The ray theory has been used for many years in underwater acoustics and historically proved to be an indispensable tool for understanding and studying sound propagation.

In 1957, Hudimac^[6] studied the sound intensity in the water due to a point airborne source by the ray theory. He expressed the intensity and lateral range as a pair of parametric equations in terms of the angle of incidence, and pointed out that the intensity increases appreciably with depth (except directly below the source).

In 1972, Urick^[7] demonstrated that in the real ocean sound produced by an airborne source reaches an underwater hydrophone in four ways, i.e., a direct refracted path, one or more bottom reflections, the so-called lateral wave, and sea scattering. With the geometry of ray acoustics shown in Fig. 1, he expressed the intensity I_w at an arbitrary point R below the surface, produced by a point source S of intensity I_a at unit distance, located at height h above a plane sea surface, as follows:

Fig. 1.

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	Fig. 1. Geometry of ray acoustics.

From Eq. (2), it can be recognized that the sound directly beneath the real source appears to come from a ‘virtual source’ situated at height nh above the surface with the intensity equal to the real source’s intensity multiplied by 4n². The above conclusion has been pointed out by Young.^[8,9] Equation (2) also indicates that the received sound intensity directly underneath the airborne source is affected by the source’s height. Equation (3) indicates that in the far field, the received sound intensity which has a cosine-squared radiation pattern in off-vertical directions is unrelated to the source’s height.

The air-to-water sound transmission was originally studied by Ma.^[10] He investigated the underwater field directly below the source by the ray theory in 1972, and pointed out that the received sound pressure radiates in a cosine direction which is consistent with Urick’s conclusion.

The underwater field calculated by the ray theory is accurate enough when the airborne source height or the receiver depth is much longer than the wavelength. However, when these assumptions do not hold true then the ray theory is no longer applicable, and the wave solution which takes the lateral wave into account is needed. The lateral wave contributes significantly if the waterborne receiver depth is within a fraction of a wavelength of the boundary. In 1960, Brekhovskikh^[11] studied the reflection and refraction of spherical wave at a plane interface based on plane wave decomposition. He decomposed the received field into the direct refracted wave and the so-called lateral wave. The amplitude of the lateral wave has an exponential decrease with increasing receiver depth, and decreases more rapidly as the frequency increases. In 1965, the wave solution for sound transmission through an air-water interface was given by Weinstein and Henney.^[12] Numerical computations were compared with the ray-theory solution, and the difference between results calculated by these two methods is obvious at low frequencies, low source altitudes, and shallow water depth. With increasing frequency and depth, this difference decreases. By combining with the effect of the bottom, they concluded that the underwater field decayed in an oscillatory manner as water depth increased, with the bottom assumed to be a semi-infinite fluid layer.

In shallow water, the ray theory is not a good choice to calculate the complex sound field because the downward-traveling acoustic energy is soon reflected back toward the surface by the seabed and begins a cycle of multiple reflections from both ocean boundaries. In 1990, Chapman and Ward^[13] reviewed the fundamental expressions for the reflection and transmission of plane and spherical waves. They pointed out that for plane waves across the air/water boundary, the intensity reflection loss is 0.005 dB and the intensity transmission loss is 29.5 dB. For a point airborne source, the expression of transmitted field has the appearance of spherical waves radiating from an effective source that has an approximate cosine radiation pattern at the air/water boundary. Then they presented the normal-mode theory of sound propagation from an airborne source into the ocean. The airborne source height is h and the ocean depth is H, the sound pressure in the far-field region is expressed as

In 1992, Chapman et al.^[14] presented that if the height of the airborne source is sufficiently small, the air-to-water transmission problem can be modeled by replacing the source in air with an underwater source at a depth which is much smaller than the acoustic wavelength in water. In principle, any standard propagation code in underwater acoustics could be modified to treat air-to-water sound transmission problems. He gave the method to model air-to-water sound transmission losses with standard underwater acoustics computer codes, such as normal mode (SNAP), multipath expansion (GSM), and parabolic equation (PE), which is shown in Fig. 2. The method was validated by comparing with the air-to-water sound transmission loss calculated by the wavenumber integration approach (SAFARI). However, in the normal mode theory, the complex contour integration is used to reduce the integral representation to a sum of residues, with neglecting the branch line integrals which represent the lateral wave, so it is applicable to the far field acoustic pressure.^[15]

Fig. 2.

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	Fig. 2. Modeling air-to-water transmission losses with standard numerical codes of underwater acoustics.

The wavenumber integration approach is basically a numerical implementation of the integral transform technique for horizontally stratified medium. As this approach makes the least approximation, it can be used to obtain a benchmark result. According to the wavenumber integration approach, Schmit^[16] developed a computer program referred to as OASES, which can be used to predict the air-to-water transmission.

One of the problems in the study of the air-to-water sound transmission is the rapidly moving airborne source. A moving source in free space results in a frequency Doppler shift. In a stratified ocean, because of effects such as refraction, boundary reflection, and the coherent summation of multiple arrivals at a receiver, moving source problems are more complicated. Clark et al.^[17] and Jacyna and Jacobson^[18] studied the effect of a moving source on the total acoustic field at a fixed receiver in a deep ocean with the ray theory in 1976 and 1977, respectively. Guthrie et al.^[19] and Hawker^[20] presented the acoustic field of a horizontally stratified ocean generated by a monochromatic point source moving slowly with constant horizontal velocity in terms of the normal-mode theory in 1974 and 1979, respectively. When the source speed is much smaller than the mode group velocity, Hawker^[20] presented a method of achieving a stationary phase which is widely used in matched-field localization. In 1994, Lim and Ozard^[21,22] obtained the acoustic field for a source moving with an arbitrary but small horizontal velocity by expressing the field as a sum of modes in terms of retarded times for the range-independent environment and the weakly range-dependent environment. In 1994, Schmidt and Kuperman^[23] derived a spectral formulation of the Doppler-shifted field in a stratified waveguide by combining both source and receiver motion, and presented a more complete formulation of the Doppler effect than earlier modal formulation. In 2006, Buckingham and Giddens^[24] developed a theory for the acoustic field in a three-layer waveguide (atmosphere, shallow ocean and sediment) based on the wavenumber integral. The unaccelerated source, such as a line source and a point source, moved horizontally in the atmosphere.

If an airborne source moves with a higher speed, the methods mentioned above are no longer applicable. In 1994, Kazandjian and Leviandier^[25] presented a normal-mode theory for evaluating the acoustic pressure field generated by a uniformly moving airborne source, but this method costs a large amount of computation time. In 2004, Lv and Wu^[26] presented an approach based on the Doppler frequency of a virtual source determined by the ray theory, and combined it with normal mode theory. They obtained a more accurate stationary phase point than Hawker’s solution for an airborne source moving at a high speed. Li^[27] presented a modal expression for the acoustic pressure in a stratified waveguide generated by a moving airborne source in 2007, and this modal has some advantages of simple formula, definite physical meaning and easy implementation. In 2007, Zhang^[28] developed a fast field program (FFP) to calculate the shallow water acoustic field generated by an airborne source, and this program has been extended to treat the elastic bottom environment and moving medium. With this model, the effects of the wind speed profile, the sound speed profile and the airborne source height together with the lateral wave are studied. He pointed out that the wind speed profile and the sound speed profile in air have a very significant effect on the sound propagation in air but a little effect on the transmitted sound under water. When the depth of the shallow water is much larger than the wavelength of sound, the lateral wave is greatly affected by the wind speed profile and sound speed profile. For the rapidly moving airborne source, he developed a two-dimensional wavenumber integration model which can be approximated by a one dimensional wavenumber integration model at the far field and studied the acoustic field in deep water and shallow water. In deep water, both the amplitude and the frequency of the received signal change with time. In shallow water, the amplitude of the received signal fluctuates quickly, the Doppler shift is evident, and the spectrum broadening is significant. For the lateral wave, its Doppler frequency shift is consistent with that of the signal in the air, and is much larger than that for the refracted wave.

Another problem in the study of the air-to-water sound transmission is the rough interface between the air and the water. In the real ocean, the sea surface is randomly rough, and the sound in air refracting into water will be random too. The rough sea surface also induces both the incident angle and the insonifying cone to change.

In 1972, Medwin and Hagy,^[29] and Medwin et al.^[30] formulated the transmission problem for a randomly rough interface between two dissimilar fluids, based on the Kirchhoff-Helmholtz integral. They pointed out that as the roughness increases, the coherent transmission term decreases while the incoherent term increases. However, there is a great difference in the mean square transmitted pressure between the experimental value and the theoretical value. In 1976, Meecham^[31] studied the air-to-water sound transmission with a rough ocean surface, based on the high-frequency approximation. As the surface roughness increases, the mean square transmitted pressure at a small grazing angle increases evidently, and the enhancement reaches up to 10 dB. In 1976, Lubard and Hurdle^[32] conducted an experiment and pointed out that the received energy increases with source frequency and ocean roughness increasing. Kuperman and Schmit^[33–36] presented a self-consistent perturbation approach to treat scattering at rough interfaces in stratified fluid-elastic media, and a series of researches was published in 1976, 1989, and 1995. In 2003, Yan and Zhang^[37] studied the air-to-water sound transmission in shallow water by using the BDRM method and they also studied the effect of a randomly rough boundary on the sound field by the coupling normal-mode method. The comparison with the field of a smooth sea surface shows that in a low wind speed scenario, the rough boundary slightly influences the mean field as well as the mean square field; whereas, in the strong wind case, the mean field decreases while the mean square field increases. This method can be used in the cases of high wind-speed and frequency, but it requires a huge calculation time, so the statistical result cannot be obtained easily.

In 2014, Zhang and Peng^[38] analyzed the effects of rough interface on the air-to-water sound transmission in a horizontally stratified environment as shown in Fig. 3, with the self-consistent perturbation approach.

Fig. 3.

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	Fig. 3. Horizontally stratified environment.

Assuming that the interface at depth z_i, separating the layers i and i + 1, is not smooth, but randomly rough with an elevation γ(r), γ(r) = z − z_i, and with the mean zero, 〈γ(r)〉 = 0. The total field χ_i is decomposed into mean field 〈χ_i〉 and scattered field s_i, and the mathematical expression is

After some algebraic operations, the Fourier spectrum of the scattered field and the mean field at interface i can be expressed as

They programmed the self-consistent perturbation approach by adding a calculation model which is used to calculate the scattered field to the internationally popular OAST. This model can reduce the calculation so it is easy to obtain the statistical result. The total underwater field generated by an airborne source is calculated by the model in the cases of different wind speeds and frequencies. Based on simulation results and their comparison with the transmission loss for the smooth boundary, the conclusions are obtained as follows: rough sea surface causes obvious fluctuations of the transmission losses of the total field and the fluctuation increases as the propagation range, frequency and wind speed increase; rough sea surface causes the mean field to decrease but the mean square field to increase, and the influences become larger as frequency and wind speed increase. Generally speaking, the greater wind speed and frequency show positive influences on the underwater field.

In the same year, according to the self-consistent perturbation approach, Zhang and Peng^[39] analyzed the effect of the airborne source height on the air-to-water sound transmission in shallow water with a randomly rough sea surface. The numerical simulation results show that the fluctuation of the scattered field decreases as the source’s height increases. In contrast, the averaged energy of the total field is hardly influenced by the source’s height in the statistical sense.

Spatial correlation is an important characteristic and directly relevant to the detection of an underwater target and underwater communication. It is also a useful and efficient tool for predicting the performance of large acoustic arrays. Spatial correlation of the sound field generated by the waterborne source^[40–43] has received lots of attention but there have been few researches about the airborne source. According to the normal-mode theory,^[13,14] Zhang et al.^[44] derived the mathematical expression for longitudinal horizontal correlation of an underwater sound field generated by an airborne source in 2016. They employed a simulation and the result demonstrates that the oscillation period of longitudinal horizontal correlation coefficient for a source in air is almost the same as that for a source in water. The mathematical expression derived gives a physical explanation for the longitudinal horizontal correlation’s oscillation structure.