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On guided-wave propagation in laminar inhomogeneous media
Journal of Sound and Vibration (1975) 39(4), 527-529
ON GUIDED-WAVE PROPAGATION IN LAMINAR INHOMOGENEOUS MEDIA An approach has been presented [1] to simplify the problem of generating the characteristic equation governing the frequencies of vibration in a dynamic system, ttere, it is shown that guided-wave propagation in laminar inhomogeneous media partakes in the tractability of this approach. i. INTRODUCTION In a recent paper, the characteristic equations for free dynamic vibration systems have been generated by using a novel approach. That same approach applies also to guided-wave propagation in laminar inhomogeneous media, bounded by two plane-parallel boundaries. This propagation model is realized in many applications. In the sea, for instance, a simplified model consists of an index of refraction varying in depth only, along with a perfectly rigid bottom and a free surface boundary. In this note, the study of such guided-wave propagation is set forth in a form where the approach to the vibrational problem applies. 2. ANALYSIS AND DISCUSSIONS
Within a wave-guide, an acoustic field, tp, obeys the reduced wave equation, along with source conditions at z = z', p = 0, surface and bottom boundary conditions at z = z,,zb, and a radiation condition at p = oo. When using the separation o f variables method, ~(p,z) = V(p) U(z), the solution in cylindrical co-ordinates is governed by two ordinary differential equations, one for V, where V(p) represents the range behavior of the field, and the other for U, where U(z) represents the depth behavior of the field; the separation constant is 2 [2]. For the range behavior of the field, the solution is given by H~(2p), the Hankel function of zeroth order and first kind. In the far field, the first term in the asymptotic expansion of H~().p) could be sufficient. The allowed values or eigenvalues, 2,., of 2 are determined in the solution for the depth variation of the field. Few profiles in the index of refraction, n(z), however, allow for U-solution which can be written in terms of known functions. Also, in many instances, approximate solutions fail to provide sufficient accuracy [2]. In those cases, the resulting equation for U must be converted into a system of algebraic equations for further treatment on the computer. To generate the characteristic equation governing the allowed values, 2m, the iterative approach in reference [I] lends itself to easy implementation. First, to conserve fidelity of representation through the conversion process, it is helpful to split the index of refraction squared, n2(z), into two parts: one of these parts, no2(Z),is chosen to be sufficiently simple in form that the resulting Helmholtz equation, Lo U = 0, is solvable analytically; the other part, n~(z), is considered as a perturbation on no2(z).Then, the differential equation for Ualong with its boundary conditions are transformed into an equivalent integral equation: ~b
U(z) = f G(z, z') [kZon](z') - )2] U(z')dz',
(i)
Zb
where ko is a reference wave number. The Green function, G(z,z'), satisfies the differential 527
528
LETTERS TO TItE EDITOR
equation, Lo U = 0, plus the b o u n d a r y conditions at z = Zb,Z,, and the continuity conditions at the point source, z = z'. Its general form is
~,,,(z')u~(z) z' ~. W , < z
'
a(~,z ) =
.
(2)
In an ocean with a pressure release surface b o u n d a r y at z = zs, and admittance, Y, at the b o t t o m b o u n d a r y (z = zb), the elements in the 'above G r e e n function are u,(z) =
~,(z) + ~(z) ~ , ( z , ) + ~r
'
u~(z) = ~,(z) 4,~(~s) - ,/,~ (~) ~,(z.),
(3)
where
d~,(z) dz
Yc~,(z)
(X--
d~2(z) dz
=--z b
W = Wronskian =
2(z')
~(z' Ig.lg t
The functions q~,(z) and ,;b2(z) are the k n o w n solutions of the differential equation Lo U = 0; their exact nature is dictated by the selected variation for no(z). For instance, when no(z) is assigned a zero value, the functions qS,(z ) and qS~(z), have a simple form, where
ul(z)=(z-zb+
l/Y),
u2(z) = (z - z,)
(4)
and w = (z,, - z, - l / r ) .
F o r the preceding formulation, the characteristic equation m a y be easily obtained [1, see equation (20)], after an interchange oftq and u2 for the G r e e n functions'[" to correspond: l-
:2] 7 C.(z')dz'=O,
f :Cb
(5)
n- 1
where
C,(z) = f dz' K(z, z ' ) . . . z
and
L
K(z C"-2),z c"-l>) u2(z <"-1>) dz c"-1)
z(
-
K (z, z') = - [u,(z') u2(z) - u,(z) u2(z')] [k2on~(z') - )2]/W.
(6)
F o r repetitive calculations as required, for instance, with changing sea conditions, equation (5) lends itselfto easy mechanization [1 ]. The zeros o f e q u a t i o n (5) are the allowed values 2,.. 1"Note that the Green function symbols, ~< and/>, are misplaced in reference [1] (see equation (6)], and should be interchanged.
LETTERSTO THE EDITOR
529
The corresponding eigenfunctions or standing modes, U,.(z), may be obtained by using standard procedures [3]. The normal modes, ~,,.(z), have the form •,,(z) ~ { U,.(z) Hg(2,, p)}.
(7)
Analogous to the vibration problem [3], the wave problem may not be to determine a complete solution. Often, the only requirement is to specify from equation (5) whether or not, for a given frequency, the wave guiding structure can support certain propagation modes. This is of interest in channel propagation, or dueting formed by a local extremum in n(z), where the trapped modes could propagate to abnormally long ranges. Naval Underwater Systems Center, Newport, Rhode Island02840, U.S.A. (Received 24 December 1974)
J. C. HASSAB
REFERENCES
1. J. C. HAssAB ~9 74 J~urnal ~f S~und and Vibrati~n 33~ 59-66. A n~ve~ generati~n ~f the characteristic
equation with application to free vibration. 2. L. M. BREKHOVSKIKH1960 Waves in Layered Media. New York: Academic Press. See Chapter VI. 3. E. C. PESTELand F. A. LECI,:IE1963 Matrix Methods in Elastomeehanics. New York: McGraw-Hill Book Company, Inc. See pp. 71-76.
ON GUIDED-WAVE PROPAGATION IN LAMINAR INHOMOGENEOUS MEDIA An approach has been presented [1] to simplify the problem of generating the characteristic equation governing the frequencies of vibration in a dynamic system, ttere, it is shown that guided-wave propagation in laminar inhomogeneous media partakes in the tractability of this approach. i. INTRODUCTION In a recent paper, the characteristic equations for free dynamic vibration systems have been generated by using a novel approach. That same approach applies also to guided-wave propagation in laminar inhomogeneous media, bounded by two plane-parallel boundaries. This propagation model is realized in many applications. In the sea, for instance, a simplified model consists of an index of refraction varying in depth only, along with a perfectly rigid bottom and a free surface boundary. In this note, the study of such guided-wave propagation is set forth in a form where the approach to the vibrational problem applies. 2. ANALYSIS AND DISCUSSIONS
Within a wave-guide, an acoustic field, tp, obeys the reduced wave equation, along with source conditions at z = z', p = 0, surface and bottom boundary conditions at z = z,,zb, and a radiation condition at p = oo. When using the separation o f variables method, ~(p,z) = V(p) U(z), the solution in cylindrical co-ordinates is governed by two ordinary differential equations, one for V, where V(p) represents the range behavior of the field, and the other for U, where U(z) represents the depth behavior of the field; the separation constant is 2 [2]. For the range behavior of the field, the solution is given by H~(2p), the Hankel function of zeroth order and first kind. In the far field, the first term in the asymptotic expansion of H~().p) could be sufficient. The allowed values or eigenvalues, 2,., of 2 are determined in the solution for the depth variation of the field. Few profiles in the index of refraction, n(z), however, allow for U-solution which can be written in terms of known functions. Also, in many instances, approximate solutions fail to provide sufficient accuracy [2]. In those cases, the resulting equation for U must be converted into a system of algebraic equations for further treatment on the computer. To generate the characteristic equation governing the allowed values, 2m, the iterative approach in reference [I] lends itself to easy implementation. First, to conserve fidelity of representation through the conversion process, it is helpful to split the index of refraction squared, n2(z), into two parts: one of these parts, no2(Z),is chosen to be sufficiently simple in form that the resulting Helmholtz equation, Lo U = 0, is solvable analytically; the other part, n~(z), is considered as a perturbation on no2(z).Then, the differential equation for Ualong with its boundary conditions are transformed into an equivalent integral equation: ~b
U(z) = f G(z, z') [kZon](z') - )2] U(z')dz',
(i)
Zb
where ko is a reference wave number. The Green function, G(z,z'), satisfies the differential 527
528
LETTERS TO TItE EDITOR
equation, Lo U = 0, plus the b o u n d a r y conditions at z = Zb,Z,, and the continuity conditions at the point source, z = z'. Its general form is
~,,,(z')u~(z) z' ~. W , < z
'
a(~,z ) =
.
(2)
In an ocean with a pressure release surface b o u n d a r y at z = zs, and admittance, Y, at the b o t t o m b o u n d a r y (z = zb), the elements in the 'above G r e e n function are u,(z) =
~,(z) + ~(z) ~ , ( z , ) + ~r
'
u~(z) = ~,(z) 4,~(~s) - ,/,~ (~) ~,(z.),
(3)
where
d~,(z) dz
Yc~,(z)
(X--
d~2(z) dz
=--z b
W = Wronskian =
2(z')
~(z' Ig.lg t
The functions q~,(z) and ,;b2(z) are the k n o w n solutions of the differential equation Lo U = 0; their exact nature is dictated by the selected variation for no(z). For instance, when no(z) is assigned a zero value, the functions qS,(z ) and qS~(z), have a simple form, where
ul(z)=(z-zb+
l/Y),
u2(z) = (z - z,)
(4)
and w = (z,, - z, - l / r ) .
F o r the preceding formulation, the characteristic equation m a y be easily obtained [1, see equation (20)], after an interchange oftq and u2 for the G r e e n functions'[" to correspond: l-
:2] 7 C.(z')dz'=O,
f :Cb
(5)
n- 1
where
C,(z) = f dz' K(z, z ' ) . . . z
and
L
K(z C"-2),z c"-l>) u2(z <"-1>) dz c"-1)
z(
-
K (z, z') = - [u,(z') u2(z) - u,(z) u2(z')] [k2on~(z') - )2]/W.
(6)
F o r repetitive calculations as required, for instance, with changing sea conditions, equation (5) lends itselfto easy mechanization [1 ]. The zeros o f e q u a t i o n (5) are the allowed values 2,.. 1"Note that the Green function symbols, ~< and/>, are misplaced in reference [1] (see equation (6)], and should be interchanged.
LETTERSTO THE EDITOR
529
The corresponding eigenfunctions or standing modes, U,.(z), may be obtained by using standard procedures [3]. The normal modes, ~,,.(z), have the form •,,(z) ~ { U,.(z) Hg(2,, p)}.
(7)
Analogous to the vibration problem [3], the wave problem may not be to determine a complete solution. Often, the only requirement is to specify from equation (5) whether or not, for a given frequency, the wave guiding structure can support certain propagation modes. This is of interest in channel propagation, or dueting formed by a local extremum in n(z), where the trapped modes could propagate to abnormally long ranges. Naval Underwater Systems Center, Newport, Rhode Island02840, U.S.A. (Received 24 December 1974)
J. C. HASSAB
REFERENCES
1. J. C. HAssAB ~9 74 J~urnal ~f S~und and Vibrati~n 33~ 59-66. A n~ve~ generati~n ~f the characteristic
equation with application to free vibration. 2. L. M. BREKHOVSKIKH1960 Waves in Layered Media. New York: Academic Press. See Chapter VI. 3. E. C. PESTELand F. A. LECI,:IE1963 Matrix Methods in Elastomeehanics. New York: McGraw-Hill Book Company, Inc. See pp. 71-76.