felsen / Local intrinsic modes
intrinsic
mode.
[V,,S(&
Written
as
Y)12= 52(x, y)+ T2k Y) = 1-
K2(X,
y)
=
n*(X,
spectra,
subject
to the modal
structed
directly
function,
in this local
(3)
y)
dinate
y( 1) = 0 and
(1) is nothing
but
K =
tive
shown
dashed.
The
(along
positive
x)
invariant horizontal
n(x, y).
In
coordinate
form, [VS(p)12= vector coordinate,
independent
n’(p), p the eikonal
can be solved by any of the conventional Local modes propagating along lateral
being a equation methods. rays were
actually introduced previously by Keller [ 1 l] who called them ‘horizontal rays’ (see also Pierce [12] and Weinberg and Burridge [13]), but by an entirely different approach based on direct solution of the wave equation and boundary conditions through an asymptotic ansatz, and not allowing access to the bottom medium. Thus, these earlier models cannot describe the important phenomenon of energy leakage into the bottom by intrinsic mode transition from the trapped to the radiating regime through cutoff. Applying localized spectral concepts to constructing local intrinsic mode fields requires not only a knowledge of the modal ray trajectories but also of the amplitude and phase variation of the local plane wave spectra along these trajectories. Since the local plane wave spectra in three dimensions are generally specified by two independent (see (1)) spectral parameters, a double integration in the local intrinsic mode integral is required, in contrast to the single integration for the two-dimensional case r] = 0. Moreover, determination of the spectral integrands is aided substantially by a proper parametrization of the local spectra. Therefore, before attempting the general case, it is advisable to look first at the twodimensional problem, but with nonplanar bottom profile h(x’) instead of the planar profile for the prototype wedge. Local coordinates are now chosen to fit the local geometry directly, instead of the apex centered cylindrical coordinates which were employed for the wedge [ 1,2], and the local
Although
(Fig.
l(a)).
all the lateral
coordinate
x (Fig.
cutoff, eneigy leaks into the bottom ;
1(b)). Beyond
hence the propagation are complex,
in (l)),
K(X)
are con-
on the third coor-
to the range
the eikonal equation for rays in an equivalent two-dimensional medium with equivalent refracindex
frame
Because there is no dependence rays are parallel
where S is the eikonal
invariant,
angles
of the local spectra
and the ray trajectories
the
local
are therefore
direction
of
propagation
shown
by
the
is
plane
wave
spectra
arrows. can
be
reflected back toward negative x, this would occur at a range xt> x0 at which we assume that the modal energy has been transferred almost entirely into the bottom. Therefore, reflection is neglected in this case. Reflection may, however, be important even in a two-dimensional configuration if the local
a)
Crass
s&anal
view
ye-;--,
I bl
--C
I :,
Top view and horizontal
ray
trajectories
Fig. 1. Two-dimensional local intrinsic mode propagation upslope in a two-dimensional layer with weakly sloping penetrable bottom. The mode is trapped inside the layer for x < x,, where x, is the horizontal range at cutoff. The coordinate I’ is measured along the bottom boundary, Y’ is perpendicular to that boundary, and h(x’) is the local height in a perpendicular cross section. (a) Cross seational view; (b) Top view and horizontal ray trajectories. Solid: trapped regime. dashed: radiating regime.
4
J.M. Arnold, L. B. Felsen / Local intrinsic modes
intrinsic
mode is launched
to the x-axis;
at an angle with respect
this implies
field distribution
that the initial
at x = 0 has a linear
phase
ation exp(ik7y) along y, 7 being constant of the geometrical invariance. Solutions problem
are constructed
trivially
Y
modal vari-
because for this
from the strictly
two-dimensional one on replacing 5 in the latter by ([‘- 72)1’2 when solving (1) or (2) with K = K(X).
The lateral
tical curves,
rays now form a family
differing
y (Fig. 2). Three
only by displacement
of idenalong
cases can be distinguished:
0)
7
small
b)
7
lorge.
(a) The ray trajectories do not deviate strongly from the x direction, and the local intrinsic mode therefore passes through cutoff (Fig. 2(a)); this case is similar to the one in Fig. l(a) and reflections can be neglected (unless the contrast between interior and exterior media is extreme). (b) The ray trajectories deviate strongly from the x direction and turn back before mode cutoff (Fig. 2(b)). A ray caustic is located at the turning range x, where 5 = 0, and a reflected spectrum must be included; for x > x,, the intrinsic mode field is evanescent (spectral wavenumbers are complex) but may exhibit small leakage due to tunneling into the radiating regime. (c) The ray trajectories are such that cutoff occurs near the turning range (Fig. 2(c)) ; this poses a new and more difficult transition problem which requires forward and reflected spectra. When a point source at x = y = 0 provides the excitation, local modes are launched in all directions, and the ray configuration shown in Fig. 3 contains all of the situations depicted in Fig. 2. Ray caustics do now not occur at the turning range but they are the envelope of the reflected ray system. These propagation problems in the twodimensional configuration of Fig. 1(a) provide the preliminary setting for analyzing the threedimensional configuration where h = h(x, y). It should be noted that intrinsic mode construction even in the two-dimensional model is far from trivial and constitutes, in fact, a new class of hitherto unsolved propagation problems that can accommodate the leakage phenomena depicted in
c) 7 in between. Fig. 2. Lateral projections of model ray trajectories for threedimensional local intrinsic mode propagation in the twodimensional configuration of Fig. l(a). (a) 17 small. Cutoff before turning; (b) 7 large. Turning before cutoff. Caustic at x,; (c) q in between. Turning near or at cutoff (x,=x,).
\ b
5
J.M. Arnold, L.B. Felsen / Local intrinsic modes
caustic
\
Y
J ’
source
0’
I
1’
---wx
XC
Fig. 3. Lateral modal ray trajectories for local intrinsic modes excited by a point source in the two-dimensional configuration of Fig. l(a). Rays 0, Q and @ correspond qualitatively to cases in Figs. 2(a), 2(c) and 2(b), respectively. A symmetrical ray picture for y < 0 is omitted.
Figs. 2 and 3. The sequence outlined followed in the presentation below.
lim
above is
2. Formulation of the problem Again keeping in mind the shallow ocean, we consider the two-layer configuration depicted in Fig. 1, where B, represents a planar and perfectly reflecting top boundary while B2 represents an arbitrarily shaped bottom boundary, which separates the upper medium with constant refractive index n, and density d, from the semi-infinite lower medium with constant refractive index n2 and density d2. The top layer (medium 1) and bottom layer (medium 2) will be denoted by X1 and X,, respectively. An as yet unspecified source with suppressed harmonic dependence exp( - id) is located in X1, and we seek a solution of the resulting scalar field u(x) satisfying at observation points x away from the source the scalar wave equations x E X,,
(44
(V’+n;k’)u=O,
XEX,
(4b)
subject to the boundary conditions = 0,
(5a)
.(5c) (5b)
Here, k is the wavenumber in a reference medium with refractive index n = 1, and (a/au) expresses differentiation with respect to the surface normal on B2. The solution for u is to be constructed by superposition of plane wave spectra satisfying the wave equations in (4a) and (4b). The boundary condition (5a) on the perfectly reflecting planar upper surface B, can be satisfied trivially by specular reflection of an incident (upgoing) spectral component in X1. The boundary conditions (5b) and (5~) on the lower surface B2 cannot easily be satisfied globally because of the assumed arbitrary shape of B2. Consequently, we restrict variations (changes in slope) on B2 to occur over length scales which are large compared to the wavelengths in X1 and X2. Then the boundary conditions at x on B2 can be satisfied locally at x’ E B2 by Fresnel (or Rayleigh) reflection and refraction of a downgoing spectral component in the plane interface tangent at x’ to BZ; for a.planar bottom boundary with constant slope (wedge configuration), the spectra constructed in this manner actually satisfy the lower boundary conditions exactly [2]. Satisfying both boundary conditions on B, and B2 simultaneously requires self-consistency of the upgoing and downgoing spectra which, when imposed locally, then generates a solution valid asymptotically for weakly changing bottom contours. This strategy is pursued below, first, for the twodimensional and then for the three-dimensional configurations.
3. The two-dimensional
(V2+ n:k2)u = 0,
lim Lx,+~,
&Y~-B~,
xeX,+B?
I,/’
@
=
lim
I d I / ,I
0
lim u] XCX,-rBZ
case
To implement the scenario described above, we consider first the two-dimensional problem where the contour B2 is described by the y-independent height function h = h(x’), and excitation is from a uniform source parallel to the y-axis. The spectral
J. M. Arnold, L.B. Felsen / Local intrinsic modes
6
representation
of the field u(x) is taken
u(x) =
;(x’)‘P(x,x’)dl’,
as
To satisfy we impose
XEX,
(6)
J X’in where I’ is the tangential v’ is the combination
coordinate
along BZ, and
of downgoing
and upgoing
plane wave spectra that satisfies the local boundary conditions at x’ on B,,
+exp[i4(8)]
‘P(x”, x’) = 0,
condition
(10)
where x” on B, is the point normal to x’ on B,,
intersected
XI’= x’+ h(x’)v,(x’)
along the normal
. (x-x’)].
exp[ik+
(5a) on B,,
X”E B,
Ye.
by the
(11)
and h(x’) is the local height
. (x-x’)]
V(x, x’) = exp[ik_
the boundary
of B, relative
Combining
to B2
(10) and (7)
yields (7)
Here, k-(x’)
and k+(r’) are wave vectors along the
directions of the downgoing and upgoing local plane wave spectral components in X,, respectively, with the requirement that k- and k, have identical projections on the tangent plane at x’ to satisfy the specular reflection law. Thus, points x’ on B2 parametrize the local plane wave spectra. With y and fi denoting components of k al x’, parallel to 1; and V; respectively, one has (instead of the (x, z) decomposition in (1)) k,(x’)
= l$(x’)
y2(x’)+p2(x’) where
rb = I,(x’)
f vhy(x’), = k:
and
(8)
vb = vO(x’) are unit
vectors
along the directed tangent and upward normal directions on B2. Inclusion in (7) of the reflection coefficient exp[i4( e)], where ,
(9)
exp( -iyh)
n, cos 0 = n2 cos 6“, ensures that the local reflected spectra also have the correct amplitude. The weighting function ii in (6) allows for an overall amplitude distribution in the normalized spectral combination V in (7). The domain of integration D is a subdomain of B, that will be further identified.
exp(iyh)
= 0
(12)
or (cf. (2)) I, = r,(X’)h(X’)+~{~[~(X’)]+~}= 4 = integer.
47F, (13)
Equation
(13) may be recognized as the local transcondition for the eigenvalue y= y,(x’) of the local normal mode with index q at the cross section defined by x’ on B2 and the normal Q(X)). W(x, x’) in (7) then represents the normal mode field at any point 0 < v < h(x’) along the local cross section for that value of q, and is therefore denoted by Wy(x, x’). To construct the spectral amplitude function zi,(x’) in (6), corresponding to ?Pq, we require that the total field u,(x) in the cross section be locally equal to the intrinsic mode field for the wedge geometry that approximates the local environment. Guided by the wedge solution in [2], we assume verse resonance
that the spectral 0 = 13(x’) is the angle between vector k-(x’) and the tangent to B, at x’, 0” is the angle between the refracted plane wave vector k!!(x’) and the bottom tangent, and 0 and t?” are connected by Snell’s law
+ exp(i$)
amplitude
G(x’) = a(x’) exp[ifi(x’)]
function
has the form (14)
and that stationary phase evaluation of the integral resulting from substitution of (7) (with (13)) and (14) into (6) yields the adiabatic mode field downslope from the cutoff transition region. We shall use this feature to determine a and 0 for the non-planar bottom by requiring the asymptotic approximation of the spectral integral to satisfy the local boundary conditions in the cross section. The spectral amplitudes obtained in this manner are then assumed to define u,(x) at any cross
J. M. Arnold, L. B. Felsen / Local intrinsic modes
section, provided validity
even in and beyond that the integral
the cutoff transition,
in (6) is kept intact. The
of this proposition
is verified
exactly
local plane
it is convenient
wave spectra
r:,*(X:-x’)=o,
for
the special case of the wedge and therefore, by the principle of locality, is taken to apply approximately to the general case. To proceed,
at x*=x:,
to decompose
$[lb
ponents along transverse and tangential directions at x’ (we omit the modal subscript q), * (XL -xl)]
= -1
and (17) becomes
the
aO(x’)
in (7) via (8) into com-
!P(x, x’) = !Pdx, x’) exp[iplh
. (xi -x’)]x,=x;
(Ha)
=
al’
(18)
P(4).
x’=x,’
As the observation
point x moves to different
cross
sections, the point xi moves on B2, and (18) generates a global relation which can be integrated to yield
where W,(x, x’) = {exp(-iyv)+exp(i@) X exp[$&
exp(iyv)}
* (x-x’)]
0(x’)
(15b)
with v=z.J:,*(x-x’),
(15c)
and xi is the point on the bottom B2 from which the normal passes through x (i.e., the point on B2 closest to x) such that x = x; + Y(Xi) Vo(X$
(15d)
The transverse variation ?PO(x, x’) changes slowly with x’, and, when x’ = x:, is precisely the local normal mode distribution for that cross section. Also treating a(~‘) in (14) as slowly varying, one obtains for the rapidly varying phase S in the integrand
of (6),
X’ p dl’. I x(,
(19)
The integration is with respect to the tangential direction on B, from X~E B,, where XL is a fixed reference. To find the amplitude a (x’) in ( 14), the boundary condition u(x:) = 0 (xz E B,) in (5a) is enforced to the second order in the asymptotic evaluation of the spectral integral (6) since, by the modal resonance condition (13), ?PO(x, x:) in (15a) vanishes on B, (see (lo)), thereby automatically satisfying (5a) to the leading asymptotic order. Writing u(x)=i
F(x,x’) I
aS(x, x’) al, exp[iS(x,
x’)] dl’
(204 where F(x, x’) = a(x’) WO(x, x’)/[i aS(x, x’)/al’],
S(x,x’)=R(x’)+plb. which has stationary
=
(x:-x’) points
I:(I) defined
(2Ob)
(16) one obtains
by
from integration
W(x,n, x’)
24(x:) = D
++5 *(x’-x’)]=O
by parts, noting
(lo),
that for X~E B,,
al’
exp[iS(xi,
x’)] dl’ = 0
(21)
(17)
where (a/al’) denotes differentiation along the tangent to BZ. Since the stationary phase evaluation is supposed to yield the local normal mode in the cross section, consistency with (15a) and (15b) requires xi to be the stationary phase point. Thus,
Stationary
phase
u(xy) - -7r”’
evaluation
e*im’4 exp[iS(xZ,
of the integral x:)]
yields
J.M. Arnold, L.B. F&en / Local intrinsic modes
8
which can be satisfied
azqx:,x’) al’
Moreover,
only when
variation =0
atx’=x:.
(23)
recognizing
that
of the local normal
!P’,=sin[k,(h-
!P,, is the transverse mode at x’, one has
u) sin 01 exp(i$/2).
Then from (24) and (25), with a convenient This implies
from (20b) that a(~‘) must satisfy the
differential
equation + g(x:)a(x:) Xi
=0
(24) Ignoring (27)-(29)
where
g(d) =
G(x;,aG(;;v x’q Xi) [
By integration variation
x’,
(25)
a(TI)=(Isexp{-[Ig(x’)dl.] where
a, is an arbitrary
constant,
and
xb is an
arbitrary reference point. As mentioned earlier, the validity of this construction of the spectral integrand in (6) can be tested on the special case of a planar bottom Bz with small slope CL Introducing p as the distance from the wedge apex and ~3as the angle of inclination of the spectral plane waves with respect to B2, one has /l = k, cos 8, and from (19), 0 = -
I
k, cos 8 dp
=-k,pcosfl-l where parts, local (12),
cx I
k,hsin@dtZ
(26)
the last equation results from integration by changing variables to 8 and introducing the height h = pa. With the resonance condition (26) becomes 0 = -k,p
cos 0
g
exp(-i+/2)
normalizations agree precisely
. I x’
and O(a) error terms, with the results in [2].
(244
xt)/a/l’
of (24), one finds for the amplitude
at arbitrary
1
4. Oblique propagation in the two-dimensional configuration
x’=x:
qocx:, x’)
G(xf,x’) = .#qxy,
choice
for a(), a(x’) = -
aa al’
(28)
The results in to accommodate phase variation This adds to the
Section 3 are easily generalized intrinsic modal fields with linear exp(ivy) along the y-direction. modal invariant in (2) or (13)
Z,[x’, h ; y(x’)] the further
= constant
(30a)
constraint
I, = 7~= constant
(30b)
subject to the dispersion relation y2 + p’+ v2 = 1. The corresponding wave phenomena depicted in Fig. 2, hitherto unexplored even for the twodimensional wedge, form a useful preliminary before attacking the general three-dimensional case. The generalization is accomplished on replacing the propagation coefficient p(V) in all results of Section 3 by [ P2(x’) - g2]“*. The modal ray paths are then defined by (3). The obliquely propagating modal field is exp(iny), %l)(x) = {n,(x)l P+~p2-1~~~~~~
(31)
where u,(x) is the local intrinsic mode field in Section 2, with n = 0. One observation is in order: as noted in Section 1, the spectral integrand must now generally include wave spectra with components in the negative as well as positive 1’ directions to accommodate reflection due to cutoff before reaching the radiating regime. These aspects are explored in greater detail in the next section.
9
J.M. Arnold, L.B. Felsen / Local intrinsic modes
5. The three-dimensional case
5.1. The spectral integral
To pass to the general three-dimensional case without translation invariance, two invariants are again employed to determine lateral ray paths I? We note that the second invariant I2 in (30b) is a global invariant induced solely by the translation invariant geometry assumed in the previous section. In the general case without translation symmetry, (30b) must be localized; this is done by observing that near any point X’E BZ, the direction tangent to a contour h =constant is the local equivalent of the former global symmetry direction. Thus we replace (30b) by the differential invariant I,
=
2
ak*(x’)
-*
al’
v&‘)=O
(32)
where v,,(x’) is the unit vector tangent to the curve h = constant at x’ E B2 (with a consistent orientation over B,; see Fig. 4).
To continue the synthesis of the intrinsic mode, we write u(x) =
I X’E D
;(x’)P(x,
x’) dB,
(33)
where D is a subset of B2, dB, is the differential area element on B,, ?P(x, x’) is the spectral plane wave pair attached to x’ E D c B, given by
. (x-x’)]
P(x, x’) = exp[ik_(x’)
+ wW(
WI
xexp[ik+(x')
. (x-x’)]
(34)
and k,(x’) are determined by the two invariants I, and Zi in the same manner as in Section 4. We again hypothesize that c = a eiR
(35)
and introduce the decomposition ly = q0 eiPJ= p0 eiP(x’)G;-x’)
(36)
where P = P(x’) = Ilf(k+(x’) + k-(x’))]]
(37)
is the magnitude of the tangential parts of k,, and I= I(x, x’) = l(J(x’) . (xi-x’)
Fig. 4. Implementing
the invariant case.
I, for the three-dimensional
The values of I, and I; provide two conditions which fix the components of k, tangential to B2 everywhere on B,; this in turn defines a vector field /? = i(k+ + k-) on B2 which can be integrated to yield trajectories P. A similar use of invariants has been suggested by Harrison [14,15] in connection with the ‘horizontal rays’ of Weinberg and Burridge [13], to which our trajectories are closely related. We consider these trajectories in more detail later, after completing the spectral construction around them.
(38)
is the component of (xi-x’) tangential to the trajectory P which passes through x’, xi being defined as the point on B2 nearest to x. Moreover, !P,,= ?PO(x,x’) is given by ?PO(x,x’) = {exp(-iyv) +exp(i+(@)
exp(iyv))
x exp[ib . (x-x:)]
(39)
with v= ZJb. (x-x’), fy=k,.d,
(40a) atx’EB2
(40b)
and 0 = arc sin( Y/kl).
(4Oc)
J. M. Arnold, L.B. F&en / Local intrinsic modes
10
As previously, requires that 0,(0
stationary
phase of the spectral
phase in the integrand
tion
+ pr) = 0
(41)
which in turn becomes,
phase
r&= &( r’)
point.
(42)
Integrating
(42) along
p dl’
= 0(x&)+
(43)
P(X’,X,;)
where P(x’, x6) is the trajectory from xh to x’, and dl’ is the differential arc length element on P, oriented from xh to x’. To determine the amplitude term a method similar to that in Section 3 is the necessary partial integration being along trajectories, similar to (25). The a(x’) = a(xA) exp[ -/p,.,.,i),
g dlf]
in (35), a used, with performed result is
(44)
for xl E P(x’, xh),
g(x:) = lim x,-x: G(x:,
x’) =
(45) w,(x:, x’) aS(xi, x’)/al”
S(x, x’) = .n(x’)+p(x’)l(x, Ef=O
are found by integraP is determined
by
k, = kn,,
+3(x’))=O
(SOa) (5Ob)
where sin 6 = (k, . VA)/ k, = y/k,
(5la)
and (a/al’) refers to differentiation
describe
geometrical
h(x:)v,(x:)
(Sib)
properties
of the wavevectors
k, and p (see Fig. 5).
When qn, then
I, is fixed as a global invariant equal to 8 is determined for all x’ E B2. Denoting
p = k, cos 0, as before,
p = PC,
(52)
we can show from (50b) that
I$(PC,)= VP
(53)
which determines I&, and hence the components of p. The trajectories P are integral curves of /I. In fact, (53) is identical to Snell’s law
(46) x’),
(47)
for rays P in a 2-dimensional inhomogeneous medium with refractive index n dependent on position in the medium; in this case, $ is the angle between the ray and the curve n = const. at any
P
atx’=x:E
4)/k,
cos+=(p.
along the trajec-
tory P passing through x’ E I&. In these equations, x: is the point on B, given by x:=x:+
path
and I
where,
r;=v,,(X’)
a
gives
0(x’)
function
Each
I, = k,h sin 0+!4(6J)+$rr,
where I; is the unit vector tangent to the trajectory through x’. Conversely, (42) implies I= 0 for a stationary
them.
two invariants
for I = 0,
V./n = pr;, = p
trajectory
along
20
k+
(49)
and is therefore a function of xi. The point x; E B, is an initial point for trajectories and is discussed further in the next section. 5.2. The trajectories
P
The trajectories P are of fundamental importante in this theory because both the amplitude and
Fig. 5. Geometrical paths.
quantities pertaining to the lateral k_, k, and v& are coplanar.
ray
J.M. Arnold,
point in the medium. We need only identify B/k, = n to draw this conclusion. The curves B = const.
of points
and h = const. are identical,
by Arnold
P are identified surface
the paths
as those of the ‘equivalent
tive index’ method. (along
and therefore
When projected
normals
‘horizontal rays’ similar Burridge [ 111.
11
L.B. Felsen / Local intrinsic modes
refrac-
from B2 to B1
x’ at which pairs of trajectories
ate. This construction inhomogeneous clarify multiple
to those of Weinberg
mode integral. Each
medium.
trajectories point
now
required
and caustics
x’ on each
to that given
propagation
We
the modifications
to B, or B2) they become and
is very similar
[16] for ordinary
degenerin an
proceed
to
to incorporate into the intrinsic
trajectory
generates
a
Initial conditions must be specified for these trajectories in order to determine them completely.
plane wave pair in the spectrum
These conditions
a stationary phase point, then contributions from this neighborhood on a single branch of trajectories generate a single asymptotic term in the spectral integral for u ; all branches together generate the full asymptotic value of the integral by superposition of all the individual contributions. This suggests that the integral (33) should be expressed in terms of an integral over trajectories to achieve a single valued representation, rather than as a superposition of several integrals over D = B2. This is easily done by expressing the area element dBz in terms of trajectory based coordin-
may be chosen
arbitrarily;
here
we choose them so that all trajectories originate from a single point xh E B2 which we call the source point. The initial condition for IA is determined for each trajectory
by its value at the source point
x&
where 1+4,is a parameter denoting the angle which a particular trajectory makes with the curve h = const. at xb (see Fig. 6). with conventional propagation in As inhomogeneous bulk media, it is possible for several trajectories to pass from the source x& to the same spectral parameter point x’. When two such trajectories degenerate, two stationary phase points in the intrinsic mode integral coalesce, and the stationary phase analysis used in Section 5.1 is invalid. However, the spectral function C remains well defined if, for each x’ E B2, 6(x’) is regarded as a superposition of contributions from all trajectories passing through x’, each contribution being constructed along a single trajectory according to Section 3. As x’ moves over B2, these individual terms in C(x’) can be regarded as branches of a multiple valued function on BZ, with various branches connected by caustics, the loci
Fig. 6. Initial
conditions
for lateral a point.
ray paths emanating
tion domain
of u. If the integra-
D is a large enough
neighborhood
of
ates (I’, I,&), where l’represents arc length to x’ from x;l along the trajectory which leaves x& inclined at an angle &, to the curve h = const. (Fig. 7). Thus dBz = J(x’) dl’ d$,
(56)
where J(x’) is the Jacobian formation given by J(x’) = ph.
5
of the coordinate
trans-
(57)
0
and ~6 is a unit vector tangent to P at x’. Allowing
to B2 and normal
0s l’
(584
0s *,<27r,
(5gb)
- h = const.
from Fig. 7. Trajectory
based
coordinates
for lateral
rays.
J. M. Arnold, L.B. Felsen / Local intrinsic modes
12
then
covers
all points
trajectories.
On the other
the spectrum medium,
on B2 which
of a point
by analogy
complex
values
tour C which passes contiguous end points.
from +ico to n-i00 integral
For this reason
D required
to be the domain
variable
on a Sommerfeld
sectors of the complex
a way that the spectral
&,-plane converges
&, con-
through in such at both
we take the integration
in the intrinsic
spanned
with
in a 2-dimensional
one might expect the angular
to assume
domain
hand,
source
lie on (real)
values
o=Gz
(59a)
(cloe C
(59b)
where C is a Sommerfeld contour and the integrand is analytically continued on C. Consequently, we are led to consider the spectral integral
The /‘-variable can be integrated out by the method of stationary phase. Stationary phase points for this integration are obtained as follows. The phase of the integrand of (60) is, from (35) and (36), S(x,x’)=n(x’)+p(x’).(x:-X’)
(61) x’)
(62)
where I(x, x’) = z; * (XL-X’),
P = PG
(63)
and Zh is the unit vector tangent to the trajectory with coordinate I,$,at x’ E B,. For a stationary value of S with fixed & we require
(>
$+ $1+p$=o.
construction
of stationary
condition.
mode integral
by the coordinate
= n(x’)+p(X’)I(X,
Fig. 8. Geometrical
(64)
Now 80/N = p, and 1= 0, al/al’= -1 (neglecting terms due to the curvature of P, which are second order in the ordering pertaining to small boundary slopes) when the vector (XL- x’) is exactly perpendicular to 16; hence the stationary phase point,
labeled xb, is the point on P closest to x: (see Fig. 8). Stationary phase integration leaves a remaining integral over &,, which is the equivalent for this problem of the uniform oscillatory integral derived in [ 161 for the field in a neighborhood of a caustic of ordinary rays in an inhomogeneous medium (this is known to have a uniform approximation in terms of Airy functions). In certain circumstances the above procedure must be modified. First, whenever a trajectory crosses a critical depth contour h = h,, where the phase, 4, of the Rayleigh Plk, = n,lnr, coefficient exhibits a branch point because the ‘incident’ wave in the spectral pair is at the critical direction for total internal reflection; this is simply the analog in three dimensions of the ‘local mode cutoff’ phenomenon which we have discussed in detail elsewhere [ 1,2] in connection with the twodimensional wedge. In this case the stationary phase integration along the trajectories will fail, and a more uniform method, following [l], is required. If, in addition, a caustic crosses the critical
depth
contour,
a compound
transition
function is required for observation points near the crossing. A second source of trouble with the spectral construction occurs on depth contours where the depth h is an extremum and dp/dl’ changes sign as the trajectory crosses the contour (see Fig. 9). The effects encountered here are analogous to those in ordinary ray propagation in inhomogeneous media near an extremum of refractive index. In particular, tunnelling across a depth minimum can occur. Further study is required before these effects can be fully incorporated into
J.M. Arnold, L.B. Felsen / Local intrinsic modes
for the special
;F’=h-in /s,+”
bounded and
case of a single homogeneous
on one side by a perfectly
on the other
which separates half space.
Fig. 9. Lateral rays paths when depth profile has a minimum.
intrinsic
no reason
mode
framework,
to suppose
although
there
is
that this will not eventually
be possible. There is little to be gained, at present, from explicit enumeration and asymptotic extraction of all the transition functions that can occur in this type of problem (with the possible exception of the simple caustic [16] and isolated local mode cutoff [l, 21 phenomena, which are both expressible in terms of Airy functions). It is more profitable to regard the intrinsic mode itself as a globally uniform asymptotic representation of u and to proceed to direct numerical evaluation. In that case, the localizing properties of the stationary phase principle can be exploited to devise numerical quadrature schemes for evaluating the integral efficiently. These aspects of the problem are currently being explored.
6. Summary Local intrinsic modes appear promising for describing propagation in layered media with weak lateral inhomogeneities on the scale of the local wavelength.
Such modes,
constructed
in principle
by plane wave spectral synthesis, are characterized by model invariants and, by methods similar to those employed for ordinary geometrical rays in an inhomogeneous environment, can be traced along lateral trajectories through guiding, cutoff transition and radiating regions. Intrinsic modes have actually been constructed for the prototype problem of a wedge-shaped layer with penetrable boundaries [ 1,2]. As a first step toward a more general implementation of the spectral method, local intrinsic modes have been developed here
side by a nonplanar
it from an exterior
a shallow
modes
dimensional
and
for
case, with an indication
can be used
interface
which
obtained
then
to synthesize
plane
homogeneous
ocean waveguide,
have been
layer
reflecting
In this configuration,
for example, two-dimensional
the intrinsic
13
models, the local
first for the the
three-
of how they
source-excited
fields.
Generalizations to accommodate impedance boundary conditions, medium variations, multiple layers, etc., are already being considered and are expected to introduce complications in detail, but not in the fundamental concepts that have been successful in this first extension of the rigorous canonical
wedge solution.
In assessing the scope of the local intrinsic mode spectral theory, one may be guided by analogous considerations for geometrical ray theory. In an extended environment with weak inhomogeneities, asymptotic ray theory (ART) provides one of the most widely used and effective analytical tools with feasible numerical implementation. The failures of ART in caustic, shadow boundary, and other transition regions can be repaired by more general spectral objects, often called generalized or uniformized rays (see [ 171 for the wedge configuration) given in the form of spectral integrals. This complicates the numerical evaluation but still renders the method tractable. ART fields are actually the stationary phase approximations formized ray integrals. In excessively media,
explicit
concern
with caustics,
of the unicomplicated etc., can be
avoided on replacing ray fields by Gaussian beam fields [18] which smooth out the more violent transitional ray behavior. While details of this procedure still remain to be clarified [19], it offers so attractive a numerical option that it has already found wide application in seismology. Intrinsic local modes may be regarded as spectral objects that play in laterally inhomogeneous layered media the same role as rays in non-layered inhomogeneous media. The analog of ordinary rays are well-guided (trapped) adiabatic local
14
J.M. Arnold. L.B. Felsen / Local intrinsic modes
modes obtained of the intrinsic
from stationary mode integrals.
phase reduction Adiabatic modes
fail in cutoff transition
regions
also in caustic
of the lateral
There,
regions
they must
spectral
integral
tracking
along
and beyond,
be expressed form. Thus,
lateral
much like tracking of adiabatic validity ing full spectral
ordinary
trajectories.
uniformly local intrinsic
trajectories
and
proceeds
by the mode very
ray fields, with regions
alternating treatment.
with those requirWhile uniform
asymptotics may be capable of reducing the spectral integrals in certain transition regions (for example, through mode cutoff [l] and at lateral ray caustics), it is not clear that this option is preferable, in general, as regards accuracy and ease of computation, to direct numerical evaluation of the integrals. The feasibility of such evaluation has already been demonstrated for the twodimensional wedge prototype [6-81, and one may infer therefrom a similar behavior for at least some of the generalizations of the simple wedge configuration. Concerning numerical aspects, an intriguing possibility may be use of the abovementioned Gaussian beam detailed treatment of adiabatic however, even more so than the effects of the consequent explored before this approach
method to avoid mode transitions; for the ray method, smoothing must be becomes a reliable
option.
Acknowledgment
This work was supported in part by the Office of Naval Research under Contract No. N-0001479-C-0013, by the Joint Services Elecronics Program under Contract No. F-49620-82-C-0084, and by the National Science Foundation under Grant. No. EAR-8213147. One of the authors (J.M.A.) also acknowledges sponsorship by the Science and Engineering Research Council, U.K.
References [l]
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1119 (1983). [2] J.M. Arnold and L.B. Felsen, “Intrinsic modes in a wedgeshaped ocean”, J. Acoust. Sot. Amer. 76, 850-860 (1984). [3] J.M. Arnold and L.B. Felsen, “Intrinsic modes and coupled mode theory”, to be published in J. Acousr. Sot. Amer., 1985. [4] A.D. Pierce, “Guided mode disappearance during upslope propagation in variable depth shallow water overlying a fluid bottom”, J. Acoust. Sot. Amer. 7.7, 523-531 (1982). [5] A.D. Pierce, “Augmented adiabatic mode theory for upslope propagation from a point source in variable-depth shallow water overlying a fluid bottom”, J. Acoust. Sot. Amer. 74, 1837-1847 (1983).. [6] L.B. Felsen, “Numerically efficient spectral representations for guided ocean acoustics”, to be published in Compur. Math. Appl. modes: numerical [71 E. Topuz and L.B. Felsen, “Intrinsic implementation”, to be published in J. Acoust. Sot. Amer. “Intrinsic 181 J.M. Arnold, A. Belghoras and A. Dendane, mode theory of tapers in integrated optics”, to be published in IEE Proc. (London), part J. “Sound propagation [91 F.B. Jensen and W.A. Kuperman, in a wedge shaped ocean with a penetrable bottom”, J. Acousr. Sot. Amer. 67, 1564-1566 (1980). [lOI J.M. Arnold and L.B. Felsen, “Theory of wave propagation in a wedge shaped layer”, to be submitted to Wave motion. waves on water of nonuniform [Ill J.B. Keller, “Surface depth”, J. Fluid Mech. 4, 607 (1958). of the method of normal modes [I21 A.D. Pierce, “Extension to sound propagation in an almost stratified medium”, J. Acoust. Sot. Amer. 37, 19-27 (1965). H. Weinberg and R. Burridge, “Horizontal ray theory for ocean acoustics”, J. Acoust. Sot. Amer. 55, 63-79 (1974). C.H. Harrison, “Acoustic shadow zones in the horizontal plane”, J. Acoust. Sot. Amer. 65, 56-61 (1979). C.H. Harrison, “Ray/mode trajectories and shadows in the horizontal plane by ray invariants”, in L.B. Felsen, ed, Hybrid Formulation of Wave Fropagarion and Scaitering, Martinus Nijhoff, The Hague, Netherlands (1984). integral theory for uniform rep[I61 J.M. Arnold, “Oscillatory resentation of wave functions”, Radio Sci. 17, 1181-l 191 (1982). [I71 Y.H. Pao and F. Ziegler, “Theory of SH-waves in a wedgeshaped layer”, Geophys. J. Roy. Ast. Sot. 71,57-77 (1982). [I81 V. Cerveny, M.M. Popov and 1. Psencik, “Computation of wave fields in inhomogeneous media-Gaussian beam approach”, Geophys. J. Roy. Asrr. Sot. 71,109-128 (1982). theory of diffraction, evanes[I91 L.B. Felsen, “Geometrical cent waves, complex rays and Gaussian beams”, Geophys. J. Roy. Asrr. Sot. 79 (1984).