### ASA 125th Meeting Ottawa 1993 May

## 4pPA12. Burgers' equation generalized for absorption obeying a power law.

**Thomas L. Szabo
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*Imaging Systems, Hewlett Packard, 3000 Minuteman Rd., Andover, MA 01810
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Burgers' equation applies to finite-amplitude waves propagating in a
medium with absorption that has a quadratic frequency dependence. Numerical
solutions of a modified Burgers' equation have been obtained in the frequency
domain for other types of losses; however, a complete set of time domain
nonlinear equations corresponding to power law attenuation has not been
available. Power law attenuation is defined by the equation,
(alpha)((omega))=(alpha)[sub 0]|(omega)|[sup y], where (alpha)[sub 0] and y are
arbitrary real constants, and (omega) is angular frequency. Blackstock [J.
Acoust. Soc. Am. 77, 2050--2053 (1985)] has suggested that Burgers' equation
could be generalized if an appropriate operator L could be found such that the
equation could become p[sub z]-L*p=Bpp[sub (tau)], where p is pressure, B is a
constant, (tau) is delayed time, and the subscripts denote derivative
operations. The L operators have been derived based on a new causality
principle and parabolic wave equations for power law loss. The resulting time
domain equations extend Burgers' approach to finite amplitude propagation in
media with arbitrary power law absorption.