A broad enhancement of the form function f(ka) for steady-state backscattering by thin spherical shells in the midfrequency range has been noted by several investigators. Consequences of this enhancement on the backscattering of tone bursts are investigated with a Fourier synthesis of the temporal response from the exact f(ka). The emphasis is on incident bursts that are sufficiently short so that scattering consists primarily of distinct echoes associated with the specular reflection and different circumnavigations of a dominant surface guided wave. That wave lies on the subsonic branch of the lowest antisymmetric Lamb mode for a fluid-loaded shell designated as ao_ by some authors. A ray method, previously verified for echo amplitudes from leaky (or supersonic) Lamb waves on thick shells [S. Kargl and P. Marston, J. Acoust. Soc. Am. 85, 1014-1028 (1989) ], is generalized to subsonic waves. The ray method well approximates echo amplitudes from the Fourier synthesis provided the incident burst is sufficiently long that effects of dispersion are weak. Over a broad frequency range, the amplitude of the earliest ao_ echo is enhanced relative to the specular echo. For the 2.5% thick stainless steel shell in water considered, the enhancement factor peaks near ka • 46 where the amplitude ratio • 3.1. The ray theory suggests that the peak ratio depends only weakly on shell thickness and material parameters provided certain conditions hold. The ao_ wave is also found to contribute a prominent wave packet to the far-field impulse response of the shell. The radiation damping and velocity of relevant surface waves were computed from the complete elastic equations and those results may be helpful for testing thin shell approximations near the coincidence frequency.

Abstract-In this paper, the coupled mode theory is used to analyze apodized fiber Bragg gratings (FBGs). Since the profile of gratings varies with the propagation distance, the coupled mode equations (CMEs) of apodized FBGs are solved by the fourth-order Runge-Kutta method (RKM) and piecewise-uniform approach (PUA). We present two discretization techniques of PUA to analyze the apodization profile of gratings. A uniform profile FBG can be expressed as a system

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Abstract-The coupled mode theory (CMT) is used to analyze uniform Fiber Bragg gratings. The multi-mode CMT is expressed as the first-order vector ordinary differential equations (ODEs) with coefficients depending on the propagation distance. We show in this paper that by changing variables, the original couple mode equations (CMEs) can be re-casted as constant coefficient ODEs. The eigenvalue and eigenvector technique (EVVT), the analytic method for solving constant coefficient ODEs, is then applied to solve the coupled mode equations. Furthermore, we also investigate the application of RungeKutta method (RKM) to the calculation of the global transfer-function matrix for CMEs. We compare the transmission and the reflection

A simple ray approximation is demonstrated for the backscattering of sound by a thin empty cylindrical shell in water at normal incidence. The approximation is applied to the region near the coincidence frequency and is used to interpret prominent features in that region associated with a subsonic guided wave. With increasing ka, the features exhibit a transition from resonance behavior dominated by interference with the specular reflection. Another feature is the interference of resonance contributions of the leaky s0 Lamb wave which may produce maxima or minima in the amplitude depending on the ka region. The cylinder studied is 2.5% thick stainless steel for ease of comparison with a similar spherical shell [P. L. Marston and N. H. Sun, J. Acoust. Soc. Am. 92, 3315–3319 (1992)] and the guided wave properties were computed from elasticity theory rather than shell approximations. A novel geometric derivation is given for the relative phases of the coupling coefficients of subsonic waves on cylinders and spheres and some implications for scattering by slightly eccentric spheroids are noted.