The problem of energy conservation in one-way wave equations

Abstract

A recent attempt to define and numerically solve range-dependent benchmark problems in ocean acoustics [J. Acoust. Soc. Am. Suppl. 1 81, S39–S40 (1987)] revealed some inherent problems with energy conservation for one-way wave equations when applied to sloping bottom environments. The problem turned out to be associated with the use of a stair-step approximation to a sloping interface, and, in particular, with the rise of the stair steps, where a one-way equation allows only one out of two necessary interface conditions to be fulfiled (continuity of pressure or horizontal particle velocity). The loss of energy increases with increasing slope and density contrast at the water/bottom interface, and it can become quite significant (3–6 dB) for slopes of a few degrees and a density contrast of 2. A simple fix to this problem is to solve for a density-reduced pressure (p/ρ) instead of pressure (p), a trivial modification to existing one-way parabolic equation codes. Numerical results for both up- and downslope propagation show that the modified wave equation performs very well with errors being reduced by 75% for realistic slopes and density contrasts.