We derive a model equation for nonlinear medical ultrasound.
Unlike the existing models, which use spatial coordinates,
we use material coordinates and are hence able to derive a model
for a heterogeneous medium.
The equation is a generalization of the Westervelt equation,
and includes the nonlinearity, relaxation, and heterogeneity of soft tissue.
We discuss the validity of the generalized Westervelt equation as a
model equation for a Piola-Kirchoff acoustic pressure and
as an equation for the acoustic pressure.
In the second case it turns out that the model follows from
two geometric approximations which are valid
when the radius of curvature of the phase fronts are
much larger than the particle displacements.
The model is exact for plane waves and includes arbitrary
nonlinearity in the stress-strain relation.
In an alternative approach
we derive an exact nonlinear wave equation
formulated in the conventional spatial description.
The equation is a generalization of the well known nonlinear wave equation
for an ideal gas.
The equation model arbitrary nonlinearity and can be modified to include
relaxation effects to linear order,
but the modeling of heterogeneous nonlinear media such as soft tissue
turns out to be awkward in the spatial description.
We demonstrate how arbitrary order equations follows from the above and
a power series representation for the equation of state,
for instance as given by the virial equation of state.
We comments on these results in relation to two conflicting derivations of a
third order
KZK equation.
Helge Holden <holden@math.ntnu.no>
2000-09-13 10:06