Hidden depths: Acceptable ignorance about ocean bottoms

Normal-mode analysis of underwater sound propagation in principle requires knowledge of pertinent physical parameters at all depths in the water and the bottom material—an unattainable omniscience. We present a method for determining the maximum depth to which this knowledge is necessary in order to hold the fractional errors in mode eigenvalues to prescribed limits. Let hn represent a vertical distance below the lower turning point of the nth-mode solution. Insertion or removal of a horizontal plane reflector, at this depth, alters the mode eigenfunction and therefore the eigenvalue En. The fractional error ΔEn/En is a calculable function of hn; this error being stipulated, hn can be found. The calculation need be made only for the highest mode that contributes significantly. Conversely, if all parameters are known to depth h, the consequent errors can be found. Two examples are analyzed, with simplifying restrictions: deep isovelocity water, low frequencies, many modes, bottoms that are isovelocity (the Pekeris case) or have a positive gradient of sound speed. For fractional errors of 10−4 to 10−6, h is a few acoustic wavelengths. In each example, bottom absorption has little effect on the result. Subject Classification: [43]30.20, [43]30.50, [43]30.25.