Helmholtz Wave Equation Research Articles

UTILIZAÇÃO DA COMPONENTE VERTICAL DE CAMPOS MAGNÉTICOS ALTERNADOS NA INTERPRETAÇÃO GEOLÓGICA

The theoretical solutions for the behavior of the sinusoidal electromagnetic fields of a plane wave within a two quarter spaces earth are established. The results for the fields, induced by the lateral contrast of conductivity, are given in the form of inverse Fourier cosine transforms. They have been obtained by solving Helmholtz wave equations subject to the proper boundary conditions. Two parameters, the modulus and the argument of the normalized vertical component of the magnetic field intensity, have been computed for five values of the contrast of electrical conductivity, and as a function of induction number. The analysis of the curves obtained for the two parameters shows that: (i) the modulus may be used in the geological interpretation of structures with one vertical contact, and it presents the advantage of being accurately and rapidly measured;(ii) though the argument does not render any aditional information, it may be combined to the modulus to calculate the tilt angle of the polarization ellipse, which is the parameter normally measured in both the AFMAG and VLF methods surveys.

On the Synthesis of Waveguides and Cavities Realized with Nonseparable Solutions of Helmholtz Wave Equation (Short Papers)

This short paper shows how nonseparable solutions of the Helmholtz wave equation can be used in the synthesis of waveguides and cavities with nonconventional cross section, and also investigates the attenuation and Q factor.

Resonant scattering from inhomogeneous nonspherical targets

A method is proposed for solving problems in which either scalar or vector waves impinge at an arbitrary angle on an inhomogeneous nonspherical target whose size is comparable to the wavelength of the incident radiation. This method reduces a partial differential equation, the Helmholtz wave equation, to an ordinary differential equation through selection of angular trial functions composed of weighted sums of spherical harmonics. The wave equation then becomes a coupled set of radial differential equations which are discretized and solved by matrix methods, enforcing boundary conditions on the surface of the smallest sphere which completely encloses the target. The method is an extension of partial wave expansion and reduces to it exactly when the target is spherically symmetric.

Use of nonseparable solutions of Helmholtz wave equation in waveguides and cavities

The letter shows how some recently discovered nonseparable solutions of the Helmholtz wave equation can be used in the synthesis of waveguides and cavities with nonconventional cross-sections, and also investigates some interesting properties.

Origin of the Sound Speed Gradient

An approach more fundamental than that usually taken in underwater acoustics is needed to explain one of the more striking features of the deep ocean, the constant sound speed gradient. This gradient is induced by the earth's gravitational field. The differential equation satisfied by the acoustic pressure is not the usual scalar Helmholtz wave equation. It is shown, however, that the propagation speed is given by the usual formula and, furthermore, the high-frequency behavior is the same as that predicted by the usual scalar Helmholtz equation with variable sound speed. Calculation of the sound speed gradient from the equation of state yields a gradient of 0.019 which compares favorably with the observed value of 0.018. [This work was supported by Contract N00039-71-C-0307].

Approximation of Natural Frequencies for Membranes of Arbitrary Shape

The method employs the nonseparable solutions to the Helmholtz wave equation discovered by D. S. Moseley [Quart. Appl. Math. 27, 451–459 (1970)] as the elements of an expansion for membrane deflection. The boundary conditions for a membrane of any shape are then approximated by selecting the coefficients for the expansion so as to minimize the mean square error over a number of points on the boundary greater than or equal to the number of terms in the series representation. This procedure leads to an eigenvalue problem, and the minima in the eigenvalues lead to approximations of the natural frequencies. The resulting approximate solution satisfies the differential equation exactly and approximates the boundary condition. One advantage of the method is that it furnishes approximations to the frequencies of higher modes, as well as the first. When the method was applied to shapes for which the exact solutions are known, the results showed excellent agreement.

Spheroidal Wave Functions: Their Use and Evaluation

The Helmholtz or scaler wave equation (▿2 + k2) ψ = 0 is separable in both oblate and prolate spheroidal coordinates. Solutions to this equation constitute an essential element in the numerical calculation of the diffraction, radiation, and scattering of acoustic and electromagnetic waves by spheroids. They also find considerable application in other areas such as signal processing and nuclear models. FORTRAN computer programs have been developed to evaluate both efficiently and accurately the radial and angular parts of the solution in both oblate and prolate spheroidal coordinates. Extensive tables of oblate and prolate radial functions containing entries for values of h up to 40.0 and for values of the prolate radial coordinate as low as 1.00000001 are being published. (h = kd/2, where d is the interfocal distance.) These tables as well as the capabilities of the capabilities of the various computer programs are discussed. Examples illustrating the use of this coordinate system in acoustic radiation calculations are presented.

Membrane Application of Second-Order Nonseparables

The Helmholtz wave equation [∇2+ (ω/c)2]W − 0 is customarily solved by the familiar, college-taught method of separation of variables. Separable solutions so obtained can be considered a special—and very important—case in an infinite set of nonseparable solutions of this equation. [See Quart. Appl. Math. 22, 354 (1965).] Known nonseparables of this set can be derived from the separable by repeated application of a generative operator. In view of this, a separable is looked upon as a nonseparable of zero order. An example in two dimensions is w = W exp iωt, where W is amplitude, W(0) = sinax sinby, W(1) = bx cosax sinby − ay sinax cosby, the operator connecting them is b∂/∂a−a∂/∂b, and the frequency eqation for both is (ω/c)2=a2+b2. Some properties of nonseparable members of the set are presented, and application to the transverse displacement of a membrane is illustrated.

Further properties of the nonseparable solutions of the Helmholtz wave equation

Nonseparable solutions W ( n ) {W^{\left ( n \right )}} of ( ∇ 2 + k 2 ) W ( n ) = 0 \left ( {{\nabla ^2} + {k^2}} \right ){W^{\left ( n \right )}} = 0 are linearly independent, but inter-related through a generative differential operator. The nonseparable of order n = 0 n = 0 is the familiar separable solution. In two cartesian coordinates, a sum of zero and second order solutions can describe transverse motion of a membrane of unique boundary contour. In three coordinates the same sum can describe acoustic pressure in a uniquely shaped cavity with pressure-release walls.

On the short-wave asymptotic solution of a problem in acoustic radiation

AbstractA study is made of a problem in acoustic radiation, with the object of developing a rigorous theory of short-wave asymptotic solutions of the Helmholtz wave equation. The particular example under consideration is that of the radiation of sound waves by a smooth convex cylinder pulsating with prescribed normal velocity V(s) exp (−iωt), where s denotes arc length of the curve C bounding the cylinder cross-section, and ω is the radian frequency.

The diffraction of the field of dipole radiation by two spheres

The diffraction of the field of dipole radiation by two spheres

Zur Theorie der ebenen, instationären Unterschallströmung um ein schwingendes Profil im Kanal

The mixed boundary value problem for the determination of the velocity potential of an unsteady perturbation flow past an oscillating thin profile in an inviscid compressible gas at subsonic speeds is treated. After the separation of the wave-like terms in the Fourier expansions of the reduced potential, satisfying the Helmholtz wave equation, the boundary value problem for the remaining potential is transformed into a functional equation of the generalized Wiener-Hopf type by means of the Laplace transform. The familiar Wiener-Hopf technique leads to a linear system of infinitely many unknowns which can be solved by iteration for not too small profile lengths. Finally, two analytic expressions for the reduced potential in terms of different kernel functions and formulae for the lift and moment acting on the profile are written down.

Non-separable solutions of the Helmholtz wave equation

A set of solutions not obtainable by the method of separation of variables is presented for the vector Helmholtz wave equation in circular cylindrical coordinates limited to non-angular dependence. These are constructed of Bessel and trigonometric functions. For example, if A is the vector, the r r -component of the simplest member of the set is \[ A r = C 1 [ m r J 0 ( p r ) cos ⁡ ( m z ) + p z J 1 ( p r ) sin ⁡ ( m z ) ] e − i w t , {A_r} = {C_1}\left [ {mr{J_0}\left ( {pr} \right )\cos \left ( {mz} \right ) + pz{J_1}\left ( {pr} \right )\sin \left ( {mz} \right )} \right ]{e^{ - iwt}}, \] where C 1 {C_1} is an arbitrary constant, m m and p p are propagation constants, and ω \omega is angular frequency. Brief reference is made to three-dimensional solutions in rectangular coordinates.

On Combined Free and Forced Convection in Channels

The heat-transfer problems of combined free and forced convection by a fully developed laminar flow in a vertical channel of constant axial wall temperature gradient with or without heat generations are approached by a new method. By introducing a complex function which is directly related to the velocity and temperature fields, the coupled momentum and energy equations are readily combinable to a Helmholtz wave equation in the complex domain. This greatly reduces the complexities of the problems. For illustrations, the cases of flows between parallel plates and in a rectangular channel are treated. It shows that this method is more direct and powerful than those of previous investigations.

THE DISTRIBUTION OF KINETIC ENERGY IN CERTAIN STEADY BAROTROPIC CURRENTS

Abstract The spreading of an energy concentration superimposed on a uniform zonal current at a particular latitude, may be characterized by the complete disappearance of the perturbation at infinity. Under this condition the vorticity equation for a nondivergent barotropic atmosphere can be linearized exactly for steady-state motion and reduces to the well-known Helmholtz wave equation. Its solution then is quite general, and is not subjected to any restrictive condition of small perturbations. Two cases of “dispersion” were studied, the problem of the deflection of a zonal current and the problem of a zonal jet stream. In the first case the northward velocity component was prescribed as a boundary condition, in the second case the eastward velocity component. Due to the assumption of nondivergent flow, both components cannot be arbitrarily prescribed since they must satisfy the continuity requirement. Prescribing the distribution of either velocity component along a fixed meridian by a function with a pr...

Mechanical and engineering progress as influenced by the mining industry

This paper presents an analysis of electromagnetic field scattering from a composite consisting of N eccentric dielectric cylinders. The scattering analysis is based on the finite element method (FEM) using triangular elements for inside the composite cylinder and the boundary element method (BEM) for the unbounded region outside the composite cylinder. The fields outside the eccentric dielectric cylinders are represented in terms of the boundary integral equation. The solution of Helmholtz wave equation inside the eccentric dielectric cylinders (inhomogeneous region) is provided using the finite element method. By applying the boundary conditions between free space and the outermost layer of the eccentric dielectric cylinders, the scattering coefficients of the scattered fields are obtained. The method is very efficient since it maintains the sparsity of the resulting matrix equation. The radar cross section (RCS) of the various geometries is presented. The available numerical results in the literature are used to validate the accuracy of this method.