The Interaction of Shocks with Dispersive Waves II. Incompressible-Integrable Limit

This is the second in a two-part series of articles in which we analyze a system similar in structure to the well-known Zakharov equations from weak plasma turbulence theory, but with a nonlinear conservation equation allowing finite time shock formation. In this article we analyze the incompressible limit in which the shock speed is large compared to the underlying group velocity of the dispersive wave (a situation typically encountered in applications). After presenting some exact solutions of the full system, a multiscale perturbation method is used to resolve several basic wave interactions. The analysis breaks down into two categories: the nonlinear limit and the linear limit, corresponding to the form of the equations when the group velocity to shock speed ratio, denoted by ε, is zero. The former case is an integrable limit in which the model reduces to the cubic nonlinear Schrödinger equation governing the dispersive wave envelope. We focus on the interaction of a “fast” shock wave and a single hump soliton. In the latter case, the ε=0 problem reduces to the linear Schrödinger equation, and the focus is on a fast shock interacting with a dispersive wave whose amplitude is cusped and exponentially decaying. To motivate the time scales and structure of the shock-dispersive wave interactions at lowest orders, we first analyze a simpler system of ordinary differential equations structurally similar to the original system. Then we return to the fully coupled partial differential equations and develop a multiscale asymptotic method to derive the effective leading-order shock equations and the leading-order modulation equations governing the phase and amplitude of the dispersive wave envelope. The leading-order interaction equations admit a fairly complete analysis based on characteristic methods. Conditions are derived in which: (a) the shock passes through the soliton, (b) the shock is completely blocked by the soliton, or (c) the shock reverses direction. In the linear limit, a phenomenon is described in which the dispersive wave induces the formation of a second, transient shock front in the rapidly moving hyperbolic wave. In all cases, we can characterize the long-time dynamics of the shock. The influence of the shock on the dispersive wave is manifested, to leading order, in the generalized frequency of the dispersive wave: the fast-time part of the frequency is the shock wave itself. Hence, the frequency undergoes a sudden jump across the shock layer.In the last section, a sequence of numerical experiments depicting some of the interesting interactions predicted by the analysis is performed on the leading-order shock equations.     

Single Ping-multiple measurements: sonar bearing angle estimation using spatiotemporal frequency filters

Presented is a mixed-signal full-custom VLSI chip designed to receive sonar return signals from an ultrasonic microphone array, and extract input bearing angles of the incoming signals. Processing utilizes simple low-power analog spatiotemporal bandpass filters to extract wavefront velocity across the array, which translates to input bearing angle. Processing uses phase information of array signals, not onset or offset of ultrasonic burst. With such synchronous processing, multiple angle readings from different returns of the same ultrasonic transmit burst are possible. Compatible microphone arrays are compact in size-test array has a total baseline of 26.5 mm. In a test with ultrasonic beacon 65 cm from a microphone array, angular precision of 1/spl deg/ was demonstrated in most instances in the range -60/spl deg/ to 60/spl deg/. Applications include sonar localization of remote objects, sonar imaging, and improved interference rejection between objects within the field of view of the sensor microphones. The chip was fabricated on a standard 3M2P CMOS process with a 0.5-/spl mu/m feature size.     

A biomorphic digital image sensor

An arbitrated address-event imager has been designed and fabricated in a 0.6-/spl mu/m CMOS process. The imager is composed of 80 /spl times/ 60 pixels of 32 /spl times/ 30 /spl mu/m. The value of the light intensity collected by each photosensitive element is inversely proportional to the pixel's interspike time interval. The readout of each spike is initiated by the individual pixel; therefore, the available output bandwidth is allocated according to pixel output demand. This encoding of light intensities favors brighter pixels, equalizes the number of integrated photons across light intensity, and minimizes power consumption. Tests conducted on the imager showed a large output dynamic range of 180 dB (under bright local illumination) for an individual pixel. The array, on the other hand, produced a dynamic range of 120 dB (under uniform bright illumination and when no lower bound was placed on the update rate per pixel). The dynamic range is 48.9 dB value at 30-pixel updates/s. Power consumption is 3.4 mW in uniform indoor light and a mean event rate of 200 kHz, which updates each pixel 41.6 times per second. The imager is capable of updating each pixel 8.3K times per second (under bright local illumination).     

Echo planar imaging of perfluorocarbons

Emulsions of perfluorotributylamine (FTBA) and perflubron were evaluated for their utility in ¹⁹F echo planar imaging. Fluorine images of the emulsions were obtained in a phantom and two mice that had been predosed. Both agents, but particularly perflubron, show potential for fluorine echo planar studies because of the long spin-spin relaxation times of the CF₃ resonances. High resolution thin slice images obtained in as little as 26.6 ms are presented.     

Book Reviews

Unsigned book reviews are by the Book Review Editor.     

Unified Interactive Multiple Objective Programming: An Open Architecture For Accommodating New Procedures

It has previously been shown that nine of the most prominent procedures of interactive multiple objective programming can be considered as special cases of a common procedure called the unified algorithm. In this paper, we show how four additional procedures can be similarly included in the unified approach. Also, the paper presents a procedure-switching matrix that depicts the possibilities for switching from one procedure to another, one iteration to the next, in the unified algorithm if so directed by a user.