Low-frequency sound transmission through rough bubbly air–water interface at the sea surface

Abstract

Transmission of a sound generated by a localized point source in the air through a realistic sea surface is studied by the use of the Kirchhoff-Helmholtz integral. An earlier approach had been based on the Kirchhoff-Helmholtz integral which only considered the effects of rough surface. In the current study, not only the effect of the rough surface is taken into account but also the effects of subsurface bubbles are included in modeling the real phenomenon more accurately. In order to include the effects of subsurface bubble population, the classic relations of the Kirchhoff-Helmholtz integral are reformulated. Accordingly, a three-phase region of air, water, and bubbly water at the sea surface is analyzed, and the rough interface of bubbly water–air is discretized. Through considering an element area A_i, the transmission coefficient ${\mathcal{T}}_{\text{i}}$ , incident angle ${\text{θ}}_{\text{li}}$ , transmitted angle ${\text{θ}}_{\text{3i}}$ , and local surface acoustical roughness R_i are investigated for each individual element. Also, the effects of subsurface bubbles, transmission change as a function of frequency f, wind speed W, incident angle $\theta$ , source/receiver position ratio (D/H), surface acoustical roughness, and subsurface bubble population are examined. Results of the modified Kirchhoff-Helmholtz integral method display good agreement against available experimental data.

Introduction

Transmission of sound through water–air interface is often considered as an example of the application of Snell’s law and Fresnel reflection and transmission coefficients ¹ in the acoustic textbooks. Studying the sound transmission between air and water is mostly conducted under the greatly simplifying assumption of smooth interface between the sea water (density: $\rho =1020\hspace{0.17em}\text{kg}/{\text{m}}^3$ , sound speed: 1500 m/s) and air (density: $\rho =1.03\hspace{0.17em}\text{kg}/{\text{m}}^3$ , sound speed: 330 m/s). The air–water interface is called a “pressure release” or “soft” surface for the underwater sound. However, if the direction of propagation is reversed, sound going from air to the ocean would find a pressure-doubling interface with significantly zero particle velocity. Viewed from the air, the same surface would be called acoustically “hard”. ²

In recent years, the anomalous transparency^1,3–5 and enhanced sound transmission^6–8 theories for sound transmission through air–water interface have been introduced. In these theories, despite the mentioning of the classical view of air–water interface, it is anticipated that most of the acoustic power in a liquid half-space for a low frequency sound generated near the interface within a fraction of wavelength can be radiated into a gas half-space. ⁵ Through some experimental studies, Calvo et al. ⁷ checked the accuracy of the mentioned theories for a smooth air–water interface. However, the anomalous transparency predictions exceed beyond the smooth air–water interface and even anticipate that rough air–water interface does not have significant effects on the anomalous transparency of the air–water interface. Although these theories suggest a new approach towards sound transmission through the air–water interface, for practical applications in the ocean, it is imperative to consider a more realistic air–water interface for obtaining accurate solutions. There are other theoretical^9–28 and experimental^{15,19,24,26,28,29–34} approaches for sound transmission or/and scattering which consider the water–air interface which focus on the acoustic field in water due to the existence of powerful airborne noise sources such as helicopters,^31–33 propeller-driven aircraft^26–28,34 and supersonic transport.^{24,25,27,28,35}

Medwin and Hagy, ³⁶ by solving Helmholtz integral and deriving the transmitted pressure to the second medium, studied sound transmission through air–water rough interface. Medwin et al., ³⁷ by conducting FLIPEX I and II experimental tests, examined the low frequency (ranging from 50 Hz to 1000 Hz) sound transmission as a function of different variables such as incident angle, frequency, source and receiver position ratio, and surface acoustical roughness. However, their theoretical approach underestimated the transmission change (TC) in some cases, compared to that of experimental data. The TC parameter is applied for defining the ratio of the transmitted pressure (in the second medium) to the incident pressure in the first medium in logarithmic unit.

Ogden and Erskine^38,39 and Nicholas et al., ⁴⁰ based on Critical Sea Tests 1–7 (CST 1–7), identified three regimes for the companion problem of the sound transmission which was sound scattering from air–water interface at the sea surface. Based on their conclusions, each regime is controlled by two mechanisms: subsurface bubble clouds and surface geometric roughness. Ogden and Erskine ³⁹ emphasized that both the stated mechanisms should be considered in order to obtain accurate scattering strength coefficients. Since scattering and transmission of sound are considered companion problems ³⁶ and their corresponding pressure fields can be derived for the same initial source pressure by the Helmholtz-Kirchhoff integral, in the current paper, both mechanisms that were identified for sound scattering from the sea surface are considered in order to examine sound transmission through a realistic air–water interface in the ocean.

As pointed out earlier, the earlier approach adopted by Medwin and Hagy ³⁶ was based on the Kirchhoff-Helmholtz integral and only considered the effects of rough surface. However, in the current study, not only the effect of rough surface is considered but also the effects of subsurface bubbles are included for a more precise modeling of the real phenomenon. In order to include the effects of subsurface bubble population, the classic relations of the Kirchhoff-Helmholtz integral are reformulated and both mechanisms of the subsurface bubble clouds and surface geometric roughness in the sound transmission are taken into consideration through a realistic air–water interface. Then, the TC equation is derived. Subsequently, an algorithm is presented in the following section for numerically solving the derived equation for TC. Then, numerical results of TC are validated against FLIPEX I and II experimental data. Finally, a set of parametric studies is conducted.

Sound transmission through air–water interface

If a continuous wave of frequency $\omega =k_a{c}_a$ ( $c_a$ and $k_a$ are sound speed and wave number in the air) which is emitted by a localized point source in the air hits the ocean surface, the geometry of this phenomenon can be considered as shown in Figure 1. In such a case, the fields $U_1$ (x, y, z, t) and $U_2$ (x, y, z, t) are solutions of the wave equation and are functions of the position, frequency, and source sound pressure. These fields have time dependency as in $exp\hspace{0.17em}(i\omega t)$ and their values are determined using Green's theorem. The general form of Green's theorem for this problem can be written as follows (the detailed procedure is presented in Medwin and Clay ² )

$$-{{\displaystyle \int }}_{\text{S}}\left({\text{U}}_1\frac{\partial {\text{U}}_2}{\partial \text{n}}-{\text{U}}_2\frac{\partial {\text{U}}_1}{\partial \text{n}}\right)\text{ds}={{\displaystyle \int }}_{\text{v}}\left({\text{U}}_1{\nabla }^2{\text{U}}_2-{\text{U}}_2{\nabla }^2{\text{U}}_1\right)\text{dv}$$

(1)

Figure 1.

Geometry of sound transmission through interface air–water. ${\text{ρ}}_{\text{a}}$ , $\hspace{0.17em}{\text{ρ}}_{\text{w}}$ , ${\text{c}}_{\text{a}}$ , and ${\text{c}}_{\text{w}}$ are air density, water density, sound speed in air, and sound speed in water, respectively. $\text{ds}$ and $\text{n}$ are interface element and normal vector of the interface element, respectively. Q is sound source and $\text{Q'}$ is the imaginary source in the second medium. ${\text{R}}_{\text{Q}}$ and ${\text{R}}_{\text{Q'}}$ are Q and $\text{Q'}$ distances from the interface element $\text{ds}$ .

The source singularity effect, Q, is omitted by enclosing it within the surface $\hspace{0.17em}{\text{S}}^{\text{′}}$ . As a result, the right-hand side of equation (1) is zero and thus leading to

$${{\displaystyle \int }}_{\text{S}}\left({\text{U}}_1\frac{\partial {\text{U}}_2}{\partial \text{n}}-{\text{U}}_2\frac{\partial {\text{U}}_1}{\partial \text{n}}\right)\text{dS}+{{\displaystyle \int }}_{{\text{S}}^{\text{′}}}\left({\text{U}}_1\frac{\partial {\text{U}}_2}{\partial \text{n}}-{\text{U}}_2\frac{\partial {\text{U}}_1}{\partial \text{n}}\right){\text{dS}}^{\text{′}}=0$$

(2)

As discussed by Medwin and Clay, ² $\text{U\hspace{0.17em}(Q)}$ , which is the scattered field in the first medium (hereby air), can be derived as

$${\text{U}}_{\text{a}}\left(\text{Q}\right)=\text{U}\left(\text{Q}\right)=\frac{1}{4\text{π}}{{\displaystyle \int }}_{\text{S}}{\left[\text{U}\frac{\partial }{\partial \text{n}}\left(\frac{{\text{e}}^{-{\text{ik}}_{\text{a}}{\text{R}}_{\text{Q}}}}{{\text{R}}_{\text{Q}}}\right)-\frac{{\text{e}}^{-{\text{ik}}_{\text{a}}{\text{R}}_{\text{Q}}}}{{\text{R}}_{\text{Q}}}\frac{\partial \text{U}}{\partial \text{n}}\right]}_{\text{S}}\text{dS}$$

(3)

The integral in equation (3) is the integral theorem of Helmholtz and Kirchhoff for the harmonic sources, and S is an element of the ocean surface or an element on the surface of a scatterer in the volume. If an artifice of transmitting a ping that is short enough to separate the direct and scattered arrivals and still long enough to permit the application of CW theory is used, then the field at Q is determined by the sound which is scattered from the surface elements alone. This is due to the fact that contribution from the source is not present at Q, when the scattered signal is observed. ² Meanwhile, it is possible to use equation (3) to calculate the transmitted sound for a surface element. The geometry is illustrated in Figure 1. The boundary condition for the outward-traveling wave is ²

$${\text{U}}_{\text{w}}\left({\text{Q}}^{\text{′}}\right)={\text{U}}^{\text{′}}\left({\text{Q}}^{\text{′}}\right)=\frac{1}{4\text{π}}{{\displaystyle \int }}_{\text{S}}{\left[{\text{U}}^{\text{′}}\frac{\partial }{\partial \text{n}}\left(\frac{{\text{e}}^{-{\text{ik}}_{\text{w}}{\text{R}}_{\text{Q′}}}}{{\text{R}}_{\text{Q′}}}\right)-\frac{{\text{e}}^{-{\text{ik}}_{\text{w}}{\text{R}}_{\text{Q′}}}}{{\text{R}}_{\text{Q′}}}\frac{\partial {\text{U}}^{\text{′}}}{\partial \text{n}}\right]}_{\text{S}}\text{dS}$$

(4)

where $k_w=\omega /c_w$ is the wave number in water, $c_w$ is the sound speed in water, and ${\text{R}}_{\text{Q′}}$ is $\text{Q′}$ distance from the interface element ds. Although the Helmholtz-Kirchhoff integral expresses the wave field $\text{U(Q)}$ or $\text{U′(Q′)}$ , the problem is still difficult because it is necessary to evaluate these quantities on the surface S. Therefore, through Kirchhoff approximation, the coefficients ℛ, which would be derived for the reflection and transmission of an incident wave at an infinite plane interface, can be used at each point of a rough surface interface. Considering U and $\text{U′}$ , respectively, as the scattered field and the transmitted field, it would be possible to obtain their quantities as a fraction of the initially generated wave field by the source, ${\text{U}}_{\text{s}}$ . ² Therefore, for the transmitted field $\text{U′}$ , it can be written that

$$\text{U′}=\mathcal{T}{\text{U}}_{\text{s}}$$

(5)

$$\frac{\partial \text{U′}}{\partial \text{n}}=-\mathcal{T}\frac{\partial {\text{U}}_{\text{s}}}{\partial \text{n}}$$

(6)

Hence, by utilizing equation (4), the transmitted wave can be expanded and written for the water medium as follows

$${\text{U}}_{\text{w}}\left({\text{Q}}^{\text{′}}\right)={\text{U}}^{\text{′}}\left({\text{Q}}^{\text{′}}\right)=\frac{1}{4\text{π}}{{\displaystyle \int }}_{\text{S}}\mathcal{T}{\left[{\text{U}}_{\text{s}}\frac{\partial }{\partial \text{n}}\left(\frac{{\text{e}}^{-{\text{ik}}_{\text{w}}{\text{R}}_{\text{Q′}}}}{{\text{R}}_{\text{Q′}}}\right)-\frac{{\text{e}}^{-{\text{ik}}_{\text{w}}{\text{R}}_{\text{Q′}}}}{{\text{R}}_{\text{Q′}}}\frac{\partial {\text{U}}_{\text{s}}}{\partial \text{n}}\right]}_{\text{S}}\text{dS}$$

(7)

where $\mathcal{T}$ is a constant or slowly varying over the surface element. It will be possible to use the mean value, and thus move transmission coefficient to outside of the integral and obtain the relation

$${\text{U}}_{\text{w}}\left({\text{Q}}^{\text{′}}\right)=\frac{\mathcal{T}}{4\text{π}}{{\displaystyle \int }}_{\text{S}}{\left[{\text{U}}_{\text{s}}\frac{\partial }{\partial \text{n}}\left(\frac{{\text{e}}^{-{\text{ik}}_{\text{w}}{\text{R}}_{\text{Q′}}}}{{\text{R}}_{\text{Q′}}}\right)-\frac{{\text{e}}^{-{\text{ik}}_{\text{w}}{\text{R}}_{\text{Q′}}}}{{\text{R}}_{\text{Q′}}}\frac{\partial {\text{U}}_{\text{s}}}{\partial \text{n}}\right]}_{\text{S}}\text{dS}$$

(8)

for the transmitted wave. By considering the wave field ${\text{U}}_{\text{s}}$ equal to the generated pressure field by the point source p, the pressure boundary condition at the air–water interface can be written as follows

$$\text{p}={\text{p}}_{1\text{s}}+\mathcal{R}{\text{p}}_{1\text{s}}=\mathcal{T}{\text{p}}_{1\text{s}}$$

(9)

$$\frac{\partial \text{p}}{\partial \text{n}}=\frac{\partial {\text{p}}_{1\text{s}}}{\partial \text{n}}-\frac{\partial \mathcal{R}{\text{p}}_{1\text{s}}}{\partial \text{n}}=\text{i}\left(1-\mathcal{R}\right)\left({\text{k}}_{\text{a}}.^\text{n}\right){\text{p}}_{1\text{s}}$$

(10)

where ${\text{p}}_{\text{1s}}$ is the incident acoustic pressure at element ds, and $\text{n}$ is the normal vector of the interface element ds, as shown in Figure 1. By applying the boundary condition of equation (10), the transmitted pressure field in equation (7) can be derived as in

$${\text{U}}_{\text{w}}\left({\text{Q}}^{\text{′}}\right)=\frac{-\text{i}2\mathcal{T}{\text{p}}_{10}{\text{e}}^{{\text{ik}}_{\text{w}}{\text{R}}_{\text{Q′}}}}{4{\text{πR}}_{\text{Q′}}}{{\displaystyle \int }}_{\text{S}}{\text{k}}_{\text{w}}.{^\text{n}\text{e}}^{\text{i}.\stackrel{r}{K}}\text{dS}$$

(11)

where ${\text{p}}_{\text{10}}$ is the incident acoustic pressure at the air–water interface. Components $\stackrel{=}{K}{\stackrel{a}{K}}_{-}{\stackrel{w}{K}}_{\hspace{0.17em}}$ and vector $\stackrel{\hspace{0.17em}}{r}$ (which is the radius vector to the interface element) are defined in Medwin and Hagy ³⁶ in more detail. For a smooth surface, uniformly illuminated over a rectangular area with width 2X and length 2Y, equation (11) can be solved in order to obtain the acoustic pressure at the receiver location ${\text{\hspace{0.17em}p}}_{\text{rec}\text{.}}$ as

$${\text{p}}_{\text{rec}.}={\text{p}}_{2\text{m}}\text{\hspace{0.17em}F\hspace{0.17em}sinc\hspace{0.17em}}\left({\text{K}}_{\text{x}}\text{X}\right)\text{\hspace{0.17em}sinc}\left({\text{K}}_{\text{y}}\text{Y}\right)$$

(12)

$$\text{sinc}\left({\text{K}}_{\text{x}}\text{X}\right)=\frac{{\text{sinK}}_{\text{x}}\text{X}}{{\text{K}}_{\text{x}}\text{X}}$$

(13)

$${\text{K}}_{\text{x}}={\text{k}}_{\text{a}}\sin {\text{θ}}_1-{\text{k}}_{\text{w}}\sin {\text{θ}}_2\cos {\text{θ}}_3$$

$${\text{K}}_{\text{y}}=-{\text{k}}_{\text{w}}\sin {\text{θ}}_2\cos {\text{θ}}_3$$

$$\text{F}\left({\text{θ}}_1,{\text{\hspace{0.17em}θ}}_2,{\text{\hspace{0.17em}θ}}_3\right)=1+\left[\frac{\left[\left({\text{c}}_{\text{w}}/{\text{c}}_{\text{a}}\right)\sin {\text{θ}}_1-\sin {\text{θ}}_2\right]\tan {\text{θ}}_2.\cos {\text{θ}}_3-\left[\sin {\text{θ}}_2\tan {\text{θ}}_2{\sin }^2{\text{θ}}_3\right]}{\left({\text{c}}_{\text{w}}/{\text{c}}_{\text{a}}\right)\cos {\text{θ}}_1-\cos {\text{θ}}_2}\right]$$

(14)

$${\text{p}}_{2\text{m}}=\left|\frac{-\text{i}2\mathcal{T}{\text{p}}_{10}{\text{e}}^{{\text{ik}}_{\text{w}}{\text{R}}_{\text{Q}}{\text{k}}_{\text{w}}}}{4{\text{πR}}_{\text{Q}}}\Delta \text{A}.{\text{\hspace{0.17em}cosθ}}_2\right|$$

(15)

$$\Delta \text{A}=4\text{XY}$$

(16)

where ${\text{θ}}_{\text{1}}$ , $\hspace{0.17em}{\text{θ}}_{\text{2}}$ , and $\hspace{0.17em}{\text{θ}}_{\text{3}}$ angles are shown in Figure 2. The average received acoustic pressure can be calculated for a statistically rough surface by taking the ensemble average and by assuming that the height distribution of the surface is Gaussian. Therefore, equation (17), which is called the coherent component of the transmitted pressure, can be obtained as follows ³⁶

$${\text{p}}_{\text{rec}.}=\left({\text{e}}^{-\text{R}/2}\right){\text{\hspace{0.17em}p}}_{2\text{m}}\text{F\hspace{0.17em}sinc}\left({\text{K}}_{\text{x}}\text{X}\right)\text{\hspace{0.17em}sinc\hspace{0.17em}}\left({\text{K}}_{\text{y}}\text{Y}\right)$$

(17)

where $\text{R}={\text{K}}_{\text{w}}^2{\text{\hspace{0.17em}σ}}^2{\left(\raisebox{1ex}{${\text{c}}_{\text{w}}$}\!\left/ \!\raisebox{-1ex}{${\text{c}}_{\text{a}}$}\right.\cos {\text{θ}}_1-\cos {\text{θ}}_2\right)}^2$ is the surface acoustical roughness for transmission and $\text{σ}$ is the sea surface RMS height. Also, by evaluating the mean received intensity, the variance of ${\text{p}}_{\text{rec}\text{.}}$ can be calculated by

$$\text{V}\left\{{\text{p}}_{\text{rec}.}\right\}=〈{\text{p}}_{\text{rec}.}{\text{p}}_{\text{rec}.}{\text{*}}^{\hspace{0.17em}}〉-〈{\text{p}}_{\text{rec}.}〉〈{\text{p}}_{\text{rec}.}{\text{*}}^{\hspace{0.17em}}〉$$

(18)

where $\hspace{0.17em}\text{V}\left\{{\text{p}}_{\text{rec}\text{.}}\right\}$ is the variance of ${\text{p}}_{\text{rec}\text{.}}$ . For a Gaussian distribution of heights, it can be stated that

$$〈{\text{e}}^{{\text{iK}}_{\text{z}}\left(\text{ξ}-{\text{ξ}}^{\text{′}}\right)}〉={\text{e}}^{-\text{R}+\text{RC}}$$

(19)

where ξ is the surface height at (x, y) and C is the correlation function, which for a Gaussian correlation length is defined as

$$\text{C}\left(\text{l}\right)={\text{e}}^{-{\text{l}}^2/{\text{L}}^2}$$

(20)

Figure 2.

Illustration of sound transmission into water from a localized point source in the air. $\text{A}$ is the insonified area, while $\text{H}$ , and $\text{D}$ are source height, and receiver depth (with respect to the air–water interface, $\text{A}$ ) respectively.

Here, L is the sea surface correlation length. By utilizing the series expansion for the exponential function and integration, the variance of ${\text{p}}_{\text{rec}\text{.}}$ is obtained as follows

$$\text{V}\left\{{\text{p}}_{\text{rec}.}\right\}={\text{p}}_{2\text{m}}^2{\text{F}}^2\frac{{\text{πL}}^2}{\Delta \text{A}}{\text{e}}^{-\text{R}}{\displaystyle \sum }_{\text{n}=1}^{\infty }\frac{{\text{R}}^{\text{n}}}{\text{nn}!}{\text{e}}^{\left(-{{\text{K}}_{\text{xy}}}^2{\text{L}}^2/4\text{n}\right)}$$

(21)

Therefore, the total relative intensity is obtained by adding the square of equations (17) to (21) and dividing by ${\text{p}}_{\text{2m}}^{\text{2}}$ as in

$$〈{\text{ρρ}}^{\text{*}}〉\equiv \frac{〈{\text{p}}_{\text{rec}.}〉〈{\text{p}}_{\text{rec}.}{\text{*}}^{\hspace{0.17em}}〉}{{\text{p}}_{2\text{m}}^2}={\text{F}}^2{\text{sinc}}^2\left({\text{K}}_{\text{x}}\text{X}\right){\text{\hspace{0.17em}sinc}}^2\left({\text{K}}_{\text{y}}\text{Y}\right)+{\text{F}}^2\frac{{\text{πL}}^2}{\Delta \text{A}}{\text{e}}^{-\text{R}}{\displaystyle \sum }_{\text{n}=1}^{\infty }\frac{{\text{R}}^{\text{n}}}{\text{nn}!}{\text{e}}^{\left(-{{\text{K}}_{\text{xy}}}^2{\text{L}}^2/4\text{n}\right)}$$

(22)

where the first and second terms in equation (22) are called coherent and incoherent terms of transmission, respectively. The TC for a localized point source in the air, which emits sound towards a smooth surface, can be obtained by utilizing the coherent term of the equation (22) as ³⁶

$$\text{TC}=-20\log \left[\left(1+\frac{\text{D}}{\text{H}}\frac{{\text{c}}_2\cos {\text{θ}}_1}{{\text{c}}_1\cos {\text{θ}}_2}\right){\left(2\cos {\text{θ}}_1\cos {\text{θ}}_2\right)}^{-1}\right]$$

(23)

where D and H are, respectively, the receiver depth and source height, and angles $\hspace{0.17em}{\text{θ}}_{\text{1}}$ and ${\text{θ}}_{\text{2\hspace{0.17em}}}$ are shown in Figure 2.

When the point source emits sound towards a rough sea surface, the incoherent component of equation (22) can be dominant. In order to obtain the incoherent term of equation (22), the insonified area on the rough air–water interface, i.e. A, should be divided into subsurfaces and all the incoherent contributions should be added. ³⁹ The requirements for each subarea are: (a) the incident pressure is approximately constant, (b) the spatial variation of the incident phase is negligible (e.g. less than one eighth of a wave length λ), and (c) the linear dimension of the subarea is several correlation lengths in extent. ³⁶ By utilizing equation (21), assumptions (a) and (b) are satisfied. Based on the assumption (c), the spatial correlation should reach a small value to accomplish infinite integral of equation (21). Hence, for a point source, equation (22) can be applied for transmission through an element of area $\Delta \text{A}$ which is now called A_i and multiplied by $({\text{p}}_{\text{2mi}}^{\text{2}}{\text{/p}}_{\text{1si}}^{\text{2}}\text{)}$ . Here, both ${\text{p}}_{\text{2mi}}$ through equation (15) and ${\text{p}}_{\text{1si}}$ (the incident pressure at element A_i) should be determined separately for all the considered interface elements. By writing the sum for many such areas, equation (22) can be determined as follows

$$\begin{array}{c}{\displaystyle \sum }_{\text{i}}\frac{〈{\text{p}}_{2\text{i}}{\text{p}}_{2\text{i}}{\text{*}}^{\hspace{0.17em}}〉}{{\text{p}}_{1\text{si}}^2}={\displaystyle \sum }_{\text{i}}\frac{{\text{p}}_{2\text{mi}}^2}{{\text{p}}_{1\text{si}}^2}{\text{e}}^{-{\text{R}}_{\text{i}}}{\text{F}}_{\text{i}}^2{\text{sinc}}^2\left({\text{K}}_{\text{xi}}{\text{X}}_{\text{i}}\right){\text{sinc}}^2\left({\text{K}}_{\text{yi}}{\text{Y}}_{\text{i}}\right)\\ +{\displaystyle \sum }_{\text{i}}\frac{{\text{p}}_{2\text{mi}}^2}{{\text{p}}_{1\text{si}}^2}\frac{{\text{F}}_{\text{i}}{\hspace{0.17em}}^2{\text{πL}}^2}{{\text{A}}_{\text{i}}}{\text{e}}^{-{\text{R}}_{\text{i}}}{\displaystyle \sum }_{\text{n}=1}^{\infty }\frac{{\text{R}}_{\text{i}}{\hspace{0.17em}}^{\text{n}}}{\text{nn}!}{\text{e}}^{\left(-{\text{K}}_{\text{xyi}}{\hspace{0.17em}}^2{\text{L}}^2/4\text{n}\right)}\end{array}$$

(24)

where ${\text{p}}_{\text{2i}}$ is the transmitted pressure through element A_i to the receiver. Considering the fact that there will be no cross-coupling between coherent and incoherent components from each subarea, an approximate solution of the problem can be obtained. In this condition, the problem becomes the separate evaluation of the terms as

$$\frac{〈{\text{p}}_2{\text{p}}_2{\text{*}}^{\hspace{0.17em}}〉}{{\text{p}}_{1\text{s}}^2}=\frac{〈{\text{p}}_{2\text{s}}{〉}^2}{{\text{p}}_{1\text{s}}^2}+\frac{\text{V\hspace{0.17em}}\left\{{\text{p}}_{2\text{s}}\right\}}{{\text{p}}_{1\text{s}}^2}$$

(25)

where the terms on the right-hand side of equation (25) represent the sum of coherent and incoherent terms on the right-hand side of equation (24), respectively. Medwin and Hagy ³⁶ state that the coherent term of equation (24) for a point source can be written as

$$\frac{〈{\text{p}}_{2\text{s}}{〉}^2}{{\text{p}}_{1\text{s}}^2}\approx {\text{\hspace{0.17em}e}}^{-\text{R}}{\displaystyle \sum }_{\text{i}}\frac{{\text{p}}_{2\text{mi}}^2{\text{\hspace{0.17em}F}}_{\text{i}}^2{\text{\hspace{0.17em}sinc}}^2\left({\text{K}}_{\text{xi}}{\text{X}}_{\text{i}}\right).{\text{sinc}}^2\left({\text{K}}_{\text{yi}}{\text{Y}}_{\text{i}}\right)}{{\text{p}}_{1\text{si}}^2}$$

(26)

This sum is the Kirchhoff-Helmholtz formulation of the relatively transmitted coherent intensity of a point source for a mirror surface. Therefore, the relative coherent transmission along the Snell direction can be obtained as

$$\frac{〈{\text{p}}_{2\text{s}}〉{\hspace{0.17em}}^2}{{\text{p}}_{1\text{s}}^2}\approx {\text{\hspace{0.17em}e}}^{-\text{R}}{\left(1+\frac{{\text{D\hspace{0.17em}c}}_2\cos {\text{θ}}_1}{{\text{Hc}}_1\cos {\text{θ}}_2}\right)}^{-2}{\left(2\cos {\text{θ}}_1\cos {\text{θ}}_2\right)}^2$$

(27)

For the incoherent term, function ${\text{S(R}}_{\text{i}}{\text{,\hspace{0.17em}K}}_{\text{xy}}\text{L)}$ is defined as follows

$$\text{S}\left({\text{R}}_{\text{i}},{\text{\hspace{0.17em}K}}_{\text{xy}}\text{L}\right)\equiv {\text{e}}^{-{\text{R}}_{\text{i}}}{\displaystyle \sum }_{\text{n}=1}^{\infty }\frac{{\text{R}}_{\text{i}}^{\text{n}}}{\text{nn}!}{\text{e}}^{\left(-{\text{K}}_{\text{xyi}}^2{\text{L}}^2/4\text{n}\right)}$$

(28)

The function $\text{S}\left({\text{R}}_{\text{i}}{\text{,\hspace{0.17em}K}}_{\text{xy}}\text{L}\right)$ curves against surface acoustical roughness R for various values of the parameter ${\text{K}}_{\text{xyi}}^{\text{2}}{\text{L}}^{\text{2}}$ can be seen in Medwin and Hagy. ³⁶ Therefore, the relative incoherent intensity in terms of the new function, S, can be rewritten as

$$\frac{\text{V\hspace{0.17em}}\left\{{\text{p}}_{2\text{s}}\right\}}{{\text{p}}_{1\text{s}}^2}={\displaystyle \sum }_{\text{i}}\frac{{\text{p}}_{2\text{mi}}^2{\text{F}}_{\text{i}}{\hspace{0.17em}}^2{\text{πL}}^2\text{\hspace{0.17em}S}\left({\text{R}}_{\text{i}},{\text{\hspace{0.17em}K}}_{\text{xy}}\text{L}\right)}{{\left({\text{p}}_{1\text{si}}\cos {\text{θ}}_1/\cos {\text{θ}}_{1\text{i}}\right)}^2{\text{A}}_{\text{i}}}$$

(29)

Considering ${\text{p}}_{\text{2mi}}$ as

$$\left|{\text{p}}_{2\text{mi}}\right|=\frac{\mathcal{T}{\text{P}}_{1\text{si}}{\text{k}}_{\text{w}}{\text{A}}_{\text{i}}}{2{\text{πr}}_{2\text{i}}}\cos {\text{θ}}_{2\text{mi\hspace{0.17em}}}$$

(30)

the relative incoherent transmitted intensity and the TC for the rough surface can be obtained through equations (31) and (32), respectively.

$$\frac{\text{Var\hspace{0.17em}}\left\{{\text{p}}_{2\text{s}}\right\}}{{\text{p}}_{1\text{s}}^2}={\displaystyle \sum }_{\text{i}}\frac{{\text{π\hspace{0.17em}A}}_{\text{i}}{\mathcal{T}}_{\text{i}}{\hspace{0.17em}}^2{\text{L}}^2{\cos }^2{\text{θ}}_{2\text{mi}}{\text{F}}_{\text{i}}{\hspace{0.17em}}^2\text{\hspace{0.17em}S}\left({\text{R}}_{\text{i}},{\text{\hspace{0.17em}K}}_{\text{xy}}\text{L}\right)}{{\text{λ}}_2^2{\text{\hspace{0.17em}r}}_{2\text{i}}^2{\left(\cos {\text{θ}}_1/\cos {\text{θ}}_{1\text{i}}\right)}^2}$$

(31)

$$\begin{array}{c}\text{TC}=10\log \frac{{\text{p}}_2{\text{p}}_2{\text{*}}^{\hspace{0.17em}}}{{\text{p}}_{1\text{s}}^2}=10\log \left[{\text{e}}^{-\text{R}}{\left(1+\frac{{\text{Dc}}_2\cos {\text{θ}}_1}{{\text{Hc}}_1\cos {\text{θ}}_2}\right)}^{-2}{\left(2\cos {\text{θ}}_1\cos {\text{θ}}_2\right)}^2\right]\\ +{\displaystyle \sum }_{\text{i}}\frac{{\text{π\hspace{0.17em}A}}_{\text{i}}{\mathcal{T}}_{\text{i}}{\hspace{0.17em}}^2{\text{L}}^2{\cos }^2{\text{θ}}_{2\text{mi}}{\text{F}}_{\text{i}}{\hspace{0.17em}}^2\text{\hspace{0.17em}S}\left({\text{R}}_{\text{i}},{\text{\hspace{0.17em}K}}_{\text{xy}}\text{L}\right)}{{\text{λ}}_2^2{\text{\hspace{0.17em}r}}_{2\text{i}}^2{\left(\cos {\text{θ}}_1/\cos {\text{θ}}_{1\text{i}}\right)}^2}]\end{array}$$

(32)

Equation (32) states that, for very low frequencies (R<<1), the summation (incoherent) term will be very small, ${\text{e}}^{\text{-R}}\cong \text{1}$ , and the TC will be given by the geometry and $\rho c$ mismatch. This pressure change may be positive or negative, i.e. a transmission can gain or lose, respectively. As the frequency (and roughness) increases in the regime where 0<R<1, the coherent transmission term will decrease but will still dominate the incoherent term. For $\hspace{0.17em}\text{R}\to 1$ , increasing the incoherent term will lead to determination of the magnitude of TC in equation (32). This will cause an increase in the transmission until a high enough frequency is reached (the value will depend on ${\text{L/λ}}_{\text{w}}$ ) and the ${\text{S(R,\hspace{0.17em}K}}_{\text{xy}}\text{L)}$ term decreases with frequency and causes a greater transmission loss, again.

Simulation of the problem

In ‘Sound transmission through air–water interface’ section, equation (32) which is the summation of coherent and incoherent terms was derived. Coherent term can be calculated easily, but to obtain the incoherent term, it is necessary to divide rough interface into subareas. On the other hand, the effects of subsurface bubble clouds should be involved in TC relation. Therefore, a third medium besides the air and water media named bubbly water medium is considered below the rough interface (Figure 3). Incoherent term contains both surface geometric roughness and subsurface bubble clouds’ mechanisms. Subsurface bubble clouds through ${\mathcal{T}}_{\text{i}}$ component and surface geometric roughness through function S_i are involved in incoherent term and should be determined separately for each surface element. In addition to the surface geometric roughness and subsurface bubble clouds, there are other variables which need to be determined for each surface element. These variables are shown in Table 1.

Figure 3.

Schematic of the sound transmission through a realistic air–water interface at the sea surface.

Table 1.

Required information for each subarea ${\text{A}}_{\text{i}}$ .

Parameter	Description
${\text{A}}_{\text{i}}$	Surface element area
${\mathcal{T}}_{\text{i}}$	Transmission coeff. of surface element
${\text{θ}}_{\text{2i}}$	Angle of transmitted sound through surface element related to normal axis
${\text{F}}_{\text{i}}$	Direction coefficient
${\text{S(R}}_{\text{i}}{\mathrm{,K}}_{\text{xy}}\text{L)}$	Function of roughness and correlation length
${\text{θ}}_{\text{1i}}$	Incident angle related to surface element
${\text{r}}_{\text{2i}}$	Surface element distance from the receiver

In order to determine the appropriate value of the incoherent term, two mechanisms of surface geometric roughness and subsurface bubble clouds should be determined for each surface element. Therefore, the modeling procedure of sound transmission through rough bubbly air–water interface is separately presented for each mechanism in the following subsections.

Modeling the rough surface

When the emitted sound from a localized point source in the air strikes the rough bubbly air–water interface, based on the incoherent term of equation (32), the transmitted sound to the next medium passes through the considered subareas on the rough interface. For each subarea A_i, the designated information in Table 1 should be determined. Each surface element depending on its features plays an individual role in sound transmission. Medwin and Hagy ³⁶ pointed out that dimensions of the surface elements should be determined in a way that two conditions are satisfied. First, the incident sound wave should have uniform phase and amplitude over the elements. Second, the correlation on the boundary of the subareas must become effectively zero. Since the rough interface is divided into elements which can be geometrically irregular, the element mean width is considered to be $\Delta \text{L}$ for each element in order to check these conditions. This matter is discussed quantitatively in the next section. The arrangement of the surface elements is shown in Figure 4. Each element over the rough surface is considered to be 2D and is at a distance ${\text{r}}_{\text{2i}}$ and angle ${\text{θ}}_{\text{1i}}$ from the localized point source in the air. Functions F_i and S_i in the incoherent term of equation (32) can be obtained by determining the individual parameters of the surface acoustical roughness R_i, incident angle $\hspace{0.17em}{\text{θ}}_{\text{1i}}$ , transmitted wave angle $\hspace{0.17em}{\text{θ}}_{\text{2i}}$ , and vector ${\text{r}}_{\text{2i}}$ for each element and overall correlation length L for the rough surface. These parameters for element i and its neighboring elements are shown in Figure 5. Therefore, as pointed out earlier, it is necessary to discretize the rough surface and determine the mentioned parameters of each element. Information is also needed for the surface elements for determining the component ${\mathcal{T}}_{\text{i}}$ which represents the effects of subsurface bubble clouds. The procedure for deriving this term based on the surface elements information is discussed in the next section.

Figure 4.

Part of the discretized rough surface and arrangement of elements. Indices $\text{i}$ and $\text{j}$ represent the horizontal and vertical position of each element over rough surface.

Figure 5.

Geometry of the rough surface transmission for a point source in the air. ${\text{P}}_{\text{1si }}$ is the incident pressure on the subarea ${\text{A}}_{\text{i}}$ _. ³⁶

Modeling the bubbles

Due to the presence of waves (rough interface of air–water, in acoustical point of view) in the real sea surface, subsurface bubble clouds are generated as a result of breaking waves. This thin layer of bubbles below air–water interface can affect the properties of the passing sound and is considered as an important factor for the sea surface in underwater acoustics. ² This layer of bubbles was not considered by Medwin and Hagy ³⁶ in studying the sound transmission through a real air–water interface in the ocean. As described in the next sections, the results of Medwin and Hagy ³⁶ underestimate the TC values compared to the experimental results. The current authors believe that this underestimation is largely due to the omission of the subsurface bubble clouds’ effects. Subsurface bubble cloud, as Ogden and Erskine^38,39 stated, is an individual mechanism in a real sea surface which should be considered in order to determine the sound quality in this region. Therefore, due to the importance of subsurface bubble clouds, this layer is considered as the third medium of the bubbly water (other than air and water media) with its own physical properties such as density ${\rho }_{\text{bw}}$ and speed of sound $c_{\text{bw}}$ .

As mentioned earlier, subsurface bubble clouds through component ${\mathcal{T}}_{\text{i}}$ in the incoherent term of equation (32) play their role in the sound transmission through rough bubbly air–water interface. Index i represents the number of surface elements for which the transmission coefficient should individually be evaluated. When the generated sound by the localized source above the sea surface is emitted towards the rough interface, it first passes the bubbly water and then enters the water medium (Figure 6). As pointed out earlier, since the rough interface is divided into subareas, the emitted sound is transmitted separately by each surface element. Therefore, transmission coefficient ${\mathcal{T}}_{\text{13i}}$ is defined for each element as follows ²

$${\mathcal{T}}_{13\text{i}}=\frac{{\mathcal{T}}_{12\text{i}}{\mathcal{T}}_{23\text{i}}\exp \left(-{\text{i\hspace{0.17em}Φ}}_{2\text{i}}\right)}{1+{\mathcal{T}}_{12\text{i}}{\mathcal{T}}_{23\text{i}}\text{exp}\left(-\text{i}2{\text{Φ}}_{2\text{i}}\right)}$$

(33)

$${\text{Φ}}_{2\text{i}}={\text{k}}_{\text{bw}}{\text{h}}_2\cos {\text{θ}}_{2\text{i}}$$

(34)

where ${\text{k}}_{\text{bw}}$ is the wave number in bubbly water medium, and ${\text{h}}_{\text{2}}$ is its depth. ${\mathcal{T}}_{\text{12i}}$ and ${\mathcal{T}}_{\text{23i}}$ are, respectively, the transmission coefficients of air–bubbly water, and bubbly water–water interfaces which should be individually computed for each surface element. Also, ${\mathcal{T}}_{\text{12i}}$ and ${\mathcal{T}}_{\text{23i}}$ are reflection coefficients of air–bubbly water and bubbly water–water for each surface element, respectively. In order to obtain the transmission and reflection coefficients, when the sound passes from medium x to medium y, equation (35) can be utilized.

$${\mathcal{T}}_{\text{nm}}=\frac{{\text{ρ}}_{\text{m}}{\text{c}}_{\text{m}}\cos {\text{θ}}_{\text{ni}}-{\text{ρ}}_{\text{n}}{\text{c}}_{\text{n}}\cos {\text{θ}}_{\text{mi}}}{{\text{ρ}}_{\text{m}}{\text{c}}_{\text{m}}\cos {\text{θ}}_{\text{ni}}+{\text{ρ}}_{\text{n}}{\text{c}}_{\text{n}}\cos {\text{θ}}_{\text{mi}}}$$

(35)

$${\mathcal{T}}_{\text{nm}}=\frac{2{\text{ρ}}_{\text{m}}{\text{c}}_{\text{m}}\cos {\text{θ}}_{\text{ni}}}{{\text{ρ}}_{\text{m}}{\text{c}}_{\text{m}}\cos {\text{θ}}_{\text{ni}}+{\text{ρ}}_{\text{n}}{\text{c}}_{\text{n}}\cos {\text{θ}}_{\text{mi}}}$$

(36)

Figure 6.

Sound transmission through elements of the rough interface and the Snell’s ray direction.

Here, n and m are the corresponding indices 1, 2, or 3 for the air, bubbly water, and water, respectively (for the bubbly water medium, index $\text{θ}$ should be considered 2m). Angles ${\text{θ}}_{\text{1i}}$ , ${\text{θ}}_{\text{2mi}}$ , and ${\text{θ}}_{\text{3i}}$ are shown in Figure 6 and should be evaluated for each surface subarea A_i, individually. As seen in equations (35) and (36), sound speed and density of the bubbly medium should also be determined.

Sound speed is determined by considering the significant effects of the subsurface bubble plumes on the sound speed and attenuation. According to Van Vossen and Ainslie, ⁴¹ sound speed in the bubbly water can be determined by the relation

$${\text{C}}_{\text{bw}}{-}^2={\text{C}}_{\text{w}}^{-2}+\frac{1}{{\text{πf}}^2}{{\displaystyle \int }}_0^{\infty }\frac{\text{an}\left(\text{a}\right)}{\left(\frac{{\text{a}}^2}{{\text{a}}_{\text{r}}^2}-1+\frac{\text{i}2\text{β}}{\text{ω}}\right)}\text{da}$$

(37)

where ${\text{C}}_{\text{bw}}$ is the complex sound speed and the imaginary part describes the attenuation. ${\text{C}}_{\text{w}}$ is the sound speed in the absence of air bubbles, $\text{ω}$ is the radial frequency, n(a) is the bubble population spectral density (PSD), $\text{β}$ is the damping factor, and ${\text{a}}_{\text{r}}$ which is related to the resonant bubble radius at frequency f is defined by von Vossen and Ainslie. ⁴¹ PSD should be determined for calculating the sound speed in the bubbly water. Different acousticians have considered various approaches to estimate the PSD. For instance, a dominant bubble radius which has the peak density is considered to estimate bubble populations beneath water surface by Von Vossen and Ainslie. ⁴¹ Also, Hall-Novarini’ model (HN) ⁴¹ considers the radius of subsurface bubbles ranging from 10 μm to 1000 μm, based on the ocean environment. It is possible to estimate the bubble populations ⁴¹ from the relation

$$\text{n}\left(\text{a},\text{z}\right)=1.6\times 10{}^4\text{\hspace{0.17em}G}\left(\text{a},\text{z}\right)\text{\hspace{0.17em}U}\left(\text{W}\right)\text{\hspace{0.17em}D}\left(\text{z},\text{W}\right)$$

(38)

where G, U, and D are the functions of dominant bubble radius $\text{a\hspace{0.17em}}(\text{μm})$ defined by Medwin and Clay, ² W (m/s) is the wind speed, and z (m) is the bubbly water depth. The bubble PSD, n (a, z), gives the number of bubbles per unit volume of ocean that have radii within a unit increment in radius and varies with depth, z, bubble radius, a, and wind speed, W. ⁴¹

Von Vossen and Ainslie ⁴¹ proposed a model called extended HN model in which bubbles with larger radius are considered through modifying the term $\text{G}\text{(a,\hspace{0.17em}z)}$ in equation (38). In the current paper, the extended HN model is utilized for estimating the subsurface bubble population. In the extended HN model, the term $\text{G}\text{(a,\hspace{0.17em}z)}$ is defined as follows ⁴¹

$$\text{G}\left(\text{a},\text{\hspace{0.17em}z}\right)=\{\begin{array}{c}{\left(\frac{{\text{a}}_{\text{ref}}\left(\text{z}\right)}{\text{a}}\right)}^4{\text{a}}_{\text{min}}\le a<{\text{a}}_{\text{ref}}\left(z\right)\\ {\left(\frac{{\text{a}}_{\text{ref}}\left(\text{z}\right)}{\text{a}}\right)}^{\text{x}(\text{z})}{\text{a}}_{\text{ref}}\left(\text{z}\right)\le a<1\text{mm}\\ {\left(\frac{{\text{a}}_{\text{ref}}(\text{z})}{1\text{mm}}\right)}^{\text{x}(\text{z})}{\left[\frac{1\text{mm}}{\text{a}}\right]}^{\text{y}(\text{z})}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1\text{mm}\le a<{\text{a}}_{\text{max}}\end{array}$$

(39)

where

$$\text{x}\left(\text{z}\right)=4.37+{\left(\frac{\text{z}}{2.55\text{\hspace{0.17em}m}}\right)}^2$$

(40)

$$\text{y}\left(\text{z}\right)=\text{p}+{\left(\frac{\text{z}}{2.55\text{m}}\right)}^2$$

The constant parameter p has been called tuning parameter by von Vossen and Ainslie ⁴¹ and can be determined by fitting the data. When the sound speed in the bubbly water is determined, it would then be possible to find the reflection and transmission coefficients based on equations (35) and (36), respectively. Van Vossen and Ainslie ⁴¹ explained the detailed process of determining the population of the bubbles based on equation (38). The same approach is adopted in the current paper and Table 2 presents the outcome of the computations based on the Van Vossen and Ainslie’s approach.

Table 2.

Wind speed information on dates of FLIPEX tests.⁴²

Date	Wind speed (m/s)
28 May 1970	3.6
29 May 1970	3
17 August 1970	2.8
18 August 1970	4

The density of bubbly water medium can be determined as follows

$${\text{ρ}}_{\text{bw}}=\left(1-\varphi \right){\text{ρ}}_{\text{w}}+\varphi {\text{ρ}}_{\text{a}}$$

(41)

where ${\text{ρ}}_{\text{bw}}$ , ${\text{ρ}}_{\text{w}}$ , and ${\text{ρ}}_{\text{a}}$ are densities of bubbly water, water, and air, respectively, while $\varphi$ is the volume fraction of bubbly medium. Since it is possible to determine the number of bubbles from equation (38), it is accordingly possible to find the volume fraction of the bubbly water medium $\hspace{0.17em}\varphi$ .

Here, the subsurface bubble properties should be determined based on FLIPEX experimental tests. FLIPEX I experiment (May 1970) and FLIPEX II experiment (August 1970) were performed at a deep sea location 4 and 5 miles NW of San Diego. ³⁷ Table 2 shows the wind speed information related to the mentioned dates.

The integrated bubble PSD at the minimum and maximum wind speeds of the FLIPEX I and II experimental tests are shown in Figure 7. Through PSD, it is possible to obtain ${\text{C}}_{\text{bw}}$ , ${\text{ρ}}_{\text{bw}}$ , and the transmission coefficient ${\mathcal{T}}_{13\text{i}}$ .

Figure 7.

Integrated bubble PSD for the extended HN model based on the conditions of FLIPEX I and II experimental tests.

Algorithm for computing the TC

The required relations and information for computing the TC were presented in the previous sections. Accordingly, it is possible to compute TC by considering two various essential mechanisms of subsurface bubble clouds and surface geometric roughness. The general procedure for calculating TC is shown in Figure 8. Based on the algorithm in Figure 8, sound transmission through air–water and bubbly water interface at the sea surface is simulated by a developed code in FORTRAN programming language which is named Sound Transmission Simulator through Sea Surface (STSSS).

Figure 8.

Flowchart of the Sound Transmission Simulator through Sea Surface (STSSS) program (i is the element’s node and n is the total number of surface elements).

Parametric study

As stated in the previous section, two different mechanisms can be considered for sound transmission through rough bubbly air–water interface. When the surface is smooth (i.e. R≪1), the coherent term based on equation (32) is important. Therefore, TC is due to geometry and the impedance mismatch, and the incoherent term can be neglected. Figure 9 presents FLIPEX II experimental tests and theoretical data of TC versus D/H ratio at a constant normal incidence. In these data sets, source height is considered to be 180 m above the sea surface and the surface acoustical roughness R for the normal incident sound is assumed to be 0.15 and $4.75\times {10}^{-2}$ for frequencies 200 Hz and 100 Hz, respectively. Also, since sound speed in FLIPEX II experimental tests has not been reported, the Mackenzie’s formula ⁴³ is applied to determine the sound speed in water at the location of the FLIPEX II tests and the resulting sound speed is 1473 m/s. It is observed that theoretical and experimental data display good agreement and neglecting the incoherent term in equation (32) seems reasonable. In Figure 10, TC is examined as a function of the incident angle at surface roughness $\text{R}\ll 1$ for three different $\text{D/H}$ ratios. Values of TC in the theoretical method at $\text{D/H}$ ratios of 0.067 and 0.5 compared to the corresponding two experimental tests show good agreement. Also, the theoretical curve for $\text{\hspace{0.17em}D/H}=10$ is presented which has the same trend as the other two theoretical curves, but with lower TC values. It is evident that by applying equation (32) and neglecting the incoherent term for a smooth surface or low roughness ( $\text{R}\ll 1$ ), the resulting TC values have good agreement with the experimental data. However, as pointed out earlier, when the surface roughness increases and becomes R>1, the incoherent term in equation (32) is dominant and controls the total TC value. In this condition, due to the unique role of each separate surface element in the incoherent term of equation (32), the total TC should be calculated by considering the effects of all elements. This calculation requires value of the surface height correlation length, L.

Figure 9.

Variation of TC as a function D/H ratio, for sound propagating at normal incidence and very small acoustic roughness $(\text{R}=1)$ .

Figure 10.

Variation of TC as a function of angle ${\text{θ}}_{\text{20}}$ for different ratios of D/H and in the case of very small acoustic roughness $(\text{R}=1)$ .

Knowing the wind speed over the sea surface makes it possible to find the surface height correlation length, L. For instance, Cox and Munk ⁴⁴ through obtaining the RMS slope of the sea surface as a function of wind speed, calculated the Gaussian correlation length, L. Since the experimental results of FLIPEX tests are considered in this paper, the corresponding correlation length $\hspace{0.17em}\text{L}\approx 1.5\hspace{0.17em}\text{m}$ is adopted for both theoretical and presented numerical approaches. Another important component of the incoherent term of equation (32) which should be determined individually is the area of surface elements A_i. Medwin and Hagy ³⁶ stated that the general limit for minimum value of A_i should be chosen in a way that satisfies two conditions. First, the incident sound wave should acquire uniform phase and amplitude over the subareas. Second, the correlation at the boundary of the subareas should effectively become zero. As seen in equation (32), the TC has two parts: (1) the coherent part and (2) the incoherent part. At higher frequencies (R > 1), only the incoherent part is modified. Also, as seen in Figure 11, variation of the correlation length, L, at the same mean element mean width, ΔL, has more effects than changing the element mean width at the same correlation length. This study is an important matter, as there should be a criterion for discretizing the rough surface. Hence, different conditions are considered here to find the best element mean width for discretizing the rough surface. Based on Figure 11, it can be seen that if the element mean width is changed from ΔL = L to ΔL = 3L and from ΔL = 3L to ΔL = 5L at the same correlation length, L, the results are exactly analogous. However, considering different correlation lengths (L = 75 cm, L = 150 cm, and L= 300 cm) at the same element mean width has significant effects on the value of the resulting TC.

Figure 11.

TC calculated by equation (35) as a function of frequency for a randomly rough ocean surface of RMS height $\text{σ}=0.13\text{m}$ and D/H = 0.067.

Therefore, in order to check these conditions, Medwin and Hagy ³⁶ studied TC values in different correlation length, L and the elements mean width $\hspace{0.17em}\Delta \text{L}$ . Figure 11 shows the TC variation as a function of frequency and for different values of correlation length L and elements mean width $\hspace{0.17em}\Delta \text{L}$ . In Figure 11, the source and receiver positions are considered to be 180 m above the sea surface and 6 m underwater, respectively. Also, the sea surface RMS height is considered to be $\text{\hspace{0.17em}σ}=0.13\hspace{0.17em}\text{m}$ . As evident in Figure 11, changing correlation length from $\text{L}=0.75\hspace{0.17em}\text{m}$ to $1.5\hspace{0.17em}\text{m}$ and then to $3\hspace{0.17em}\text{m}$ with the same $\Delta \text{L}=3\text{L}$ has the most effect on the TC. On the other hand, changing elements mean width from $\Delta \text{L}=\text{L}$ to $\hspace{0.17em}\Delta \text{L}=3\text{L}$ , or $\Delta \text{L}=5\text{L}$ for the constant correlation length $\hspace{0.17em}\text{L}=0.75\hspace{0.17em}\text{m}$ , has the smallest effect on the predicted TC values. Therefore, surface correlation length, L_, compared to the element’s mean width $\hspace{0.17em}\Delta \text{L}$ , is more important and should be properly determined. ³⁶ Since in the current study, the numerical and theoretical calculations of TC are based on FLIPEX II setups, $\text{L}=1.5$ m is adopted for the surface correlation length and the surface elements mean width is considered to be $\Delta \text{L}=5\text{L}$ .

As pointed out earlier, Medwin and Hagy ³⁶ only considered the surface geometric roughness effects on the sound transmission and in their theoretical approach, the effects of subsurface bubble clouds were neglected. Also, through FLIPEX II experimental field study, they reported TC values in different conditions, but their theoretical approach underestimated the TC in some cases. In the current paper, due to the important role of the subsurface bubble clouds in sound transmission, the influence of this thin layer on TC is taken into consideration in equation (32) through component of the transmission coefficient. Since the incoherent term of equation (32) for the surface acoustical roughness, R>1, is dominant, it is necessary to determine the TC values by considering two mentioned mechanisms effects.

In order to perform the numerical simulation based on FLIPEX II setups, the rough sea surface is discretized according to its surface correlation length, surface RMS height, and surface element mean width, as shown in Figure 12. Based on Figure 11, the variation of the correlation length has much more effect on the TC, compared to that of elements mean width. Based on the results in Figure 11, it can be seen that there is no linear relationship between the surface correlation length and the TC. In fact, since TC variation depends on other environmental conditions such as sea surface height, wind speed and also on the point source height and receiver’s position during the tests, one cannot conclude how surface correlation length variation affects the TC. However, as shown in Figure 11, it can be concluded that variation of the elements mean width $\Delta \text{L}$ at different surface correlation lengths does not have significant effects on the resulting TC values. Subsequently, surface correlation length, L, surface element mean width $\Delta \text{L}$ , and surface roughness $\text{σ}$ are considered based on the FLIPEX II experimental tests conditions which are 1.5 m, 7.5 m, and 13 cm, respectively. Air and water densities are considered, respectively, 330 $\text{kg}/{\text{m}}^3$ and 1020 $\text{kg}/{\text{m}}^3$ and sound speed in air and water are assumed to be 330 m/s and 1500 m/s, respectively. Figure 13 shows the experimental data of FLIPEX II analyzed by an analog playback and digital computers along with Medwin and Hagy’s theoretical approach ³⁶ as well as the numerical results of the current study. In the experimental tests, the sound source is considered to be 180 m above the sea surface and over the receiver position. Also, two different underwater receiver locations of 6 m and 90 m are considered in order to change D/H ratio. As evident in Figure 13, the numerical simulation in the current study, which considers subsurface bubble clouds’ effects in addition to the surface geometric roughness role in the incoherent term of equation (32), is in good agreement with the two experimental analyses. On the other hand, since Medwin and Hagy’s theoretical approach ³⁶ only considers the surface geometric roughness effects on the sound transmission and neglects the subsurface bubbles’ population influences on the incident sound, their results underestimate the TC values. Medwin et al. ³⁷ mentioned that this underestimation is due to the incoherent term of equation (32) which is the dominant component in the region R≥1. However, in the region $\text{R}\ll 1$ in which the coherent term is dominant, their theoretical approach behaves more accurately.

Figure 12.

Schematic of sea surface discretisization (x and y axes scale are ${10}^2 \text{m}$ , and z axis scale is $1\text{cm}$ ).

Figure 13.

TC as a function of surface acoustical roughness, in two different D/H ratios 0.067 and 0.5, surface correlation length and RMS wave height are considered 1.5 m and 0.13 cm, respectively.

Figure 14 shows the TC values in a condition that the point source is not directly over the receiver position. In this case, point source is in the air and is offset by 75 m from the position of receivers. Therefore, the two considered receivers in 6 m and 90 m depths in FLIPEX II setups are at angles ${\text{θ}}_{20}=85°$ and 40° with respect to the normal axis. Other required setups in this case study are similar to the mentioned setups for Figure 13. Comparison of the FLIPEX II experimental data, Medwin and Hagy’s theoretical approach, ³⁶ and numerical study of the current paper are presented in Figure 14. Since the surface roughness is considered in the region R>1, the incoherent term of equation (32) plays an important role. Compared to Figure 13, in Figure 14 which considers none normal incidence, the Medwin and Hagy’s theoretical approach ³⁶ has a better agreement with experimental data. As evidenced in Figure 14, the numerical approach presented here offers results which are in agreement with both of the other theoretical and experimental curves.

Figure 14.

TC as a function of surface acoustical roughness, $\text{R}$ , for a point source 180 m above the sea surface. Source is offset 75 m from position of receivers.

Since in this study, wind-generated bubbles are considered, the subsurface bubbles’ population is a function of wind speed (equation (37)). Also the population of bubbles in the bubbly water medium determines the density and sound speed or impedance of this medium. Therefore, in Figure 15, TC is studied at different wind speeds 5 m/s, 10 m/s, 15 m/s and for two various D/H ratios 0.065 and 0.5 in order to examine the subsurface bubble population effects. Frequency range is considered between 50 and 1000 Hz. Based on the presented results in Figure 15, it can be seen that as wind the speed increases, which implies an increase in the subsurface bubbles’ population and surface RMS height, for a constant D/H ratio, the TC decreases. Also, when D/H ratio increases at the same wind speeds and frequency range, TC decreases again. Reduction of TC values at a constant wind speeds and D/H ratios, as the acoustical roughness increases due to variation of frequency within the considered spectral region 50–1000 Hz, can be observed in the presented curves of Figure 15.

Figure 15.

TC variation at the normal incidence and frequency range 50–1000 Hz in three wind speeds 5, 10, and 15 m/s, and D/H ratios 0.067 and 0.5.

Ogden and Erskine^36,37 defined a transition regime for sound scattering from the sea surface in which both subsurface bubbles’ population and surface RMS height are involved in the scattering phenomenon. These two mechanisms are involved in transmission of sound from a realistic sea surface. In addition, the wind speed increase causes an increase in both surface RMS height (according to Pierson-Moskwitz ⁴⁵ empirical relation) and subsurface bubbles’ population, equation (37), which finally result in TC decrease and vice versa. Therefore, it will not be possible to define a separate role for each mechanism in TC variation. However, as mentioned earlier, neglecting the subsurface bubble population effects will result in an underestimation of the TC values.

Figure 16 presents the TC at frequencies 100 Hz, 500 Hz, and 1000 Hz and as shown in Figure 15 for two different D/H ratios 0.067 and 0.5. As shown in Figure 15, in Figure 16, the change in wind speeds in a range of 2–15 m/s at a constant frequency changes the acoustical roughness values, and TC results are studied at a constant frequency and D/H ratio, while the wind speed varies. As evident in Figure 16, by increasing the wind speed, which yields an increase in the subsurface bubble population and RMS wave height at a constant frequency and D/H ratio, the TC values decrease. Also, a frequency increase from 100 Hz to 500 Hz or 1000 Hz has the same effects on TC values and causes its reduction. Since frequency and wind speed, respectively, indirectly determine, through wave number and surface RMS height, the surface acoustical roughness, R, and the wind speed determines subsurface bubbles’ population, one may, based on the results in Figures 15 and 16, conclude that at the normal incidence, the increase in frequency and wind speed causes a reduction in TC values and vice versa. In Figures 15 and 16, TC is investigated at the normal incidence and in a case where receivers are exactly under the source location. However, as mentioned earlier, the angle ${\text{θ}}_{20}$ (Figure 2) is another important factor which directly affect the TC values ( ${\text{θ}}_{20}=0°$ is considered in both Figures 15 and 16).

Figure 16.

TC variation at the normal incidence and in wind speeds between 2 and 12 m/s at three frequencies 100, 500, and 1000 Hz, and D/H ratios 0.067 and 0.5.

In Figure 17, the results of TC in an underwater field with dimensions $250\text{\hspace{0.17em}m}\times 90\text{\hspace{0.17em}m}$ over the plane $y=0$ are presented (see Figure 2). This numerical study is done according to the presented algorithm of Figure 8 in which point source is located over the sea surface H, surface correlation length, L, wind speed, W, surface RMS height $\text{σ}$ are adjusted as FLIPEX II setups. The results of TC are presented in Figure 17(a) based on R=2 and in Figure 17(b) according to R=3. In the presented contours of Figure 17, each grid actually represents a hydrophone which is located at an angle ${\text{θ}}_{20}$ and the TC value is calculated by equation (32) separately for each grid. By comparing Figure 17(a) and (b), it can be seen that at exactly the same environmental surface condition (FLIPEX II setup), the same hydrophone shows a lower TC value in the lower acoustical surface roughness, R. However, in order to examine the effects of angle ${\text{θ}}_{\text{20}}$ variation, it is necessary to examine the TC values at different angles ${\text{θ}}_{\text{20}}$ . Therefore, for an arbitrary and constant depth of a hydrophone H (or grid’s vertical position z), if ${\text{θ}}_{20}$ is increased by moving along the x axis (see Figure 2), it can be seen that TC values generally decrease in both Figure 17(a) and (b). However, in Figure 17(b), which represents the TC values at a higher acoustical roughness R=3, the reduction of TC values at the same depth when ${\text{θ}}_{\text{20}}$ increases is slower compared to Figure 17(a) which presents the TC values according to R=2.

Figure 17.

TC underwater, sound source is located at x = 0 and z = 90 m: (a) $f=560 \text{Hz}$ , $R=2$ , (b) $f=690 \text{Hz}$ and $R=3$ .

Conclusion

In this paper, transmission of low frequency sound generated by a localized point source is investigated in the air through rough bubbly air–water interface. To accomplish this goal, the modified Kirchhoff-Helmholtz integral is applied. In general, two mechanisms of subsurface bubble clouds and surface geometric roughness are involved in sound transmission in a realistic sea surface. In this paper, a modified Kirchhoff-Helmholtz integral is presented in which subsurface bubbles’ population effects are accounted for. The proposed optimization of Kirchhoff-Helmholtz integral is achieved through inserting a transmission coefficient ▪ into the corresponding relation. Therefore, a three-phase region of air, bubbly water, and water at the sea surface region is considered. As a result, sound transmission from air into water passes through middle medium of bubbly water. In the considered air, water, and bubbly water media, a relation is derived for TC based on the transmitted pressure to the water from the point source localized in the air. The relation is studied versus different effective variables such as wind speed W, frequency f, source to receiver positions ratio D/H, and acoustical surface roughness R. Summation of the two components of coherent and incoherent terms forms the final TC relation. At low surface acoustical roughness R≪1, the coherent term of the TC relation is dominant, but at higher acoustical roughness R>1, the incoherent term is dominant. Since the incoherent term is formed by summation of functions of the surface elements, various terms such as element area A_i, transmission coefficient ${\mathcal{T}}_{\text{i}}$ , incident angle ${\text{θ}}_{1\text{i}}$ , transmitted angle ${\text{θ}}_{\text{3i}}$ , local surface acoustical roughness R_i, among others are needed to discretize the rough surface of air–bubbly water interface. The stated terms should be computed for each element individually in order to obtain the total incoherent term of TC.

Due to the presence of the wind-generated subsurface bubble clouds in a realistic sea condition, the propagation of sound in the middle bubbly water medium faces different impedances, compared to the air and water impedances. Therefore, transmission of sound is affected by the features of this medium. In this paper, by including the subsurface bubble clouds’ effects, the underestimation of TC values reported by earlier studies is resolved. Also, the parametric studies of TC versus various variables are conducted for different setups of the numerical simulation presented in the current paper. Based on these studies, it is concluded that at normal incidence, frequency increase at a constant wind speed and vice versa will result in the reduction of TC for a constant D/H ratio. Also, an increase in D/H ratio at a constant surface acoustical roughness R will result in the reduction of TC. However, at a constant D/H ratio, the increase in angle ${\text{θ}}_{20}$ is more effective on the reduction of the TC values at lower surface acoustical roughness $\text{R}$ and its increase at the higher surface acoustical roughness reduces the TC values, more slowly.