Systems of interacting harmonic oscillators have recently received considerable attention as models for describing a variety of physical problems. %'e have investigated the validity of the rotating-wave approximation which constitutes the traditional approach to the solution of the dynamical problem by comparing it with the exact solution. A numerical comparison has been made and the limits of validity of the rotating-wave approximation has been established in terms of the strength of oscillator interaction.In particular, the time development of dynamical operators and certain transition probabilities have been compared. In the region where the rotating-wave approximation is valid, the time evolution of the quasiprobability distribution P(e, t) of one oscillator is given for several initial conditions. A counting scheme similar to the argument given by Feynman for the driven harmonic oscillator, is proposed for the interpretation of the time-dependent transition &.mplitudes between number states.

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Generating underwater acoustic signals from a remote, aerial location by use of a high-energy pulsed infrared laser has been demonstrated. The laser beam is directed from the air and focused onto the water surface, where the optical energy was converted into a propagating acoustic wave. Sound pressure levels of 185 dB re microPa (decibel re microPa) were consistently recorded under freshwater laboratory conditions at laser-pulse repetition rates of up to 1000 pulses/s. The nonlinear optoacoustic transmission concept is outlined, and the experimental results from investigation of the time-domain and frequency-domain characteristics of the generated underwater sound are provided. A high repetition rate, high-energy per pulse laser was used in this test under freshwater laboratory conditions. A means of deterministically controlling the spectrum of the underwater acoustic signal was investigated and demonstrated by varying the laser-pulse repetition rate.

A numerical technique for the propagation of sound in the ocean where velocity is a function of both range and depth has been developed. The technique is not restricted to the narrow-angle (parabolic) approximation and it reduces to an exact solution for a homogeneous media. The given field is transformed into a sum of plane waves via an FFT. This transformed field is propagated without approximation through a homogeneous space represented by an average wave number for that space. Updating the average wave number provides for Snell’s law bending in range. The variations from the average velocity are accounted for by summing deviations from the nominal phase to develop a group of direction-sensitive phase correction masks. A weighted group of plane waves centered about each direction are inverse transformed for multiplication by the phase mask with the results summed. The choice of overlapping weights in the transformed space provides an approximate continuous phase correction mask for all directions. The functional relationship of range-step size to frequency, angular spectrum, and velocity profiles are developed. Several results for a 1500-m channel are shown.

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