Shear wave excitation of an equilateral triangular solid bar shows the existence of horizontally polarized (SH) and vertically polarized (SV) shear waves. The SH wave modes are Lamé solutions of the Neumann problem for displacement and Dirichlet problem for stress in the equilateral triangle. It is shown that Lamé's solutions are obtained from superposition of three equal plane waves and their reflections. The Neumann SH modes have cutoff wavenumber 4mπ/(3a), where m=1,2,3,…, a being the side length of triangular cross section. The first Dirichlet SH mode has a higher cutoff wavenumber 4π7/(3a). The SV modes are related to the 30°-60°-90° and 30°-120°-30° sub triangles. Their cutoff wavenumbers lie between those of the first (m = 1) and second (m = 2) Neumann SH modes. The standing wave condition in each sub triangle is related to three unequal plane waves which decompose via symmetrical component analysis into two different sets of equal plane waves. A nonlinear product solution based on the new plane wave set leads to modal SV wave solutions. Dispersion curves calculated using COMSOL confirm the correct cutoff wavenumbers of both SH and SV waves. Normal displacement mode shapes are calculated analytically and verified experimentally using laser vibrometer with good agreement for all modes.

The filtering problem of a periodically corrugated rectangular metallic waveguide of length l is used as an example to set an upper bound on the validity of the various orders of asymptotic approximation of the electromagnetic field in the waveguide. We limit the analysis to the first two orders of asymptotic expansion in powers of a small parameter δ characterising the amplitude of the periodic corrugations to minimise the detail, but the procedure is applicable to higher orders of approximation without loss of generality. The asymptotic expansions are obtained via the perturbation method of multiple scales. Following conventional practice in antenna work, a phase error of π/8 is used to set an upper bound on the phase error for each order of approximation. This bound is found by setting the difference between the results of simulation using the HFSS software package and the phase of either the first-order or the second-order approximation of the electromagnetic field equal to π/8. This bound sets an upper limit on δ for a given length I which gives the same results for a fixed (δ × I) product for the first-order expansion.

A plane wave reconstruction technique is presented for the solution of Helmholtz’s equation governing wave propagation in equilateral triangular electromagnetic waveguides in order to aid in the classification of symmetric and antisymmetric modes as well as to resolve the problem of excitation of triangular patch antennas. [...]

Symmetric solutions of the problem of wave propagation in equilateral triangular waveguides are obtained by using a combination of three plane waves. The plane wave configuration is either a delta (Δ), with waves traveling parallel to the sides of the triangle, or a wye (Y) with waves traveling perpendicular to the sides. The Y configuration corresponds to the fields of a Dirichlet problem while the Δ configuration corresponds to the fields of a Neumann problem. Longitudinal acoustic waves are examples of the Y configuration, while shear acoustic waves belong to the Δ configuration. Each of the triangle's surfaces is excited by two waves whose combination yields a partial field expressed in terms of the orthogonal coordinates of the surface. This partial field is invariant with respect to coordinate transformations through 120° rotations in view of the symmetry of the equilateral triangle. A linear combination of the three partial fields yields the eigenfunctions characterized by a single eigenvalue. In view of the boundary conditions for either compressional or SV shear acoustic waves, the two waves combine to produce a composite field at frequencies where their wavenumbers become equal. The boundary conditions on a stress free surface dictate a phase matching condition on the transverse wavenumbers wherein the SV transverse wavenumber is √3 times the transverse wavenumber of the longitudinal wave. An experiment with a longitudinal wave excitation confirmed the occurrence of the combined SV and longitudinal modes at frequencies where their longitudinal wavenumbers are equal.

Coupled-mode equations governing the amplitudes of the higher-order symmetric Lamb modes S-1 and S-2 with the antisymmetric mode A(2) in an infinite elastic plate with sinusoidal surface corrugation over a finite length are obtained via multiple-scales analysis. This phenomenon of three-mode coupling is observed when the wavenumbers k(s1) and k(s2) of the symmetric modes and k(A2) of the antisymmetric mode satisfy the simultaneous resonance conditions k(s1) - k(A2) = k(w) and k(A2) - k(s2) = k(w), where k(w) is the wavenumber of the sinusoidal corrugation. Near resonance, the coupled amplitude equations are solved exactly as an initial-value problem and it is seen that the modes are transmitted through the grating without reflection. Complete conversion from the symmetric modes into the antisymmetric mode is observed at periodic intervals along the grating when the resonance conditions are exactly satisfied. The effect of detuning away from resonance also shows propagation without reflection with periodic energy exchange. In the latter case, the modes couple without complete conversion. This phenomenon of mode conversion is confirmed by the results of an experiment on an aluminum plate with a triangular grating excited with the S-2 symmetric mode at 2.7 MHz.

In this paper, guided ultrasonic wave propagation is analyzed in an elastic plate with sinusoidal surface corrugations. The corrugated area acts as a finite-length grating which corresponds to a 1-D phononic crystal (PC). The multiple-scale perturbation technique is used to derive coupled-mode equations describing the amplitudes of interacting modes. These equations are solved exactly for the two-point boundary-value problem of the PC. The study involves the coupling of the incident symmetric Lamb wave S(0) to the reflected antisymmetric Lamb wave A(0). The influences of the depth of corrugation and length of the PC are studied. Theoretical results are compared with experimental measurements.

In this paper, a second-order multiple scales expansion is used to derive coupled-mode equations for a linearly chirped Bragg grating. This eliminates the error in the spectral response introduced by large values of the grating strength when conventional first-order coupled-mode theory is used. The autonomous and nonautonomous formulations of these equations are considered and compared in terms of accuracy and speed of the numerical solution of the resulting two-point boundary-value problem for the reflectance of the grating. These solutions are compared with the characteristic matrix solution taken as a reference. By using the fundamental matrix method, the autonomous formulation is found to be as accurate as the characteristic matrix method but faster in terms of computer CPU time.

The effect of spatial harmonics of a square-wave surface grating on the multiband frequency response of ultrasonic SH waves in an elastic plate is investigated. Stopbands are found to occur under simultaneous occurrence of first-order and higher-order Bragg resonances together with co-directional and/or contra-directional mode coupling conditions. The odd harmonic nature of the structure causes stopbands to appear above the cutoff frequencies of the odd-numbered modes. The attenuation within stopbands is found to be greatest when all propagating modes are coupled and reflected. The simultaneous resonance conditions for higher order stopbands must contain at least one co-directional mode coupling condition. The analysis is performed according to the multiple scales scheme, leading to the derivation of coupled mode equations, which are solved numerically using the fundamental matrix method.

In this paper, we study the effect of harmonics of phase-shifted square-wave corrugations on the different stopbands of a corrugated parallel-plate waveguide supporting TM waves. The stopbands are characterized by simultaneous occurrence of first-order and higher-order Bragg interactions. Coupled mode equations of two or four modes are derived via the method of multiple scales. Above cutoff of a higher-order mode, it is found that three resonance conditions must be satisfied: a first-order Bragg condition for the higher-order mode and a higher-order Bragg condition for the dominant TM10 mode, and a third condition coupling the two modes. A design of a multiple stopband filter for interference suppression in UWB applications is given showing all stopbands above cutoff of the TM10 and higher-order modes up to TM50 by including effects of higher harmonics of the structure. The bandwidth of each stopband is controllable by varying the phase shift between corrugations on the two walls of the waveguide. The fundamental matrix method is used to solve the filtering problem numerically.

A step-chirped step-apodized transmission grating is proposed to maximize compression ratio, minimize compensating length, and improve recompressed pulse shape of a 40-ps (FWHM) Gaussian pulse, broadened and chirped upon transmission in a 100 km length of optical fiber. This is accomplished by tuning a single transmission side lobe to lie within or coincide with the pulse bandwidth at the upper edge of the photonic band gap of each grating section. Analysis is via the theory of uniform cascaded gratings achieving a compression ratio of 5.625 in a four-section grating 3.422 cm long.

In this paper, the filtering problem of apodized rugates is solved by deriving first-order, as well as second-order, coupled-mode equations via the perturbation method of multiple scales. The first-order perturbation equations are the same as those of coupled-mode theory. However, the second-order perturbation expansion is more accurate, and permits the use of larger amplitudes of the periodic index variation of the rugate. The coupled-mode equations are solved numerically by using two different formulations. The first approach is a two-point boundary-value problem formulation, based on the fundamental matrix solution, that is essentially the exact solution for the unapodized rugate. The second approach is an initial-value problem formulation, that uses backward integration of the coupled-mode equations. Comparison with the characteristic matrix method is made for the case of unapodized rugate in terms of speed and accuracy, and it is found that the fundamental matrix solution is the fastest. The accuracy of the multiple scales solution is measured in terms of the amplitude error and the phase error of the filter's spectral response, taking the characteristic matrix solution as a reference for the unapodized rugate. The proposed formulations are utilized to calculate the spectral response of apodized rugates.

Analytical and numerical investigation of the TE/sub 10/ mode in a rectangular corrugated waveguide, whose sinusoidal corrugation reverses phase halfway along the waveguide, reveals the existence of a very narrow passband amid a wide stopband. An X-band experimental prototype confirmed the result.

The method of multiple scales is employed to analyze the interaction of SH modes in an elastic plate having periodically corrugated outerfaces. Two types of resonant conditions leading to two-mode as well as four-mode interactions are considered. The results of the analysis are utilized to develop ultrasonic mechanical wave filters operating on frequency bands centered at the resonant frequency. The stop-band filter frequency response is presented in terms of the power reflection coefficient. The characteristics of reflection of these filters are enhanced by imposing amplitude taper on the periodic corrugations.

A corrugated circular waveguide is proposed as a microwave filter. The analysis is carried out using the perturbation method of multiple scales for the case of TM modes. The analysis concerns the interaction of two and four modes satisfying the resonant condition (Bragg condition) imposed by the periodicity of the waveguide wall. The coupled mode equations derived via the method of multiple scales are used to formulate the filtering problem as a two-point boundary-value problem which is solved numerically. Desirable filtering characteristics may be realised by introducing tapered as well as chirped corrugations to control the frequency response of such a wave filter. In case of two-mode interaction the side ripples can be eliminated by tapering the waveguide wall. In the four-mode interaction case, a multichannel filter may be realised by combining taper and chirp.

Using a periodic corrugation at the surface of a semi-infinite elastic medium, the incident and reflected Rayleigh modes are strongly coupled when the Bragg condition is satisfied. The method of multiple scales is used to derive the coupled-mode equations describing such an interaction. The filter response is presented in terms of the reflection coefficient. An improved filter response may be obtained by imposing an amplitude taper on the corrugation, resulting in a close to ideal narrower midband response. By further introducing a chirped corrugation, an optimized wider midband response is realized.

Mode coupling of Love waves in an orthotropic thin film having periodically corrugated surfaces over an isotropic elastic half space is considered. Six modes are coupled by both surfaces by means of three simultaneous resonant conditions. On the basis of the weakness of the corrugations, the method of multiple scales is used to derive the coupled-mode equations. These equations together with relevant boundary conditions form a two-point boundary-value problem, which is solved numerically. The filter frequency response of a corrugated film designed as a stop-band filter is calculated. Enhanced filter characteristics are achieved when tapered corrugations are imposed. A narrow pass-band filter is also designed. Its high quality factor presents the fascinating features that might be realized by including the periodic corrugations in the design of SAW devices.< >

This study is concerned with the interaction of six torsional modes in a composite axisymmetric waveguide whose interfaces are sinusoidally corrugated in the axial direction. The modes are interacting when two resonant conditions on the codirectional modes and a Bragg condition occur simultaneously. In light of the weakness of the interface corrugations, the perturbation method of multiple scales is used to derive the mode coupling equations. A novel numerical scheme for two-point boundary-value problems is used to solve the coupled amplitude equations. The power reflection coefficient of a filter section is then calculated for the cases of uniform, tapered, and chirped corrugations. An optimal filter is realized by combining both taper and chirp thus producing a nearly ideal characteristics.

A simple and straightforward numerical steady-state solution of the strongly non-linear differential equations governing the differential-pair amplitude modulator is presented. It is shown that under normal biasing conditions and input signal levels exponential terms that give rise to coupling of the governing equations are very small in practice and may be dropped from the equations without influecing the circuit behaviour. The simplified equations lend themselves to a simple numerical solution by the brute force method. The results thus obtained are in excellent agreement with the harmonic balance results of Chua and Ushida (1981). The operation of the circuit in mixing an AM signal with an LO signal as well as the small signal modulator case are also treated. In the former case, four input frequencies are involved. In the latter, a multiple scales analysis is used to find the steady-state response analytically and it is found to be in excellent agreement with the numerical solution and other approximate methods of analysis.

A variable-step method is developed for the calculation of the two- and three-dimensional stability of compressible and incompressible two-dimensional boundary layers. The proposed method is compared with the computer code SUPORT and a finite-difference method. It is more efficient and requires less computer storage than both of these methods.