Non-Diffusive Westervelt Equation

• The non-diffusive Westervelt equation is a nonlinear model that captures cumulative acoustic wave propagation in media where dissipative effects are absent.
• It omits local diffusive nonlinearities to emphasize waveform steepening, derived rigorously from fluid dynamics and asymptotic analysis.
• Its formulation underpins practical applications such as ultrasound imaging, high-intensity focused ultrasound therapy, and atmospheric infrasound modeling.

The non-diffusive Westervelt equation is a nonlinear evolution equation that models the propagation of finite-amplitude acoustic waves in media where dissipative effects are minimal or completely neglected. It succinctly encapsulates the essential cumulative nonlinearities of wave propagation, omits local diffusive nonlinearities and viscous terms, and thereby represents a hyperbolic, nondissipative regime central to modern theoretical and applied nonlinear acoustics. Its derivation, mathematical properties, and practical applications extend across ultrasound modeling, high-intensity focused ultrasound (HIFU), medical imaging, nonlinear acoustic tomography, and atmospheric infrasound propagation.

1. Mathematical Formulation and Non-Diffusive Limit

The canonical non-diffusive Westervelt equation (typically in pressure, $p$ or velocity potential, $\Pi$) is given by: $\Box_c p - \beta \partial_t^2 (p^2) = 0$ or, more generally,

$$(1 - 2\beta p) p_{tt} - c^2 \Delta p = 2\beta (p_t)^2$$

where $c$ is the sound speed and $\beta$ is the nonlinear acoustic parameter quantifying the strength of the quadratic nonlinearity.

The “non-diffusive” aspect refers to the absence of dissipative terms such as

$- \delta \Delta p_t$

which represent viscous and thermal losses (diffusive damping with infinite propagation speed). As shown by multiple asymptotic limit and modeling studies (Dekkers et al., 2020, Kaltenbacher et al., 2021), the non-diffusive form arises naturally in inviscid media as the dissipation coefficient $\delta$ and, in the case of relaxation phenomena, the thermal relaxation time $\tau$ tend to zero. Analytically, the difference between the full dissipative model and the non-diffusive model is controlled linearly in $\delta$: $\|p^{\delta} - p\|_{energy} \lesssim \delta,$ for sufficiently smooth data (Kaltenbacher et al., 2021).

2. Modeling Origins: From Fluid Dynamics and Thermoelasticity

Derivations of the non-diffusive Westervelt equation begin with the Navier-Stokes or Euler equations, where acoustic waves are assumed to be small perturbations to a quiescent background. A rigorous expansion in the small parameter $\varepsilon$ (related to Mach number or amplitude) leads to the Kuznetsov equation. A crucial step is a nonlinear change of variables,

$$\Pi = u + \frac{\varepsilon}{2c^2} \partial_t (u^2)$$

which transforms Kuznetsov’s equation to a form where only the cumulative (progressive) nonlinear term remains: $$\partial_{tt} \Pi - c^2 \Delta \Pi = \varepsilon \partial_{tt} (\Pi^2)$$ Further negligible terms ($\mathcal{O}(\varepsilon^2)$) are omitted, rigorously justifying the approximation in the weakly nonlinear regime (Dekkers et al., 2020). This omits local quadratic nonlinearities (which could be diffusive) and retains only the cumulative nonlinearity responsible for waveform steepening, signature in high-intensity applications.

Recent generalizations incorporate deformation theory to extend the non-diffusive Westervelt equation to elastic and tissue media: $$\partial_{tt}p - c_l^2 \partial_{xx}p = \left(\frac{\beta}{\rho c_l^2}\right) \partial_{tt}(p^2)$$ with higher-order harmonics encoded via inverse series relating strain to pressure, linking fundamental material nonlinearity to acoustic propagation (Caruso et al., 17 Mar 2025).

3. Well-Posedness, Regularity, and Boundary Effects

Weak well-posedness and regularity results in arbitrary, fractal, or highly irregular domains have been established for non-diffusive Westervelt-type equations (Dekkers et al., 2020). The framework employs variational formulations in Sobolev spaces $H^1_0(\Omega)$, accommodating both homogeneous and nonhomogeneous Dirichlet boundary conditions via trace and extension operators, especially in $d$-set boundary geometry. Mosco convergence of associated functionals ensures that weak solutions on a sequence of smoother domains converge to the limiting solution on the irregular domain.

Regularity plays a critical role in ensuring well-posedness and stable numerical integration. For a non-degenerate and globally well-posed evolution, initial data must be sufficiently smooth and small to avoid vanishing denominators (e.g., $1-2\beta p$ bounded away from zero) (Benabbas et al., 2023), and energy estimates in appropriate Sobolev spaces certify uniform bounds and solution existence.

4. Analysis of Nonlinear Effects and Geometric Optics

In the weakly nonlinear geometric optics regime, the non-diffusive Westervelt equation admits a leading profile equation of Burgers’ type: $$(\partial_t + \omega \cdot \nabla_x) U_0 + \alpha U_0 \partial_\theta U_0 = 0$$ where $U_0$ is the oscillatory amplitude, and $\alpha$ the spatially varying nonlinearity (Eptaminitakis et al., 2022). This formulation describes the nonlinear tilting and phase modulation of high-frequency wave packets and underpins methods to reconstruct material nonlinearity from wavefront measurements. Phase shifts (tilts) are linearly related to the X-ray (Radon) transform of $\alpha$, making it feasible to recover spatial maps of nonlinear coefficients via probing wave experiments.

5. Inverse Problems and Parameter Reconstruction

Stable recovery of the nonlinear parameter $\beta$ (or similar) and the sound speed $c(x)$ from boundary data is attainable by analysis of the Dirichlet-to-Neumann (DN) map associated with the non-diffusive Westervelt equation: $$\Lambda: f \mapsto \partial_\nu u|_{\partial \Omega}$$ A second-order linearization approach expands the solution in amplitude: $u = v + \varepsilon^2 w + R$ with $v$ solving the linearized problem, and $w$ encoding the quadratic nonlinearity. An integrity identity (Alessandrini-type) connects variations in the DN map to weighted geodesic ray transforms, which are stably invertible under foliation conditions on the acoustic metric and small perturbation of the sound speed from a reference $c_0$ (Wendels, 2 Oct 2025). Numerical experiments confirm Hӧlder stability of the parameter recovery.

6. Numerical Discretization and Computational Techniques

Operator splitting methods (Lie-Trotter, Strang), mimetic finite difference schemes, and asymptotic-preserving finite element discretizations provide efficient time integration and numerical treatment:

• Splitting the nonlinear operator into hyperbolic (“wave”) and nonlinear (“reaction/diffusion”) parts yields tractable subproblems, with stability ensured by rigorous regularity theorems (Kaltenbacher et al., 2013).
• Mimetic discretizations preserve Hamiltonian structure and exact differential-geometric invariants (e.g., vorticity), employing discrete de Rham complexes and cochain spaces (Barham et al., 20 Jul 2024).
• Asymptotic-preserving FE schemes control the singular vanishing dissipation limit ($\epsilon\to 0$), yielding uniform energy and error estimates (Nikolić, 2023).

Uniform-in-parameter energy control is achieved by two-level testing strategies and suitable choices of memory kernel (for fractional damping), allowing robust modeling even for vanishingly small dissipative effects.

7. Applications, Extensions, and Future Directions

Non-diffusive Westervelt equations underpin:

• High-intensity ultrasound simulation, where nonlinear cumulative effects drive shock formation and waveform steepening (Kaltenbacher et al., 2013).
• Ultrasound-assisted drug delivery and nonlinear thermally coupled models, requiring careful analysis of time-fractional and spatially variable coefficients (Careaga et al., 10 Dec 2024).
• Infrasound modeling through planetary atmospheres, where the equation describes nonlinear propagation under arbitrary Mach number winds, enabling efficient alternatives to direct numerical simulation and improved source reconstruction for explosion monitoring and remote sensing (Tope et al., 24 Jan 2024).
• Medical imaging and tomography, via stable recovery of sound speed and nonlinear tissue parameters from boundary measurements (Wendels, 2 Oct 2025).

Extensions to deformation-theory-based models facilitate propagation in solids and tissues, accommodating higher-order nonlinearities and complex boundary geometries (Caruso et al., 17 Mar 2025).

Continuing research investigates modeling in weakly stratified, turbulent, or fractal domains, nonlinear parameter inversion under relaxed geometric conditions, efficient numerical methods for singular limits and memory-driven damping, and the refinement of elastic nonlinear models for application in therapy and quantitative imaging.

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The non-diffusive Westervelt equation thus serves as a foundational framework for nonlinear acoustic wave propagation across diverse scientific and engineering contexts, combining deep mathematical structure, efficient computational techniques, and robust modeling of cumulative nonlinearities in media where dissipation is minimal or neglected.