My name is Daniel Walsken, and I am a Ph.D. student in the Applied and Computational Mathematics research group at the University of Wuppertal. For my Ph.D., I am supervised by Matthias Ehrhardt from the University of Wuppertal and Pavel Petrov from the Instituto de Matemática Pura e Aplicada (IMPA) in Rio de Janeiro, Brazil.
Propagation problems for the wave equation have existed for as long as the equation itself and remain relevant today. In ocean acoustics, the primary field of Pavel Petrov, these problems include efficiently computing the acoustic pressure field generated by industrial activities, such as geoacoustic surveys, to predict environmental impact, especially on aquatic mammals.
As the need for minerals in the seabed increases, industrial activity in the ocean is becoming more possible and necessary. Limiting the environmental impact of such mining operations is only possible if their impact can be predicted in advance. Geoacoustic surveying also substantially impacts the underwater soundscape. This technique is used to find oil deposits and classify the composition of the seafloor. It involves triggering short, low-frequency sound pulses by igniting a combustion chamber in an underwater controlled explosion and measuring the scattered and propagated signal. Other applications of propagation problems include radio wave propagation through the Earth’s atmosphere, as well as optical propagation in laser physics.
The propagation of waves is generally governed by the wave equation. Since the wave equation is classified as a hyperbolic partial differential equation (PDE), the concept of a “parabolic” wave equation may be unfamiliar to the reader. However, when solving scattering and guided wave propagation problems, it is often unnecessary to solve the wave equation directly. Instead, the Helmholtz equation is solved. This yields a steady-state solution of the wave field—intensity in the case of electromagnetic radiation and pressure in the case of acoustic waves—from which propagated signals can be computed. The Helmholtz equation operator is split into two factors: one representing the incoming wave and the other representing the outgoing wave. In the field of long-range wave propagation, it is common to assume that back scattering can be neglected by assuming the incoming wave operator has no effect on the field. This transforms the complicated Helmholtz boundary value problem (BVP) into an initial boundary value problem (IBVP) of the new parabolic equation. This approximation involving the root of an elliptic operator was developed by Levontovich and Fock [1] in the 1940s.
In general, the square root of an unbounded operator is not well-defined. However, recently, existence, uniqueness, and well-posedness results were established [2]. Despite this, solution techniques have been developed ever since the approximation was first introduced. One important method is the split-step Padé scheme, which was developed independently by Avilov [3] and Collins [4]. In this method, the general formal solution to the parabolic wave equation involving the exponential of the square root of the Laplace operator is expanded using a Padé series. These equations can be efficiently and accurately solved for a dense grid using finite differences. However, for sparse grids, the accuracy of the results decreases to the point of producing physically nonsensical solutions.
Spectral methods [5] are an alternative class of methods that are considerably less limited by the density of the discretization. In these methods, the differential operator is evaluated analytically using a set of global basis functions. Therefore, the precision of these methods is limited by the selection of basis functions. For waves, trigonometric basis functions are used, corresponding directly to the application of a Fourier transform to the equation. Spectral methods utilizing fast transforms, such as the Cooley-Tukey FFT [6], offer exceptional performance for smooth problems. These methods converge exponentially rather than polynomially, which gives them an advantage, especially for large domains typically encountered in underwater acoustics or radiophysics. However, the drawback of such methods is that they can only be applied naively to linear equations with constant coefficients.
The initial task was to transform the subproblems generated by the Padé expansion into a linear form that could be solved using a spectral method [7]. This was accomplished by computing the inverse of the subproblem operator and expanding it using a Neumann series. This method yields good results, especially for problems with a smooth change in the speed of sound across the domain. Another significant issue in wave propagation involves unbounded, open domains. A successful approach to problems similar to the parabolic wave equation is introducing a perfectly matched layer (PML) around the domain of interest. Currently, we are modifying the Spectral Split Step Padé method to include a PML in order to solve the horizontal refraction equations, which is a parabolic wave equation for a modal ansatz in wave field propagation.
References
[1] Leontovich, M. A. and Fock, V. A., “Solution of the problem of propagation of electromagnetic waves along the earth’s surface by the method of parabolic equations”, J. Phys. of the USSR 10 (1946), 12-24.
[2] Ehrhardt, M., Glück, J., Petrov, P. and Tappe, S., “Square root operators and the well-posedness of pseudodifferential parabolic models of wave phenomena”, Appl. Math. Lett. 171 (2025), 109644.
[3] Avilov, K. V., “The calculation of the harmonic sound fields in the waveguides in the corrected wide-angle parabolic approximation”, in: Waves and Diffraction-85: IX All-Union symposium on wave diffraction and propagation}, Volume 2, 1985, pp. 236-239.
[4] Collins, M. D., “A split‐step Padé solution for the parabolic equation method”, J. Acoust. Soc. Amer. 93(4) (1993), 1736-1742.
[5] Boyd, J.P., “Chebyshev and Fourier spectral methods”, Courier Corporation, 2001.
[6] Cooley, J. W. and Tukey, J. W., “An algorithm for the machine calculation of complex Fourier series”, Math. Comput. 19(90) (1965), 297-301.
[7] Walsken, D., Petrov, P. and Ehrhardt, M., “A Spectral Split-Step Padé Method for Guided Wave Propagation”, IMACM Preprint 25/11, August 2025.