What Is Nonlinear Acoustics?

Nonlinear acoustics is a branch of acoustics concerned with the propagation of sound waves at amplitudes large enough that the superposition principle no longer holds and the medium's response depends on the instantaneous state of the wave itself. At low amplitudes, the linear wave equation accurately describes how pressure disturbances travel through a fluid or solid. As amplitude increases, the dependence of local sound speed on instantaneous pressure, combined with convective nonlinearity from particle motion, causes the wave to distort progressively as it propagates. Nonlinear acoustics characterizes and exploits these finite-amplitude effects.

The discipline draws on fluid mechanics, thermodynamics, and classical wave theory, and is distinct from linear acoustics primarily in the governing equations: the nonlinear Euler and continuity equations replace their linearized counterparts, and the equation of state must be expanded beyond the first-order term. The field has applications from underwater sonar and medical ultrasound to industrial sonochemistry.

Acoustic Distortion and Harmonic Generation

The most direct consequence of nonlinear propagation is waveform distortion. A sinusoidal wave launched into a nonlinear medium gradually transfers energy from the fundamental frequency into its higher harmonics. The local sound speed at a pressure peak is slightly higher than at a trough, causing the peaks to advance relative to the troughs; the waveform steepens until the wave front becomes near-discontinuous. This process is quantified by the coefficient of nonlinearity B/A, which expresses the ratio of the second-order to first-order terms in the Taylor expansion of the equation of state. In water, B/A is approximately 5; in soft biological tissue it ranges from about 6 to 11, a range documented in the Introduction to Nonlinear Acoustics by Bjørnø in Physics Procedia. Second harmonic generation, the transfer of energy from the fundamental to twice its frequency, is routinely exploited in nonlinear imaging modalities such as tissue harmonic imaging in medical ultrasound, where the second harmonic component provides superior contrast and spatial resolution compared to the fundamental.

Shock Wave Formation

When nonlinear distortion is sufficiently strong, a finite-time singularity forms in the wave profile: the multivalued waveform is replaced by a discontinuity in pressure known as a shock front. The Goldberg number quantifies the relative importance of nonlinear distortion versus absorption; for Goldberg numbers well above unity, shock formation occurs before absorption dissipates the wave. In underwater acoustics and in therapeutic ultrasound, shock waves carry intense positive and negative pressure excursions that can fragment kidney stones by cavitation and mechanical stress, a technique called shock wave lithotripsy. In atmospheric acoustics, sonic booms from supersonic aircraft are a familiar manifestation of shock wave arrival at the ground.

The Burgers equation, derived from the nonlinear Euler equation with an absorption term, is the canonical one-dimensional model for shock formation and attenuation in lossy media. Extended to three dimensions and including diffraction, it yields the KZK (Khokhlov-Zabolotskaya-Kuznetsov) equation, which forms the basis for most quantitative nonlinear ultrasound field modeling.

Parametric Arrays and Acoustic Beaming

A distinctive application of nonlinear acoustics is the parametric array, in which two closely spaced high-frequency primary beams mix in a nonlinear medium to generate a highly directional difference-frequency beam at much lower frequency. Because the difference-frequency beam is generated continuously along the primary beam path, its effective aperture is very large, producing directional performance normally achievable only with physically large transducers. Parametric sonar systems exploit this to achieve narrow beamwidth in sub-bottom profiling and underwater communications.

Applications

Nonlinear acoustics has applications in a wide range of fields, including:

• Medical ultrasonics, including tissue harmonic imaging and high-intensity focused ultrasound (HIFU) tumor ablation
• Lithotripsy, for non-surgical fragmentation of kidney and gallbladder stones
• Underwater acoustics, for sub-bottom profiling using parametric sonar
• Sonochemistry, where acoustic cavitation drives chemical reactions in liquids
• Nondestructive evaluation, using harmonic generation to detect micro-damage in materials