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Asymptotics of the Helmholtz equation solutions in a two-layer medium with a localized right-hand side

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Abstract

We construct the leading term of the semiclassical asymptotic solution of the Helmholtz equation with a small parameter in the localized right-hand side. This equation arises, for example, in the problem of ocean acoustics, in which the small parameter is given by the ratio of the characteristic scale of the “vertical ” coordinate to that of the other coordinates. The equation is considered in the region bounded in the “vertical ” coordinate; it is divided into two layers, with the coefficient in the Helmholtz equation and the derivative of the solution having fixed jump discontinuities at the interface. The technique for constructing the asymptotics involves the operator separation of variables (adiabatic approximation) and the use of the recently developed method for constructing asymptotics of equations with localized right-hand sides in the equations obtained after the variable separation.

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Notes

  1. For \(\zeta> 0\), this follows from \(\sqrt{\zeta} > \sin \sqrt{\zeta}\); for \(\zeta \leq 0\), this can be seen from the signs of the power series terms.

  2. Strictly speaking, the symbol \(\phi_\varkappa(x, p, z, h)\), which depends parametrically on \(z\), must be understood as a map \((x, p, h) \mapsto \phi_\varkappa(x, p, \cdot, h)\) with values in the domain (4) of the operator \(\mathcal{H}(x, p)\) for given \((x, p)\).

  3. For example, in the case of dimension \(n=2\), we have the circle \(\mathbb{S}^{1} \cong \mathbb{R}/2\pi \mathbb{Z}\), and we can take \(Q(\psi)=\sqrt{\mathcal{E}_\varkappa(x^0)}(\cos\psi,\sin\psi)\).

  4. For example, for \(n=2\) and \(Q(\psi)=\sqrt{\mathcal{E}_\varkappa(x^0)}(\cos\psi,\sin\psi)\), we have \(J^\varepsilon(\psi,\tau)=-2\det\frac{\partial (\mathcal{X}-i\varepsilon \mathcal{P})}{\partial (\psi,\tau)}\).

  5. For \(\varepsilon = 0\), at the points where this argument is nonzero.

  6. The choice of such an atlas depends on some properties, and such atlases are called canonical (see [23]). Also see [24], where more general formulas for the Maslov canonical operator are described for more general classes of atlases.

  7. A chart \(V_\beta \subset \Lambda_+\) is called nonsingular if its projection onto the configuration space \(V_\beta \to \mathbb{R}^n_x\) defined by the map \((\psi, \tau) \mapsto \mathcal{X}(\psi,\tau)\) is an embedding.

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Acknowledgments

The authors are grateful to S. Yu. Dobrokhotov and V. E. Nazaikinskii for the attention to this work and the valuable discussions.

Funding

This work was supported by the Russian Science Foundation (grant No. 21-11-00341), https://rscf.ru/en/project/21-11-00341/.

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Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia

    A. Yu. Anikin & A. I. Klevin

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  1. A. Yu. Anikin
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  2. A. I. Klevin
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Correspondence to A. I. Klevin.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 216, pp. 148–168 https://doi.org/10.4213/tmf10421.

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Anikin, A.Y., Klevin, A.I. Asymptotics of the Helmholtz equation solutions in a two-layer medium with a localized right-hand side. Theor Math Phys 216, 1036–1054 (2023). https://doi.org/10.1134/S0040577923070103

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