Rigorous Series Solution Research Articles

Response of a shallow asymmetric V-shaped canyon to antiplane elastic waves.

This study focuses on the theoretical aspect of topographic scattering induced by a shallow asymmetric V-shaped canyon under plane shear horizontal-wave incidence. An analytical approach, based on the region-matching technique, is applied to derive a rigorous series solution, which is more general than that in a previous study. For the wave functions constrained in two angular directions, a novel form of Graf's addition formula is derived to arbitrarily shift the local coordinate system. Barrier geometry, angle of incidence and wave frequency are taken as the most significant parameters in exploring the topographic effects of localized concave free surfaces on ground motions. Both surface and subsurface motions are presented. Comparisons with previously published results and boundary-element solutions show good agreement. Frequency-domain results indicate that, for the high-frequency case at a low grazing angle (corresponding to the potential case in teleseismic propagation), the high levels of amplified motions occur mostly on the illuminated side of the canyon. When the windward slope is steeper, the peak amplitude values, at least 2.4 times larger than those of free-field responses, tend to increase. Time-domain simulations display how a sequence of scattered waves travel and attenuate at regional distances.

Ground motions around a semi-circular valley partially filled with an inclined alluvial layer under SH-polarized excitation

A simplified mathematical model, composed of a semi-circular valley partially filled with an inclined alluvial layer under plane SH-wave incidence, is presented. To evaluate the site response theoretically, a rigorous series solution is derived via the region-matching technique. For angular wavefunctions constrained by an inclined free surface, the original form of Graf's addition formula is recast to arbitrarily shift the local coordinate system. The valley geometry, filling material, angle of incidence, and wave frequency are taken as significant parameters in exploring the site effect on ground motions. Also included are the frequency- and time-domain computations. Two canonical cases, the semi-circular vacant canyon and the fully filled semi-circular alluvial valley, with exact analytical solutions, and the partly horizontally filled case previously studied, are taken to be particular cases of the proposed general model. Steady-state results show that the peak amplitudes of motion may increase at low frequencies when the filling layer inclines to the illuminated region. At low-grazing incidence, the phenomenon of wave focusing becomes evident on the shadow side of the filling layer. Transient-state simulations elucidate how a sequence of surface waves travel on the topmost alluvium along opposite directions and interfere with multiple reflected waves within the filling layer.

Scattering and Focusing of SH Waves by a Lower Semielliptic Convex Topography

Abstract The SH -wave scattering induced by a lower semielliptic convex topography is studied here. A rigorous series solution is obtained via the region-matching technique. The method of separation of variables in elliptic coordinates is adopted to express the pertinent wave fields in terms of an infinite series containing products of radial and angular Mathieu functions with unknown coefficients. Steady-state responses for some parameters are calculated and discussed. Because the present surficial configuration may collect wave energy in much the same way as a concave mirror does with converging light waves, the potential focusing effects are further probed in the frequency domain. Numerical results demonstrate that the geometry under consideration is indeed capable of causing a localized high concentration of wave energy underground, which may have a great influence on the structures below the convex topography such as mountain tunnels.

TE-wave scattering from a shallow circular channel in a perfectly conducting plane

The problem of TE-wave scattering by a shallow circular channel in a perfectly conducting plane is considered and its rigorous series solution is also derived. By the region-matching technique, a semi-circular auxiliary boundary is introduced to divide the analyzed region into two subregions. The magnetic field of each subregion is represented in terms of an infinite series of wavefunctions with unknown coefficients. To include more practical applications, the wave source is assumed to be a Gaussian beam. The Kozaki's approximate expansion for Gaussian-beam incidence is adopted, which can convert into the plane-wave expansion with a special condition. With the enforcement of interface continuity conditions and the satisfaction of the channel surface boundary condition through the Graf's addition formula, the unknown expansion coefficients are determined. Comparisons for the semi-circular channel case with available data in the literature indicate excellent agreement. Pronounced effects of the depth-to-half-width ratio of the channel on backscattering width, far-field radiation pattern and equi-amplitude distribution of the magnetic field are illustrated by graphical results. The flexible and tunable features of the presented geometry make it easily applicable to the design and fabrication of multiple finite channels for optical devices.

Electromagnetic plane-wave scattering from a sectorial groove in a perfectly conducting plane

Abstract Scattering characteristics of plane waves by a sectorial groove in a perfectly conducting plane are investigated. Both the transverse magnetic (TM) and transverse electric (TE) polarizations of the incident wave are considered. Judicious use of the region-matching technique provides a rigorous series solution to the problem. The analyzed region is separated into two sub-regions by choosing a semi-circular auxiliary boundary. Thefield in each sub-region is expanded as a summationof proper wave functions with unknown coefficients. Enforcing the matching of conditions on the auxiliary boundary and of boundary condition on the circular-arc surface of the groove leads to a linear set of equations and the unknown coefficients are then determined. Numerical results demonstrate the influence of central angles of the sectorial groove on echo width, far-field pattern and near-field distribution. The presented geometry is easily applicable to the design and fabrication of a grating structure for optical switches and tunable filters.

Transverse electric scattering by a dielectric biconvex cylinder loaded with a shallow circular trough in a ground plane

The scattering problem of transverse electric wave from a dielectric biconvex cylinder buried in a shallow circular trough of a ground plane is investigated and a rigorous series solution is also derived. Based on the region-matching method, the analysed region is decomposed into two subregions by introducing a semi-circular auxiliary boundary. The magnetic field of each subregion is expressed in terms of cylindrical wave functions with unknown expansion coefficients. After imposing the matching conditions and the boundary condition on the trough surface with the aid of Graf's addition theorem, the unknown coefficients are determined. Comparisons with published data for a dielectric circular cylinder case show very good agreement. Visible effects of depth-to-half-width ratios of a dielectric biconvex cylinder on echo width, far-field pattern and near-field distribution are illustrated in graphical form.

A Rigorous Series Solution for Diamond Heat Spreaders with Temperature-Dependent Thermal Conductivity Used in Microwave Power Devices

In this paper we derive a rigorous series solution of the nonlinear heat transfer equation using the modified Kirchhoff transformation for a cylindrical diamond heat spreader with temperature-dependent thermal conductivity, situated on the top of a copper heat sink. We point out what we believe to be errors in two analytical approaches applied to the same problem which were published recently [Proc. R. Soc. Lond. A 441 (1993) 181; Proc. R. Soc. Lond. A 445 (1995) 375; and IEEE Trans. Microwave Theor. & Technol. 42 (1994) 573]. The basic difference between our work and these previous studies lies in the boundary condition (b.c.) of the transformed heat potential over the spreader-sink interface–we use nonlinear b.c. whereas the boundary conditions in the previous studies have been assumed to be linear. Calculations of change in the thermal resistance with the geometrical dimensions of the spreader, the input power, and the ambient temperature show up the profound influence which the assumed b.c. have on the problem. Our study has also revealed errors in numerical results presented in another paper [IEEE Semi-Therm Proc. (4th Annu. Semiconductor Thermal and Temperature Measurement Symp.), p. 113].

Three-Dimensional Analysis of a Thermal Dissipation System with a Rectangular Diamond Heat Spreader on a Semi-infinite Copper Heat Sink

We consider the 3-D problem of conductive heat dissipation from a rectangular patch source of constant heat flux, through a finite rectangular parallelepiped heat spreader into a semi-infinite heat sink. Using the Fourier analysis, we have developed a rigorous series solution to this 3-D boundary value problem. Considering the case of a diamond heat spreader with a copper heat sink, we have investigated the effects of the geometrical dimensions of the spreader and the shape of the rectangular heat patch source on the thermal behavior of the system. The main results we have obtained are: (1) the commonly-used assumption of isothermal spreader-sink interface is valid provided that the spreader thickness is relatively large compared with the heating-patch's length and width; (2) for a square heat spreader with a fixed area ratio of rectangular heat patch to the spreader's cross section, the maximum temperature occurs for square heat patches as a result of the minimum perimeter; (3) for the same thickness and cross-section area of the spreaders, circular systems can be used to predict an upper bound of thermal behaviors for the rectangular heat dissipation systems.

A Rigorous Series Solution for a Thermal Dissipation System with a Diamond Heat Spreader on an Infinite Slab Heat Sink

We consider the problem of conductive heat dissipation from an infinite stripe of constant heat flux, through an infinitely-long rectangular rod diamond heat spreader and an infinite slab heat sink. Using the Fourier cosine series and the Fourier transform, we have developed series solutions for both the temperature distributions in the system and the average temperature over the heating stripe. The series coefficients are determined through an infinite set of linear algebraic equations. Considering the case of diamond heat spreader and copper heat sink, we have investigated the effects of the geometrical dimensions and the thermal conductivity of the spreader on the thermal behavior of the system for a given sink thickness. The main results are: (1) there exists a spreader thickness for achieving the minimum temperature at the top spreader surface; (2) the commonly-used assumption of isothermal spreader-sink interface is valid for a relatively thicker spreader compared to the heating-stripe width; (3) increasing the ratio of the heat spreader width to the heating-stripe width beyond 30 does not reduce the average temperature significantly, indicating the existence of an effective spreader width; (4) for lowering the surface temperature, the spreader width is the most effective and practical design parameter over the thickness and the thermal conductivity.

Sound Scattering by Fluid Cylinders

The scattering of plane waves of sound from an infinitely long circular cylinder has been investigated both theoretically and experimentally. Wave motion in the cylinder cannot be neglected because the properties of the fluid in the cylinder are comparable to the properties of the external fluid. A rigorous series solution for the scattered radiation has been obtained, taking into account the possibility of compressibility, density, and attenuation differences between the two fluids. At high frequencies the series solution is difficult to apply because of the many terms that must be included. Consequently, approximate closed-form solutions have been developed which are useful for high frequencies and which agree essentially with the rigorous series solution in the limit of low frequencies. The approximate solutions follow from an integral equation formulation which has application in other coordinate systems and which is similar to the results of some recent work of Montroll and Hart. Measurements have been performed on cylinders containing mixtures of tertiary butyl alcohol and water immersed in water for wavelengths approximately 18 the diameter. The approximate theory predicts the angular positions of the minima of the scattered radiation within experimental error and the magnitude of the maxima of the scattered radiation within a few decibels.