Field representations in spherically stratified regions

Marcuvitz, N.

doi:10.1002/cpa.3160040204

Cited by 54 publications

(21 citation statements)

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“…the characteristic (resolvent) Green's function of the one-dimensional transverse problem 313 in the ^-direction. Consequently, all the poles of f («•) may be found conveniently by using the transverse resonance principle, 30 which, in the present case, means that these poles are given by the roots of Z,(K) + Z 0 (/c) = 0 . .…”

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confidence: 99%

Guided complex waves. Part 1: Fields at an interface

Tamir¹,

Oliner²

1963

Proc. Inst. Electr. Eng. UK

221

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The variety of waves which can be supported by a plane homogeneous interface includes surface waves of the forward and backward type, and several kinds of complex wave, the latter being characterized by wave numbers which are complex even though the media involved are not necessarily lossy. The present study views all these waves as contributions due to poles in several alternative integral representations of a source-excited field, and places particular stress on the steepest-descent representation. The pole locations, field distributions and power-transport properties are explored in detail for all the wave types. Distinctions are made between proper (spectral, modal) and improper waves, and between lossy and lossless structures; complex waves along lossless structures are shown to appear always in degenerate pairs consisting of a forward and a backward wave, with interesting power-flow characteristics. The different wave types are grouped into the general category of guided complex waves which propagate without attenuation as inhomogeneous slow plane waves at some angle to the interface. Power-transport considerations via the steepest-descent representation show that these waves either carry power to compensate for losses in the system or account for a transfer of energy into the radiation field. kList of symbols c -Velocity of light in free space f(«0 -Complex amplitude function for the /c-mode G(x, z) -•-Total solution for the field in Cartesian co-ordinates G P (A', Z) = Pole contribution to G(x, z) G(r, 6) = Total solution for the field in polar co-ordinates G p (r, 8) = Pole contribution to G(r, 8) G s (r, 8) -Space-wave contribution to G(r, 8) G 0 ((f) p ) -Field contribution at the origin due to a p o l e a t = p t(ju. o e o ) l/2 = Wave number of plane wave in free space. P x , P z --Components of power flow r •-Radial field co-ordinate v g -Group velocity v p = Phase velocity x, y, z = Cartesian field co-ordinates ZJ(K) = Impedance at the interface ZO(K) -Characteristic impedance of transmission line representing free space a,. ~ Radial attenuation factor P,--Radial phase factor = Absolute permittivity of free space = Wave number in the z-direction -Location of a pole in the complex £-plane r) = Imaginary component of (f> p -Imaginary component of p 8 -Angular field co-ordinate ' c = Angle of definition for pole contributions = Wave number in the jc-direction +JK p i -Location of a pole in the complex /c-plane /x 0 = Absolute permeability of free space g = Real component of £ p = Real component of p

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confidence: 99%

Guided complex waves. Part 1: Fields at an interface

Tamir¹,

Oliner²

1963

Proc. Inst. Electr. Eng. UK

221

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“…For the exp (-iwt) time factor, these solutions can be found in Morse and Feshbach [1953] and Marcuvitz [1951], respectively, and are given by…”

Section: Formal Solutionmentioning

confidence: 97%

A Normal Mode Theory of an Underwater Acoustic Duct by Means of Green's Function

Deavenport

1966

Radio Science

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The resolvent Green's function technique is used to find the acoustic field due to a CW point source located in a laminar inhomogeneous medium. Results are in terms of a residue expansion (normal modes) and branch line integrals. This approach is then used to solve the wave equation for an underwater acoustic duct described by an Epstein layer. Only the residue expansion is evaluated. From the residue expansion, propagation loss is found as a function of range. Application to a realistic velocity-depth profile indicates a pattern of beats ·resulting from a superposition of the normal modes. A general procedure is next described for obtaining exact solutions of the wave equation for a large variety of velocity-depth profiles. This procedure may prove useful in developing future normal mode models.

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“…Following the new understanding provided by Watson and Sommerfeld, the different expressions available for the solution of a separable BVP were systematically derived for many important BVPs [76], [51,Chapters 4 and 5], [33], [34,Chapters 3 and 6], [19], [18]. However, the important question about the completeness of the associated transforms arose, namely the basic question of whether an arbitrary function (from a suitable class) can be expanded in the appropriate eigenfunctions and whether the relevant series or integral converges.…”

Section: Establishing Completeness Of a Transformmentioning

confidence: 99%

Synthesis, as Opposed to Separation, of Variables

Fokas¹,

Spence²

2012

SIAM Rev.

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Abstract. Every applied mathematician has used separation of variables. For a given boundary value problem (BVP) in two dimensions, the starting point of this powerful method is the separation of the given PDE into two ODEs. If the spectral analysis of either of these ODEs yields an appropriate transform pair, i.e., a transform consistent with the given boundary conditions, then the given BVP can be reduced to a BVP for an ODE. For simple BVPs it is straightforward to choose an appropriate transform and hence the spectral analysis can be avoided. In spite of its enormous applicability, this method has certain limitations. In particular, it requires the given domain, PDE, and boundary conditions to be separable, and also may not be applicable if the BVP is non-self-adjoint. Furthermore, it expresses the solution as either an integral or a series, neither of which are uniformly convergent on the boundary of the domain (for nonvanishing boundary conditions), which renders such expressions unsuitable for numerical computations. This paper describes a recently introduced transform method that can be applied to certain nonseparable and non-selfadjoint problems. Furthermore, this method expresses the solution as an integral in the complex plane that is uniformly convergent on the boundary of the domain. The starting point of the method is to write the PDE as a one-parameter family of equations formulated in a divergence form, and this allows one to consider the variables together. In this sense, the method is based on the "synthesis" as opposed to the "separation" of variables. The new method has already been applied to a plethora of BVPs and furthermore has led to the development of certain novel numerical techniques. However, a large number of related analytical and numerical questions remain open. This paper illustrates the method by applying it to two particular non-self-adjoint BVPs: one for the linearized KdV equation formulated on the half-line, and the other for the Helmholtz equation in the exterior of the disc (the latter is non-self-adjoint due to the radiation condition). The former problem played a crucial role in the development of the new method, whereas the latter problem was instrumental in the full development of the classical transform method. Although the new method can now be presented using only classical techniques, it actually originated in the theory of certain nonlinear PDEs called integrable, whose crucial feature is the existence of a Lax pair formulation. It is shown here that Lax pairs provide the generalization of the divergence formulation from a separable linear to an integrable nonlinear PDE.

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Field representations in spherically stratified regions

Cited by 54 publications

References 1 publication

Guided complex waves. Part 1: Fields at an interface

Guided complex waves. Part 1: Fields at an interface

A Normal Mode Theory of an Underwater Acoustic Duct by Means of Green's Function

Synthesis, as Opposed to Separation, of Variables

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