Mathematical Analysis of Deep-Ocean Acoustical Effects of Receiver and Source Motions at Short and Intermediate Ranges.

A mathematical investigation is conducted to develop general results which describe the deep-ocean acoustical effects of receiver and source motions on a CW signal. Using ray theory, analyses are presented for constant-depth motions for both short and intermediate ranges, and the effects of depth-varying source motion are examined at short ranges. An analytical approach is used to determine general results for the effects of both receiver and source motions on a CW signal transmitted through a deep ocean channel at short ranges. A bilinear sound-speed profile is used. The receiver and source are restricted to the surface, and only SRBR rays are relevant. Time-dependent expressions for the total-field amplitude and phase are developed for appropriately limited time intervals, and numerical results are presented. General analytical expressions for the total field are derived and demonstrated to approximate closely numerical results. These expressions provide the basis for a study of the acoustical effects of varying motion parameters and initial range. It is demonstrated that effects of differences in range on total-field phase rate and the time interval between amplitude maxima are significant at short ranges and diminish as range increases. Effects on total field due to receiver motion are shown to be both significant and widely varying, depending on receiver and source directions and speeds. The effects of receiver and source motion are examined for a CW signal transmitted through a deep ocean at ranges of tens to hundreds of km. Ray theory is used to develop results for multipath signals consisting of a wide variety of combinations of SRBR and RSR ray arrivals. A bilinear sound-speed profile is assumed for which bottom and surface sound speeds need not be equal, and receiver and source are chosen to move on the surface. Numerical results are presented using time-dependent total-field expressions, valid for suitably limited time intervals. Analytical expressions are developed which closely approximate numerical results and which provide general conclusions regarding acoustical effects of receiver-source motion at different ranges. When only SRBR rays can occur, total fields are shown to have significantly different characteristics depending on range, in contrast to the virtually range-independent total fields which contain RSR rays. When total-field phase is interpreted in terms of an approximate Doppler shift, the frequency change shows relatively wide variations with both range and total-field composition. Thus, a given frequency shift at the receiver may be the result of considerably different receiver-source directions and speeds. Using ray theory the combined effects of time -dependent changes in source depth and receiver-source range are examined for a CW signal transmitted over relatively short ranges. Approximating deep-ocean sound speed with a bilinear profile, general results are obtained when the receiver is taken fixed on the surface, while the source moves on an arbitrary constant-velocity path above the SOFAR axis. Time-dependent expressions for the amplitude and phase of the received multipath signal are used to present numerical data for suitably restricted time intervals. Then, the effects of source path and speed are analyzed using convenient formulas which closely approximate numerical results. For strictly horizontal motion, total-field phase rate remains approximately proportional to time, source frequency and speed, and horizontal receiver-source range, but virtually independent of source depth. However, when source depth varies with time, overall linear phase patterns are interrupted by regularly spaced, brief changes in phase rate. The periodicity of these changes, and the accompanying amplitude fades of up to 35 dB, are virtually proportional to vertical speed, source frequency and range, but invariant with changes in horizontal speed and direction and initial source depth.