Lines of equal phases and phase invariant in the sound field of the deep sea

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Abstract

The spatial-frequency characteristics of the amplitudes and phases of sound pressure in the deep sea have been studied. Analytical relationships have been obtained that allow one to calculate and compare the amplitude-phase structures of water, leaky and trapped modes, as well as the sound pressure field formed by the sum of the modes. The calculations have been performed using the modified WKB (Wentzel–Kramers–Brillouin) approximation. It has been shown that in the deep sea, as in the shallow sea, there are stable lines of equal phases along which, under certain conditions, coherent summation of complex Fourier components is possible. To describe the lines of equal phases, a differential equation has been obtained that uses the phase invariant, already studied in the shallow sea, as a basic parameter. This has made it possible to study the properties of the phase invariant corresponding to water, leaky and trapped modes in all zones of the sound field for the deep sea as well. It is established that at different distances in the field constructed from the sum of all modes, invariant properties are manifested, first of all, those modes that dominate at these distances. It is shown that the leaky modes formed in the near illumination zone and in the shadow zone, formed by steep rays reflected from the bottom, have invariant properties only at large distances from the source. Water and trapped modes have invariant properties in full and at all distances. Recommendations are given on the use of equal phase lines and the phase invariant in processing experimental data and modeling.

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About the authors

S. P. Aksenov

A.M. Prokhorov General Physics Institute of the Russian Academy of Sciences

Author for correspondence.
Email: skbmortex@mail.ru
Russian Federation, Moscow, 119991

G. N. Kuznetsov

A.M. Prokhorov General Physics Institute of the Russian Academy of Sciences

Email: skbmortex@mail.ru
Russian Federation, Moscow, 119991

A. N. Stepanov

Samara National Research University named after academician S.P. Korolev

Email: skbmortex@mail.ru
Russian Federation, Samara, 443086

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Supplementary files

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2. Fig. 1. The WRSZ adopted in the calculations (a simplified version of the WRSZ for the Mediterranean Sea). The source and receiver depths zs = 80 m and z = 60 m are indicated by red lines.

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3. Fig. 2. Attenuation of the ZD amplitude with distance and contributions of individual mode groups.

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4. Fig. 3. (a–g) — Surfaces of the ZD amplitudes in the BZAO for different mode groups: (a) — all water modes, (b) — trapped modes, (c) — leaky modes, (d) — the sum of all modes. (e–h) — Surfaces of the phases of sound pressure in the BZAO for different mode groups: (e) — all water modes, (e) — trapped modes, (g) — leaky modes, (h) — the sum of all modes.

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5. Fig. 4. (a, d) — Amplitude and phase surfaces of the mode group of the first type, (b, d) — mode groups of the second type, and (c, e) — sums of water modes of the first and second types.

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6. Fig. 5. (a, b) — Contour graphs of phase surfaces of water mode groups of the first and second types, and (c) — sums of all water modes. (d, d) — FI graphs along equal phase lines of water mode groups of the first and second types, and (e) — sums of all water modes.

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7. Fig. 6. Contour graphs of phase surfaces in BZAO (distances of 1000–2500 m): (a) — sum of water modes, (b) — trapped modes, (c) — leaky modes, and (d) — sum of all modes. Thickened black lines represent one of the equal phase lines.

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8. Fig. 7. Frequency dependences of the phase invariant along the selected equal-phase line in BZAO (distances of 1000–2500 m): (a) — sum of water modes, (b) — trapped modes, (c) — leaky modes, and (d) — sum of all modes.

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9. Fig. 8. Contour graphs of phase surfaces in ZT1 (distances of 4000–10500 m): (a) — sum of water modes, (b) — trapped modes, (c) — leaky modes, and (d) — sum of all modes. The thickened black lines represent one of the equal-phase lines.

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10. Fig. 9. Frequency dependences of the phase invariant along the selected equal-phase line in ZT1 (distances of 4000–10500 m): (a) — sum of water modes, (b) — trapped modes, (c) — leaky modes, and (d) — sum of all modes.

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11. Fig. 10. Contour plots of phase surfaces in ZT2 (distances of 14,000–30,000 m): (a) — sum of water modes, (b) — trapped modes, (c) — leaky modes, and (d) — sum of all modes. Thickened black lines represent one of the equal-phase lines.

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12. Fig. 11. Frequency dependences of the phase shifters along the equal-phase line in ZT (14,000–30,000 m): (a) — sum of water modes, (b) — trapped modes, (c) — leaky modes, and (d) — sum of all modes.

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13. Fig. 12. Contour plots of phase surfaces in the DZAO (distances 35,000–45,000 m): (a) — sum of water modes, (b) — trapped modes, (c) — leaky modes, and (d) — sum of all modes. Thickened black lines represent one of the equal phase lines.

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14. Fig. 13. Frequency dependences of the phase shifters along the equal-phase line in DZAO (35,000–54,000 m): (a) — sum of water modes, (b) — trapped modes, (c) — leaky modes, (d) — sum of all modes.

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