Weakly nonlinear ray tracing approximations for focused ultrasound propagation

Focused ultrasound is used in a therapeutic treatment (HIFU) and uses ultrasound waves to non-invasively destroy malignant cells inside the human body. The technique works by sending a high-energy beam of ultrasound into the tissue using a focused transducer. Numerically modeling HIFU presents a problem due to nonlinear effects leading to the formation of harmonics of the source frequency. Each harmonic requires a finer grid to resolve, rapidly increasing computational complexity. This work will focus on the derivation and benefits of two ray tracing methods using weakly nonlinear ray theory formed by different asymptotic expansions of the governing acoustic equations. The first has the ray equations identical to those from linear ray theory while the amplitude equation is a nonlinear transformation of the Burgers’ equation. In the second method the Eikonal and transport equations are coupled which results in ray trajectories which depend on the amplitude.