  An ideal contrast-enhanced ultrasound image should display
microbubble-induced nonlinearities while avoiding wave propagation
nonlinearities. One of the most successful ultrasound pulse sequences to
disentangle these nonlinear effects relies on the transmission of
cross-propagating plane waves. However, theory describing the noncollinear
nonlinear interaction of two finite plane waves has not been fully developed
and a better understanding of these effects would improve contrast-enhanced
ultrasound imaging further. Here, local nonlinear interactions at the
intersection of two plane-waves are investigated by extending the Westervelt
equation with a term including the Lagrangian density. The Iterative Nonlinear
Contrast Source (INCS) method is employed to numerically solve this full
nonlinear wave equation for two 3D finite cross-propagating pulsed plane waves.
In addition, analytical expressions for the cross-propagation of two infinite
continuous plane waves are derived. Numerical results obtained with INCS show
good agreement with the analytical expressions. Overall, the generated results
show that the pressure associated with local nonlinear effects is two orders of
magnitude lower than the pressure associated with global nonlinear effects.
Local nonlinear effects are therefore expected to be negligible in the context
of single-shot ultrasound imaging, but they may influence approaches that
subtract pressure fields such as amplitude modulation or pulse inversion.
