---
title: Westervelt Equation Overview
url: https://www.emergentmind.com/topics/westervelt-equation
type: topic
---

# Westervelt Equation Overview

The Westervelt equation is a quasilinear model of finite-amplitude sound propagation in viscous or thermoviscous media. In a standard pressure formulation with acoustic pressure fluctuation \(u(t,x)\), sound speed \(c>0\), diffusivity \(b>0\), and nonlinearity parameter \(k\), it is written as
\[
u_{tt}-c^2\Delta u-b\,\Delta u_t=k\,(u^2)_{tt},
\]
or equivalently,
\[
(1-2ku)\,u_{tt}-c^2\Delta u-b\,\Delta u_t=2k\,(u_t)^2.
\]
This equivalent form exhibits the quasilinear dependence of the principal time-derivative coefficient on the acoustic amplitude, a feature that governs both analysis and computation. Across recent literature, the equation appears in pressure, velocity-potential, periodic, absorbing-boundary, thermoacoustic, and time-fractional variants, and it serves as a central model for nonlinear acoustics, ultrasound simulation, and parameter identification [1502.05816].

## 1. Classical formulations and model variants

In nonlinear acoustics, the pressure formulation is often expressed as
\[
p_{tt}-c^2\Delta p-b\,\Delta p_t=\partial_t^2\!\left(\frac{\beta}{\rho c^2}p^2\right),
\]
or, after setting \(k=\beta/(\rho c^2)\),
\[
u_{tt}-c^2\Delta u-b\,\Delta u_t=k\,(u^2)_{tt}.
\]
Using \((u^2)_{tt}=2u\,u_{tt}+2(u_t)^2\), one obtains the quasilinear form \((1-2ku)u_{tt}-c^2\Delta u-b\,\Delta u_t=2k(u_t)^2\), which makes the effective inertia explicit [2204.09630].

A complementary formulation uses the acoustic velocity potential \(\psi\). In dimensionless variables, one form studied in the literature is
\[
\partial_{tt}\psi-\Delta\psi=\alpha\,\Delta(\partial_t\psi)+\beta\,\partial_t(\partial_t\psi)^2,
\]
with acoustic velocity \(v=-\nabla\psi\) and pressure variation \(p=\partial_t\psi\). This formulation is particularly convenient for energy-preserving discretization, because it admits a canonical gradient-system structure and an associated continuous energy identity [1912.07037].

The equation is closely related to the Kuznetsov equation but is not identical to it. The Westervelt model arises as a reduced nonlinear acoustics equation under appropriate simplifications, including the neglect of convective terms and retention of strong damping through a \(-b\Delta u_t\) contribution [2204.09630]. In periodic settings with Robin boundary conditions, the equation is also written as
\[
p_{tt}-c^2\Delta p-b\Delta p_t=\eta(x)(p^2)_{tt}+h,
\]
with time-periodic conditions \(p(0)=p(T)\), \(p_t(0)=p_t(T)\), and impedance-type boundary terms \(\beta p_t+\gamma p+\nabla p\cdot n=0\), thereby linking nonlinear propagation to multiharmonic analysis [2407.17043].

Although the classical Westervelt equation is rooted in fluid acoustics, a recent derivation from deformation theory extends the model to nonlinear lossy solids and tissues. In that framework, the pressure equation takes the generalized form
\[
\partial_{tt}p-c_l^2\partial_{xx}p
=
-\sum_{n\ge 2}\frac{b_n}{b_1}\partial_{tt}(p^n)
-\sum_{n\ge 1}\nu b_n\,\partial_{ttt}(p^n),
\]
with coefficients determined by elastic moduli, geometric nonlinearity, and dissipative parameters. This places the classical Westervelt equation as a leading-order member of a broader hierarchy of nonlinear pressure-wave models [2503.14544].

## 2. Quasilinearity, damping, and energy structure

The defining structural feature of the Westervelt equation is the amplitude-dependent coefficient in front of \(u_{tt}\). In the pressure form
\[
(1-2ku)\,u_{tt}-c^2\Delta u-b\,\Delta u_t=2k(u_t)^2,
\]
or, in a normalized variant,
\[
(c^{-2}-2\gamma u)\,u_{tt}-\Delta u-\beta\,\Delta u_t=2\gamma (u_t)^2,
\]
the equation loses parabolic-type coercivity when \(1-2ku\) or \(c^{-2}-2\gamma u\) approaches zero. For this reason, smallness conditions that keep the coefficient strictly positive are not merely technical conveniences; they are structural conditions preventing degeneracy [1603.02097].

The strong damping term \(-b\Delta u_t\) or \(-\beta\Delta u_t\) gives the equation a mixed hyperbolic-parabolic character. This is the mechanism behind maximal-regularity results, temporal smoothing, and exponential decay in several analytical works. It also explains why the Westervelt equation behaves differently from nondiffusive weakly nonlinear wave models [1502.05816].

Energy identities make this structure transparent. In the velocity-potential formulation, one energy is
\[
\mathcal{E}(\psi,\partial_t\psi)
=
\int_\Omega \frac12 |\nabla\psi|^2
+
\left(\frac12-\frac{2\beta}{3}\partial_t\psi\right)|\partial_t\psi|^2\,dx,
\]
and under homogeneous Neumann boundary conditions it satisfies
\[
\frac{d}{dt}\mathcal{E}(\psi,\partial_t\psi)
=
-\alpha\int_\Omega |\nabla(\partial_t\psi)|^2\,dx.
\]
Thus, exact conservation occurs when \(\alpha=0\), while \(\alpha>0\) yields viscous dissipation [1912.07037].

For absorbing-boundary pressure formulations, a natural energy is
\[
\mathcal{E}(t)
=
\frac12\int_\Omega \Big((c^{-2}-2\gamma u)|u_t|^2+|\nabla u|^2\Big)\,dx,
\]
and the boundary law
\[
\partial_\nu(u+\beta u_t)+u_t\sqrt{c^{-2}-2\gamma u}=0
\]
produces boundary dissipation in addition to the interior term \(\beta\int_\Omega |\nabla u_t|^2\,dx\) [1603.02097].

In the nondiffusive regime, the weakly nonlinear geometric-optics limit is different. For
\[
\partial_t^2 p-\Delta p-\alpha(x)\partial_t^2(p^2)=0,
\]
the leading profile equation is of Burgers type,
\[
(\partial_t+\omega\cdot\nabla_x)U_0+\alpha\,U_0\,\partial_\theta U_0=0,
\]
so wave steepening, rather than parabolic smoothing, governs the principal asymptotics [2208.13945].

## 3. Initial-boundary value problems and long-time dynamics

For bounded \(C^2\)-domains with homogeneous Dirichlet boundary conditions, the Westervelt equation admits a global small-data theory in an optimal maximal-\(L_p\)-regularity framework. If the initial data are sufficiently small and satisfy the appropriate compatibility conditions, there exists a unique global solution
\[
u\in W_p^2(0,T;L_p(\Omega))\cap W_p^1(0,T;W_p^2(\Omega)),
\]
and the solution converges to zero exponentially fast as \(t\to\infty\) [1502.05816].

A more refined analysis with nonlinear absorbing boundary conditions of order zero establishes global well-posedness for small initial data, instantaneous temporal regularization, and convergence to equilibria. In that setting the full problem is
\[
\begin{cases}
c^{-2}u_{tt}-\Delta u-\beta\Delta u_t=\gamma (u^2)_{tt} & \text{in } J\times\Omega,\\
\partial_\nu(u+\beta u_t)+u_t\sqrt{c^{-2}-2\gamma u}=0 & \text{on } J\times\partial\Omega,\\
u(0)=u_0,\quad u_t(0)=u_1,
\end{cases}
\]
and each equilibrium belongs to
\[
\mathcal{E}=\{(r,0): r\in\mathbb{R},\ |r|<1/(2\gamma c^2)\}.
\]
Solutions starting sufficiently close to an equilibrium converge exponentially to a possibly different equilibrium, reflecting the one-dimensional center manifold generated by constants [1603.02097].

Instantaneous smoothing is a recurring feature. In the absorbing-boundary theory, for every \(k\in\mathbb{N}\) and \(\tau\in(0,T)\),
\[
u\in W_p^{k+2}(\tau,T;L_p(\Omega))\cap W_p^{k+1}(\tau,T;W_p^2(\Omega)),
\]
hence \(u\in C^\infty(0,T;W_p^2(\Omega))\) [1603.02097]. A parallel parameter-trick argument in the non-isothermal Westervelt–Pennes system yields analogous instantaneous temporal regularization for both acoustic and thermal variables [2204.09630].

These analytical results also clarify a recurrent misconception. The equation is sometimes described as a weakly nonlinear wave equation with lower-order corrections, but the main highest-order coefficient depends on the solution itself, and the damping term is analytically decisive. The small-data restriction therefore preserves the equation’s admissible regime rather than merely simplifying the proofs [1502.05816].

## 4. Boundary conditions, periodic forcing, and coupled thermoacoustic models

Artificial boundary modeling is a major part of Westervelt-equation analysis. A family of nonlinear absorbing boundary conditions was constructed by pseudo-differential calculus in one and two space dimensions. The zero-order condition
\[
u_n^\beta+\sqrt{c^{-2}-2\gamma u}\,u_t=0,
\qquad
u^\beta:=u+\beta u_t,
\]
acts as a nonlinear amplitude-dependent impedance. First-order conditions introduce tangential derivatives and memory-type terms, and numerical experiments showed that the pseudo-differential conditions were more accurate than classical Engquist–Majda absorbing boundary conditions in the tested regimes [1408.5031].

Periodic excitation leads naturally to multiharmonic formulations. For a bounded domain with Robin boundary conditions,
\[
\beta p_t+\gamma p+\nabla p\cdot n=0,
\]
and time-periodic forcing, the nonlinear periodic Westervelt equation generates higher harmonics through quadratic frequency autoconvolution. The corresponding truncated multiharmonic system consists of coupled Helmholtz-type equations for harmonic amplitudes \(\hat p_m\), and an iterative harmonic-by-harmonic solution scheme converges for sufficiently small excitation [2407.17043].

The equation also appears in thermoacoustic couplings. In the non-isothermal Westervelt–Pennes model,
\[
u_{tt}-c^2(\theta)\Delta u-b(\theta)\Delta u_t=k(\theta)(u^2)_{tt},
\]
the acoustic coefficients depend on temperature, and the pressure couples to a bioheat equation through a source term \(Q(u_t)\). Local and global well-posedness, instantaneous regularization, and exponential decay near equilibrium were established in an \(L_p\)-\(L_q\) maximal-regularity framework [2204.09630].

Replacing Fourier heat conduction with the hyperbolic Cattaneo law yields the Westervelt–Pennes–Cattaneo model. Local well-posedness and nondegeneracy under small pressure data were proved together with a singular limit \(\tau\to 0\), in which the Cattaneo system converges to the Fourier-coupled Westervelt–Pennes model [2304.02881]. A later global theory showed existence and exponential decay under small lower-order Sobolev norms even when higher-order norms are arbitrarily large, provided the acoustic diffusivity \(b>0\) is present [2404.03920].

Time-fractional attenuation provides another extension. In the fractionally damped model
\[
p_{tt}-c^2\Delta p-b\,\Delta(D_t^\alpha p)=k(x)(p^2)_{tt}+f,
\]
inhomogeneous Neumann boundary control and distributed control were analyzed, and existence of globally optimal controls together with adjoint-based first-order optimality conditions were derived [2511.15382].

## 5. Numerical discretization and computational methods

Several numerical frameworks are now established for the Westervelt equation. For homogeneous Dirichlet problems, a semi-discrete \(P_1\) finite element approximation yields optimal a priori spatial error bounds in \(L^2\)-based norms under sufficiently small data and mesh size. The analysis combines a linear variable-coefficient wave estimate with a Banach fixed-point argument and uses inverse estimates together with Scott–Zhang interpolation to prevent degeneration of the discrete coefficient \(1-2ku_h\) [1901.08510].

Structure-preserving discretization has also been developed in the velocity-potential formulation. A canonical gradient-system reformulation supports space-time Galerkin–Petrov approximations of formally arbitrary order, and with a consistent quadrature-based inner product the scheme preserves a discrete energy identity exactly. In the linear case \(\beta=0\), the method coincides with Lobatto–IIIA; for \(\beta>0\), it preserves the nonlinear energy identity whereas Lobatto–IIIA, Gauß–Runge–Kutta, and Newmark do not [1912.07037].

Time integration by operator splitting exploits the hyperbolic-parabolic decomposition induced by the Westervelt nonlinearity and damping. For the first-order Lie–Trotter splitting applied to the velocity-potential form, a global error estimate confirms stability and first-order convergence under sufficient regularity, and numerical experiments show that the favored decomposition reduces computational cost by separating a nonlinear diffusion subproblem from a simpler wave-like subproblem [1311.1224].

For curved geometries and manifolds, discrete exterior calculus provides a coordinate-free spatial discretization. The DEC formulation approximates the Laplace–Beltrami operator on simplicial complexes and leads to explicit, implicit, and semi-implicit schemes for
\[
\Delta p-\frac{1}{c_0^2}p_{tt}+\frac{\delta}{c_0^4}p_{ttt}+\frac{\beta}{\rho_0c_0^4}\partial_t^2(p^2)=0.
\]
This extends finite-difference-type ultrasound simulation to triangulated and tetrahedralized manifolds [1001.2082].

For nonlocal attenuation, a semi-discretization in time combines the trapezoidal rule with A-stable convolution quadrature. Because the fractional kernel produces a weak singularity at \(t=0\), an initial correction is added to restore the optimal rate compatible with the weak-start behavior. Without correction the energy-norm error is \(O(\tau)\), whereas the corrected scheme achieves \(O(\tau^{1+\mu})\) for kernels behaving like \(t^{\mu-1}\) near zero [2210.16349].

## 6. Inverse problems and imaging

The Westervelt equation has become a model for nonlinear ultrasound imaging because harmonic generation carries information about material parameters not present in linear wave data. In a lossless variable-speed model,
\[
\frac{1}{c^2(x)}p_{tt}-\Delta p-\beta(x)\partial_t^2(p^2)=0,
\]
the coefficient of nonlinearity \(\beta(x)\) can be recovered uniquely from the nonlinear Dirichlet-to-Neumann map. The proof uses second-order linearization and Gaussian beam solutions to reduce the problem to a weighted geodesic ray transform [2105.05423].

In the nondiffusive weakly nonlinear regime,
\[
\partial_t^2 p-\Delta p-\alpha(x)\partial_t^2(p^2)=0,
\]
the transmitted high-frequency wave packets acquire a measurable tilt. That tilt is proportional to the X-ray transform of the compactly supported nonlinearity \(\alpha\), and the second harmonic coefficient is proportional to the same line integral in the small-\(\alpha\) regime [2208.13945].

Simultaneous identification problems are more demanding. For a Westervelt model with spatially varying sound speed \(c_0(x)\) and nonlinearity coefficient \(\kappa(x)\), the problem can be reformulated in terms of squared slowness \(\mathfrak{s}=1/c_0^2\) and
\[
\eta=\frac{B/A+2}{\varrho_0 c_0^4}.
\]
A frozen Newton method with a range-invariance identity was devised, and linearized uniqueness for simultaneous recovery was proved under suitable two-measurement conditions [2210.08063].

The periodic setting admits even broader identification results. For the Westervelt operator
\[
\mathcal{W}(u)=\partial_t^2(u-\alpha u^2)-\nabla\cdot(\gamma\nabla u)-\nabla\cdot(\beta\nabla \partial_t u),
\]
complex-valued time-periodic solutions excited at high frequency generate first and second harmonic boundary data. Knowledge of those two harmonic Cauchy data sets is sufficient to determine simultaneously the wave-speed coefficient \(\gamma(x)\), the diffusivity \(\beta(x)\), and the nonlinearity \(\alpha(x)\) [2509.10718]. In a related periodic framework with Robin-type boundary conditions, Fréchet differentiability of the forward operator and linearized uniqueness for an all-at-once frozen Newton scheme were established for the simultaneous reconstruction of sound speed, diffusivity, and nonlinearity from boundary measurements [2605.15946].

These developments suggest a broad shift in Westervelt-equation research: the model is no longer used only for forward simulation of nonlinear acoustics, but also as a parameter-sensitive imaging equation in which harmonic content, boundary traces, and periodic responses encode recoverable information about the medium.

Source: https://www.emergentmind.com/topics/westervelt-equation