Exact dispersion relation for linear surface waves on arbitrary vertical shear

Abstract: We derive the formal solution to the dispersion relation for linear surface waves on a horizontal mean current with arbitrary vertical dependence. The problem is cast in a Green's function framework for the Rayleigh equation, neglecting viscosity but making no further approximations about the mean velocity profile. The solution is the dispersion relation in the form of a single, implicit equation relating -- and containing only -- the velocity profile, wave frequency, and wavenumber. By isolating curvature effects in a path-ordered exponential, we obtain a solution that serves as a natural starting point for systematic approximations. We demonstrate that our solution reduces to the expression found by Shrira (1993, J. Fluid Mech. 252, 565--584) in the deep-water limit, yields known asymptotic approximations, and recovers known analytical solutions in special cases.

• The paper derives an exact implicit dispersion relation by inverting the Rayleigh equation with a Green’s function and combining it with the free-surface boundary conditions.
• The formulation isolates velocity-profile curvature in a path-ordered exponential, enabling systematic Magnus-expansion approximations for weak shear, finite depth, and strong profile variation.
• The result recovers deep-water, constant-vorticity, and Stewart–Joy limits, providing a benchmark for approximate models while excluding viscosity, critical-layer effects, and horizontally varying currents.

Overview

Linear surface gravity waves propagating on a horizontally uniform current with vertical shear are governed by the Rayleigh equation, a second-order ODE in the vertical coordinate whose coefficients depend on the mean velocity profile $U(z)$. For arbitrary $U(z)$ this boundary-value problem admits no closed-form solution, and the literature has relied on approximate dispersion relations valid in restricted regimes—weak shear expansions of Stewart–Joy type [(2107.00000)-style asymptotics; see also (2604.24484) references to Stewart & Joy (1974), Kirby & Chen (1989), Skop (1987), Ellingsen & Li (2017)] or special profiles such as constant vorticity (Burns 1953, Dalrymple 1974). The paper by Heinrich and Ellingsen derives an exact, implicit dispersion relation for inviscid linear surface waves on a current with arbitrary vertical dependence $U(z)$, formulated as a single equation containing only the profile, the wave frequency $\omega$, and the wavenumber $k$.

Formulation as a Green's function problem

The authors cast the Rayleigh boundary-value problem in a Green's function framework. The Rayleigh operator acting on the vertical wave amplitude is inverted via its Green's function, so that the kinematic and dynamic free-surface boundary conditions combine into one scalar condition on $\omega$ and $k$. The result is an exact dispersion relation: an implicit equation relating frequency and wavenumber through functionals of $U(z)$ alone, with no approximation on the profile beyond inviscidity. This places the problem on the same footing as the classical irrotational dispersion relation $\omega^2 = gk$, which it reduces to when $U \equiv 0$.

A central structural feature of the solution is that curvature effects of the mean profile—terms involving $U(z)$0 and higher derivatives—are isolated inside a path-ordered exponential (time-ordered exponential in the analogy with quantum mechanics, where the Rayleigh equation maps onto a Schrödinger-like problem). The construction borrows machinery from product integration and path-ordered exponentials developed for time-dependent quantum systems (Dyson 1949; Dollard & Friedman 1979; Blanes et al. 2009), which the authors adapt to the spatial "propagation" variable of the Rayleigh problem.

Systematic approximations from the exact result

Because curvature contributions are segregated into the ordered exponential, the exact relation serves as a natural generating object for systematic perturbation schemes. Truncating the Magnus expansion of the path-ordered exponential yields controlled approximations organized by powers of profile curvature, recovering known results as leading orders:

• Weak-shear limit: the expansion reproduces the standard Stewart–Joy-type surface-current approximation used in HF radar interpretation of near-surface currents.
• Deep-water limit: the formula reduces exactly to Shrira's (1993) solution of the boundary-value problem for surface waves on deep water with shear, providing a nontrivial consistency check.
• Constant vorticity: the classical analytical dispersion relations (Burns 1953; Ellingsen & Brevik 2014) are recovered for linear $U(z)$1, where the Rayleigh equation has elementary solutions.
• Finite-depth and strongly sheared flows: the framework subsumes the regime targeted by earlier approximate relations (Kirby & Chen 1989; Banihashemi & Kirby 2017, 2019), which were constructed for wave-action conservation in vertically sheared mean flows rather than derived from an exact kernel.

The practical significance is that prior approximate dispersion relations can now be assessed against an exact benchmark, and their domains of validity quantified rather than assumed.

Relation to prior work

Shrira (1993) solved the shear-current surface-wave problem for deep water but his formulation does not extend transparently to finite depth or serve as a systematic approximation scheme. Ellingsen and Li (2017) and Smeltzer and Ellingsen (2017) developed approximate dispersion relations for arbitrary shear, accurate in weak-shear regimes but uncontrolled for strong curvature. Maxwell and Ellingsen (2020) computed exact dispersion curves numerically by path-following on the Rayleigh equation, establishing that exact solutions exist computationally but not analytically. The present work supplies the missing analytic object: an exact implicit relation from which both the numerical approaches and the analytic approximations follow as limiting procedures. Applications motivating this line of work include wave focusing on shear currents (Ellingsen et al. 2024), weakly nonlinear wave statistics on sheared currents (Zheng et al. 2023), and oceanographic problems where near-surface shear measurably affects transport, breaking, and wind-stress parameterization (Laxague et al. 2018; Zippel & Thomson 2017; Ortiz-Suslow et al. 2025).

Limitations and open questions

The derivation neglects viscosity entirely, so critical-layer physics associated with the resonance $U(z)$2—relevant for wave generation by wind (Miles 1957) and for short waves on strong shear (Zhang 2005)—lies outside the theory; the treatment of singularities at critical layers within the Green's function inversion is not addressed in the abstract-level formulation. The inviscid assumption also excludes viscous attenuation over depth, which matters for short capillary–gravity waves. Second, while the path-ordered-exponential structure suggests convergent Magnus-type truncations, convergence criteria for the specific operator norms arising here (cf. Moan & Niesen 2008) are not established, so the radius of validity of low-order curvature expansions remains to be quantified. Finally, the theory assumes a steady, horizontally uniform current; extension to slowly varying bathymetry and currents (as in Li & Ellingsen 2019) using this exact kernel is left open.

Conclusion

Heinrich and Ellingsen provide an exact, implicit dispersion relation for inviscid linear surface waves on a current of arbitrary vertical shear, expressed through a Green's function solution of the Rayleigh equation with curvature effects isolated in a path-ordered exponential. The result reproduces Shrira's deep-water solution, constant-vorticity analytics, and weak-shear asymptotics as special or limiting cases, and supplies a rigorous foundation against which the widely used approximate dispersion relations for sheared flows can be systematically evaluated.