# Dielectric and Radiative Properties of Sea Foam at Microwave Frequencies: Conceptual Understanding of Foam Emissivity

^{*}

## Abstract

**:**

## 1. Introduction

_{f}.

_{f}at microwave frequencies is high, close to that of a blackbody [14,19–22]. To explain and model e

_{f}, one needs to understand the radiative processes taking place in foam. Knowledge of the mechanical and dielectric properties of sea foam is necessary for such an understanding. While extensive oceanographic research facilitates the characterization of the foam mechanical properties (Chapters 4.4 and 4.5 in [23]), the knowledge of the foam dielectric properties is incomplete.

_{f}, the formulation of relevant assumptions and/or simplifications, and the interpretation of the model results.

_{f}. Anguelova and Gaiser [25] focused on the skin depth of foam layers with a vertical void fraction profile. In the present paper, we consider the foam properties at microwave frequencies, which control fundamental radiative processes such as reflection, scattering, and transmission in vertically structured foam layers. The parameters used for this consideration are foam impedance η, size parameter x, and refractive index m (Section 3). Analysis of these dielectric properties helps to identify the unique traits, which the fundamental radiative processes acquire when microwave radiation interacts with sea foam (Section 4). Finally, we put all our findings together to formulate a concept of the high foam emissivity (Sections 5.1 and 5.2) and discuss scattering regimes in foam (Section 5.3). The information, findings, and generalizations in this series of papers form the physical basis on which we have built our foam emissivity model (briefly described in [16] and to be detailed in a forthcoming paper).

## 2. Background

#### 2.1. Sea Foam as a Medium

_{a}(defined as the fraction of a unit volume of seawater that is occupied by air) and foam layer thickness t can describe this mechanical structure. A set of microscopic quantities such as bubble dimensions (radius a and wall thickness w) and concentration or size distribution N(a) are useful when one considers the characteristics of the bubbles forming the foam.

_{a}(from 0 to 1) needs to be considered [24]. It was shown that various functional forms could represent the shape of the void fraction profile in the foam depth [25]. A review of oceanographic data from laboratory and field experiments [25] established a plausible range of foam layer thicknesses encountered in the ocean, from 1 cm to more than 12 cm in active whitecaps and from 0.1 to 1 cm when the whitecaps decay.

#### 2.2. Bubbles in a Foam Layer

#### 2.3. Definitions of Electromagnetic Properties

_{f}at microwave frequencies F from 1.4 to 37 GHz. The formulae used are valid for scatter free medium, but are applicable to sea foam because a survey of experimental and modeling observations has shown that at the frequencies of interest the scattering in foam is weak to negligible (Section 2.3 in [24]).

_{ref}, where σ is the standard deviation of the surface height (or RMS height) and the reference value is ${\sigma}_{\mathit{ref}}=\frac{\lambda}{32\hspace{0.17em}\text{cos}\hspace{0.17em}\theta}$, with λ being the wavelength and θ the incidence angle of the EM radiation. The other criterion expresses σ in electromagnetic units with the quantity kσ where k = 2π/λ is the radiation wavenumber.

## 3. Foam Dielectric Properties

#### 3.1. Sea Foam Permittivity

_{f}at microwave frequencies [24]. For consistency with the study on the foam skin depth [25], here we use ε

_{f}obtained with the Polder-van Santen (PS) mixing rule (Equation (1) in Table 1). The calculations use a double Debye model for the complex dielectric constant of seawater ε [33]. Refer to [24] for the dependencies of ε

_{f}obtained with the PS mixing rule on the frequency, sea surface temperature (SST), and salinity. All results are illustrated with an exponential profile for the foam void fraction f

_{a}(z) with values at the air-foam and foam-water boundaries of 99% and 1%, respectively. Refer to [25] for comparison of exponential f

_{a}(z) to other functional forms of the void fraction profile.

#### 3.2. Foam Skin and Penetration Depths

_{a}(z) profile (Section 3.1), d could vary from 0.17 cm to no more than 7 cm. Analysis of the relationship between d, t, and the f

_{a}range (upper to lower values at the foam layer boundaries) showed that the thermal emission from foam-covered surfaces could be from the entire foam layer, from part of the foam layer, or from both the foam layer and the seawater.

_{a}values. For f

_{a}= 90%, δ ranges from a few centimeters to a few meters; for f

_{a}< 60%, δ is mostly less than 1 cm.

#### 3.3. Foam Impedance

_{a}= 98%, dash-dot line) and wet (constant profile of f

_{a}= 10%, dashed line) foam. The impedances of the seawater (solid line) and air (dotted line) are shown for comparison. Figure 2(b) depicts how the foam impedance changes as a function of f

_{a}in a vertically structured foam layer for three frequencies. Over the range of considered frequencies, foam impedance varies from 1 to less than 0.2 as f

_{a}decreases from 100% to 0%.

#### 3.4. Wavelength Changes in Foam Layers

_{f}and propagation constant k

_{f}as compared to that in air λ

_{0}(Equations (5) and (6) in Table 1). Table 2 shows values of λ

_{f}in dry and wet foam represented with f

_{a}= 98% and 10%, respectively, for all considered frequencies. The corresponding values in seawater λ are also included for reference.

_{f}from λ

_{0}to λ as it propagates through foam with varying void fraction. In the figure, for each frequency we discern three regions in the trend of λ

_{f}based on the rate of change ∂λ

_{f}/∂f

_{a}. In dry foam, λ

_{f}varies slowly remaining close to λ

_{0}; ∂λ

_{f}/∂f

_{a}changes by a factor of 1.4 (1.1) at 1.4 (37) GHz as void fraction changes from 80% to 100%. For void fractions below 40%, λ

_{f}is quite close to λ and also changes relatively slowly. The slope ∂λ

_{f}/∂f

_{a}changes by a factor of 2.6 on average over the range of frequencies when f

_{a}varies from 40% to 10%. A steeper transition between the air-like and seawater-like behavior takes place in the void fraction range of 80% to 40% with ∂λ

_{f}/∂f

_{a}changing on average by a factor of 4.8. The lower the frequency the more clearly these three regions are seen.

_{a}values with vertical lines in Figure 3. While we base this conceptual representation on different rates of change of λ

_{f}values, Raizer (Figure 1(a) in [39]) gives a similar schematic of a vertically stratified foam based on experimental observations [28].

#### 3.5. Scattering Parameters of Foam

#### 3.5.1. Characteristic Bubbles for Each Foam Sublayer

_{a}and λ

_{f}) are related to those of a separate bubble (e.g., f

_{b}and λ

_{b}).

_{a}values (Section 3.4) could be alternatively differentiated by expected specific, predominant bubble radius a (Figure 3).

_{a}of a collection of N bubbles with a fixed outer radius a and a wall (shell) of seawater w = a – qa, where qa is the inner radius of a bubble, is

_{b}≈ 97%. Since a bubble with such dimensions (large a, thin wall w) is characteristic for the dry foam sublayer represented with f

_{a}= 98%, f

_{b}can characterize the mixture’s f

_{a}. Similarly, the void fractions of smaller bubbles, e.g., a = 0.3 mm and 0.1 mm, having walls with a thickness w of 0.05 mm (i.e., q = 0.83 and 0.5 respectively) are representative of foam with f

_{a}≅ f

_{b}= q

^{3}≈ 60% and 12%, respectively.

#### 3.5.2. Size Parameter in a Foam Layer

_{f}and the dominant bubble size a change.

_{a}of a foam mixture comprising bubbles with a fixed radius a = 0.3 mm (a size within the peak of the bubble size distribution, Section 2.2). Changes of f

_{a}in the depth of foam comprising bubbles with the fixed radius a are realized via variations of bubble wall thickness w which yield q

^{3}= f

_{a}(Section 3.5.1).

_{f}values shown in Table 2. For instance, at 37 GHz the value of x at f

_{a}= 98% changes by a factor of 3.8 compared to the values of x at f

_{a}= 10%, while at 10.7 GHz this change is by a factor of 6.6. Table 3 summarizes the ranges of variation of x in foam thickness using the values of the predominant bubble size for each sublayer (Columns 1 and 2 in Table 3).

#### 3.5.3. Refractive Index of Foam

_{a}= 98%) m′ ≈ 1 and m″ features the lowest values of O(10

^{−3}). As the seawater content increases in foam depth, so does the refraction index (Table 3): at f

_{a}= 60%, m′ = O(3) and m″ = O(1); at f

_{a}= 10% m′ = O(5 to 9) and m″ = O(3).

#### 3.6. Roughness of Foam Layer Interfaces

_{af}= 1.3 mm. At the foam-water interface, we estimate σ

_{fw}= 0.2 mm using the small-size end of the bubble data, radii from 0.05 mm to 1 mm. For σ

_{ref}and kσ (Section 2.3) we use λ

_{0}and λ

_{f}at a void fraction of 10% (Table 2) for the air and water interfaces, respectively.

_{ref}(curved solid line) of the Fraunhofer criterion changes from about 10 mm at 1.4 GHz to 0.4 mm at 37 GHz. Compared to the rms height due to bubbles at this boundary, σ

_{af}= 1.3 mm (horizontal solid line), the Fraunhofer criterion identifies dry foam as a smooth surface for frequencies below approximately 12 GHz (vertical solid line). For higher frequencies, the dry foam would be a rough surface. Approximately the same frequency limit of 12 GHz divides the wet foam into smooth and rough surfaces for the foam-water interface (dashed lines).

## 4. Radiative Processes in Foam

#### 4.1. Foam Reflection and Transmission

#### 4.1.1. Foam as Impedance Matching

_{0}= 1 − i.0) and seawater, which leads to very different intrinsic impedances (Figure 2(a), solid and dotted lines). Because the impedance difference between two media determines the reflectivity at their interface, r ∝ Δη (e.g., p. 228 in [35]), the reflectivity at the air water boundary is large. Thus, only a small portion of down-welling atmospheric radiation impinging on the air water interface would pass and be available for absorption within the foam-seawater system. As Kirchoff’s law holds, a foam free water surface has low emission. The cartoon in Figure 7(a) illustrates this.

_{a}≅ 98%) at the top of the foam layer has dielectric properties close to those of air, while wet foam (f

_{a}≅ 10%) at the bottom of the foam layer has dielectric properties similar to those of seawater [25]. Accordingly, the intrinsic impedances for wet and dry foam are close to those of seawater and air, respectively, Figure 2(a) (dashed and dash-dotted lines). Therefore, as the foam void fraction gradually changes from the top of the foam layer to its bottom, it provides a seamless matching of the dielectric properties of air and seawater, Figure 2(b). This action of a foam layer on the water surface is the natural counterpart of the technology used to reduce reflection in optical systems ([42], Chapter 2, §16). That is, the foam layer acts as an antireflective coating on the sea surface allowing effective transmission of down-welling atmospheric radiation into the foam-seawater system. As more of the transmitted radiation is absorbed in the foam-water system, more is emitted, ultimately resulting in high emissivity from foam-covered water surfaces. The cartoon in Figure 7(b) illustrates this situation.

#### 4.1.2. Effectiveness of Impedance Matching

_{a}range shifted toward lower values, e.g., f

_{a}∈ (<60% to 0%) ([25], Section 4.4). Figure 2(b) suggests, however, that even foam with a limited f

_{a}range could provide impedance matching. It may not be the most effective, but still it could mediate some decrease in the impedance mismatch between air and seawater. The most important requirement therefore for the impedance matching is the foam to float on the surface.

#### 4.2. Surface Scattering of Foam

_{af}and σ

_{fw}used in Figure 6 to the rms height of a wind-roughened sea surface σ

_{u}. We obtain σ

_{u}from the variance of the sea surface displacement h about the mean surface at a given point. It is defined as the zeroth moment of the wave spectrum S(K), ${\sigma}_{u}\equiv {m}_{0}=\underset{{K}_{\mathit{low}}}{\overset{{K}_{\mathit{up}}}{\int}}S(K)dK=\langle {h}^{2}\rangle $ [43,44], where K = 2π/Λ is the wave number of an ocean wave with wavelength Λ. Note that σ

_{u}is a measure of the geometric roughness of a wind-driven sea surface, not the aerodynamic roughness usually represented by the roughness length z

_{0}[45,46].

_{low}and K

_{up}above determines the wavelength scales contributing to σ

_{u}. But these length scales, determined mostly by geometric consideration, also need to be tied to length scales important for the aerodynamic surface roughness because this is the variable used for air-sea interaction studies. Our reasoning in choosing K

_{low}and K

_{up}is thus as follows. First, note that remote sensing of the sea surface is effective when the probing EM wavelengths λ

_{0}are comparable to the length scales of the ocean waves. For example, for scatterometers, ocean length scales ranging from about 40 cm to less than 2 cm correspond to λ

_{0}from 25 cm to 3 cm at incidence angles of 20° and 60° ([36], p. 1705). On this premiss, one can expect the probing wavelengths λ

_{0}for the range of frequencies investigated here (Table 2) to be most relevant to ocean lengths scales from about 40 cm to a few millimeters. These length scales represent short gravity and capillary water waves. Next, note that analysis of the mean square slope of wave number spectrum, resulting from oceanographic surface wave measurements, has shown that intermediate-scale waves with Λ between 6 m and 2 cm are the dominant contributors to the aerodynamic surface roughness in oceans [47]. Combining these two considerations, we restrict our estimates to ocean length scales from 40 cm to 2 cm. The corresponding wavenumbers K range from 3.14 rad·cm

^{−1}to 0.16 rad·cm

^{−1}. To cover this range of K values, we obtained m

_{0}for three portions of the wave spectrum: S

_{2}(K), S

_{3}(K), and S

_{4}(K) as defined in [43]. In the estimates we used a friction velocity u

_{*}of 24 cm·s

^{−1}(U

_{10}≈ 7 m·s

^{−1}), a value close to the globally averaged wind.

_{u}ranges from about 1 mm for the short end of water waves (Λ of 2 cm) to 5.4 mm for longer water waves (Λ of 40 cm). We obtain similar estimates for σ

_{u}if an empirically formulated wave spectrum in terms of mean square slope is used [47] instead of S(K).

_{af}= 1.3 mm implies that there could be cases in which the roughness of the dry foam could be comparable to the roughness created by the shortest capillary waves. But the dielectric contrast between dry foam and air is almost none existing (Figure 2 and Section 4.1). This leads ultimately to lack of surface scattering at the air-foam boundary.

_{fw}= 0.2 mm) is an order of magnitude below that of the shortest capillary waves (1 mm). This implies that wet foam, whether at the surface or as a sublayer in the foam thickness, is usually smoother than the sea surface roughness. We thus deem the wet foam boundary as incapable of providing significant surface scattering as well.

#### 4.3. Weak Volume Scattering Throughout a Foam Layer

#### 4.3.1. Expected Scattering in Foam Layers

_{a}values the integral effect of a vertical f

_{a}profile is to limit d [25], we infer from Figure 1 that significant volume scattering could be expected in active whitecaps as they involve more dry foam and thus have the largest penetration depth. Because the penetration depth of intermediate-wet and wet foam (f

_{a}≤ 60%, e.g., residual whitecaps) is much smaller, the inhomogeneities within could be assumed to contribute little to volume scattering.

_{f}<< a or x >> 1) the more effective the scattering ([37], p. 68); and (ii) the larger the dielectric contrast between the refractive indices of the particle and the surrounding medium the stronger the scattering. That is, in the case of particles in air, there is no scattering for m = 1 and scattering is small for m near 1 ([37], p. 172). We apply these rules for particles to foam on the basis of the connection between foam layers and bubbles (Section 3.5.1).

#### 4.3.2. Altered Scattering in Foam Layers

_{f}and the number of bubbles with radii effective for scattering keeps the x values relatively low and confined in a narrow range. As a result, the scattering in foam remains less effective. This is compounded with the effect of the dielectric contrast, which is expressed via m increasing in foam depth yet staying adjacent to sublayers with similar m values. The scattering in wet foam is somewhat more effective than the scattering in dry foam, however the thickness, thus the penetration depth, of wet foam restricts this further. The overall outcome is weak scattering throughout the foam.

#### 4.4. Strong Absorption by Wet Foam Revisited

_{f}) of foam is a complex number (m = m′ – im″, Section 3.5.3). Media with even very small m″ values have been called strongly absorbing. On the basis of m″ ≠ 0 and m – 1 << 1, which means that a negligible fraction of the radiation is reflected, one could conjecture that even dry foam (Figure 5, solid lines) would be absorptive. In fact, a combination of refractive index values similar to that of dry foam with large size parameter (e.g., x ≥ 100) could represent a medium whose behavior approaches that of a black body ([37], p. 269).

## 5. Discussion

#### 5.1. Concept for the High Foam Emissivity

_{f}of sea foam. Rather, all foam properties and all radiative processes engage in an emergent behavior (i.e., the whole is greater than the sum of its parts [53]) and contribute in some way to bring about the overall effect of a high e

_{f}. Moreover, we argue that what makes the sea foam an exceptional emitter is the combination of these properties and processes in a vertically structured foam layer floating on seawater surfaces. The following is a summary of our conceptual understanding of how this happens.

_{f}could not be guaranteed. The potential of the foam-seawater system for strong absorption and emission cannot be realized without effective transmission and propagation of radiation across layer boundaries into the foam; and vice versa—effective transfer across the layer boundaries is not enough if a highly absorptive agent is not in place to carry on high absorption and emission.

#### 5.2. Foam as a Dynamic System

#### 5.3. Modeling Scattering in Foam

#### 5.3.1. Scattering Regimes in Foam

#### 5.3.2. Variations of the Scattering Regimes in Foam

_{a}≤ 100%) to Rayleigh scattering in wet foam (0% ≤ f

_{a}≤ 40%) with a gradual transition between these two in the intermediate-wet foam (40% ≤ f

_{a}≤ 80%). Other possible reasons that could change the values of the scattering parameters are: (i) Changes in m due to full account of the scattering in ε

_{f}; (ii) changes in x due to SST effect (at fixed salinity) on ε

_{f}and with this on λ

_{f}; and (iii) changes in x due to salinity effect (at fixed SST) on a and ε

_{f}(thus λ

_{f}).

_{e}by overestimating or underestimating it. Because of the direct proportionality ${k}_{e}\propto \left|\text{Im}\left(\sqrt{{\epsilon}_{f}}\right)\right|={m}^{\u2033}$ (Equations (2), (3), and (8) in Table 1), m″ of intermediate-wet and wet foams (Table 3) would increase or decrease. Meanwhile the values of Re{m} are expected to change little compared to those reported in Table 3.

_{f}may increase λ

_{f}noticeably at lower SST ([24], Figure 6). Such a result would decrease size parameter x, which would confine the scattering in foam to the transitional (R-RG in Figure 9) or Rayleigh regimes.

_{f}(via the salinity dependence of ε

_{f}) and the bubble size a. Seawater with lower salinity (at fixed SST) breaks into fewer but larger bubbles (e.g., [58], Figure 5). Meanwhile, Anguelova ([24], Figure 8) shows that, as salinity decreases, the real part of foam permittivity increases, negligibly so for higher frequencies (18 GHz to 37 GHz) and more noticeably for lower (1.4 GHz to 10 GHz). Both changes—shift of bubble size distribution toward larger sizes and shorter wavelengths—lead to an increasing x at lower salinity. In addition, lower salinity values decrease seawater conductivity thus the imaginary part of foam permittivity and refraction index m. The real parts of ε

_{f}and m would, however, keep the m − 1 values no higher than about 10. Therefore, at lower salinity the scattering regime would shift more toward the Rayleigh-Gans (Born approximation) regime. Similar reasoning points out that higher salinity reinforces conditions for predominantly Raylegh scattering in foam.

## 6. Conclusions

_{f}. To model e

_{f}, knowledge of mechanical and dielectric properties of sea foam is necessary. Here we present a full physical description of the dielectric properties of sea foam layers with a vertical profile of void fraction. These include foam dielectric constant, foam skin depth, foam impedance, wavelength variations in foam depth, roughness of foam layer interfaces with air and seawater, and foam scattering parameters such as size parameter, and refraction index.

- Weak or negligible reflection at the foam layer interfaces with air and seawater due to foam impedance matching.
- Floating foam layers provide the most effective impedance matching.
- Weak or negligible surface scattering at the foam layer interfaces due to small roughness and low dielectric contrast of foam boundaries.
- Weak volume scattering throughout the foam due to simultaneous decrease in the foam depth of radiation wavelength and bubbles’ radii effective for scattering.
- Absorption losses are predominantly confined to the wet portion of the foam layer.

## Acknowledgments

## Appendix: Scattering Regimes Mapped in m − x Domain

_{a}, Q

_{s}, and Q

_{e},) are formulated in terms of size parameter x and refractive index m. Values of x could be from 0 to ∞; m varies from 1 to ∞ for scatterers in vacuum or could be less than 1 in other surrounding media.

**Table A.**Approximate scattering theories for boundary regions of the m − x domain (modified from Table 8 in van de Hulst ([37], p. 133).

Region | x | m-1 | x(m − 1) | Regime |
---|---|---|---|---|

RG (1) | arb | s | s | Rayleight-Gans |

AD (2) | l | s | arb | Anomalous Diffraction |

GO (3) | l | arb | l | Geometric optics |

TR (4) | arb | l | l | Total Reflector |

OR (5) | s | l | arb | Optical Resonance (Mie) |

RS (6) | s | arb | s | Rayleigh scattering |

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**Figure 1.**Foam penetration depth as a function of frequency δ (F) obtained with Polder-van Santen mixing rule with exponential void fraction profile ranging from 99% at the air-foam interface and 1% at the foam-seawater boundary at fixed seawater temperature (T

_{s}= 20 °C) and salinity (S = 34 psu) at three values of foam void fraction f

_{a}.

**Figure 2.**Normalized intrinsic impedance of sea foam obtained from foam permitivity (Equation (4) in Table 1) at fixed seawater temperature (T

_{s}= 20 °C) and salinity (S = 34 psu): (

**a**) as a function of frequency for seawater, air, wet foam (f

_{a}= 10%), and dry foam (f

_{a}= 98%); (

**b**) as a function of foam void fraction at three frequencies.

**Figure 3.**Wavelength in foam λ

_{f}(cm) (Equation (5) in Table 1) as a function of foam void fraction at fixed seawater temperature (T

_{s}= 20 °C) and salinity (S = 34 psu) and different frequencies. Vertical lines divide the f

_{a}range into three regions whose values can be associated with bubble diamaters predominant for each region.

**Figure 4.**Size parameter in foam x (Equation (7) in Table 1) as a function of foam void fraction at various frequencies and bubble radius a = 0.3 mm at the peak of the bubble size distribution obtained from oceanographic measuremetns.

**Figure 5.**Foam refractive index m (Equation (8) in Table 1) as a function of frequency at foam void fractions representing dry foam (f

_{a}= 98%), wet foam (f

_{a}= 10%) and intermediate stage foam (f

_{a}= 60%) at fixed seawater temperature (T

_{s}= 20 °C) and salinity (S = 34 psu): (

**a**) Real part, Re{m}; (

**b**) Imaginary part, |Im{m}|.

**Figure 6.**Criteria for surface smoothness (Section 2.3) at the air-foam boundary (solid lines) and foam-water boundary (dashed lines): (

**a**) Fraunhofer criterion; (

**b**) kσ criterion.

**Figure 7.**Schematic representation of reflection and transmission of incident (left) and emitted (right) radiation under different conditions at air-sea interface: (

**a**) foam-free interface; (

**b**) foam-covered interface; (

**c**) bubbly mixture in the water without foam on the surface.

**Figure 8.**Size parameter in foam x (Equation (7) in Table 1) as a function of frequency and foam void fraction representing: (

**a**) dry foam (f

_{a}= 98%); (

**b**) intermediate-wet foam (f

_{a}= 60%) and (

**c**) wet foam (f

_{a}= 10%). For each f

_{a}region the size parameters for three possible bubble radii are shown. The radii least likely for each type of foam are in gray. Fixed seawater temperature (T

_{s}= 20 °C) and salinity (S = 34 psu).

**Table 1.**Basic relationships from the electromagnetic theory used to obtain foam dielectric properties.

Equation # | Property | Symbol (Units) | Formula | Reference | Notes |
---|---|---|---|---|---|

Dielectric properties | |||||

(1) | Dielectric constant | ε_{f} |
$$\frac{{\epsilon}_{f}-\epsilon}{{\epsilon}_{f}+2\epsilon +2\left({\epsilon}_{f}-\epsilon \right)}={f}_{a}\frac{1-\epsilon}{1+2\epsilon +2\left({\epsilon}_{f}-\epsilon \right)}$$
| Equation (9.7) in [34] | Seawater is environment with permittivity ε; Bubbles are inclusions with void fraction f _{a} |

(2) | Skin depth | d (mm) |
$$\begin{array}{c}\underset{0}{\overset{d}{\int}}\alpha (z)dz=1\\ \alpha (z)=\frac{2\pi F}{c}.\left|\text{Im}\sqrt{{\epsilon}_{f}(z)}\right|\end{array}$$
| p. 847 in [32] | α field attenuation coefficient F frequency (Hz) c speed of light (cm s ^{−1}) |

(3) | Penetration depth | δ (mm) |
$$\begin{array}{c}\underset{0}{\overset{\delta}{\int}}{k}_{a}(z)dz=1\\ \delta =d/2\hspace{0.17em}@\hspace{0.17em}{\epsilon}_{f}(z)=\mathit{const}\end{array}$$
| ditto | Scattering ignored Extinction ≅ Absorption i.e., k _{e} ≅ k_{a} = 2α |

(4) | Intrinsic impedance | η |
$$\eta =1/\sqrt{{\epsilon}_{f}}$$
| p. 226 in [35] | Normalized (relative), complex |

(5) | Wavelength in foam | λ_{f} (cm) |
$${\lambda}_{f}={\lambda}_{0}/\sqrt{{{\epsilon}^{\prime}}_{f}}$$
| p. 1453 in [36] | λ_{0} free-space wavelength |

(6) | Propagation constant (wave number) | k_{f} (cm^{−1}) |
$${k}_{f}=2\pi /{\lambda}_{f}$$
| p. 116 in [37] | |

Scattering parameters | |||||

(7) | Size parameter | x |
$$x=a\cdot \left(2\pi /{\lambda}_{f}\right)={k}_{f}a$$
| p. 128 in [37] | a bubble radius |

(8) | Refraction index | m |
$$m=\sqrt{{\epsilon}_{f}}={m}^{\prime}-i{m}^{\u2033}$$
| p. 116 in [37] | m′ and m″ real and imaginary parts of m |

**Table 2.**Wavelengths at fixed SST (20 °C) and salinity (34 psu) in air (λ

_{0}), in seawater (λ), in dry and wet foam (λ

_{f}).

F GHz | λ_{0} cm (in air) | λ_{f} cm | λ cm (in seawater) | |
---|---|---|---|---|

f_{a} = 98% | f_{a} = 10% | |||

1.4 | 21.4 | 20.8 | 2.762 | 2.55 |

6.8 | 4.4 | 4.28 | 0.603 | 0.56 |

10.7 | 2.8 | 2.72 | 0.414 | 0.38 |

18.7 | 1.6 | 1.56 | 0.286 | 0.26 |

23.8 | 1.3 | 1.22 | 0.253 | 0.23 |

37.0 | 0.8 | 0.79 | 0.207 | 0.19 |

**Table 3.**Ranges of variation of size parameter (x), and foam refractive index (Re{m} and |Im{m}|) in foam depth.

Foam Mixture | x (Figure 4) | Re{m} (Figure 5(a)) | |Im{m}| (Figure 5(b)) |
---|---|---|---|

f_{a} = 98%a = 10 mm | 0.2–10 | ≅1 | 6 × 10^{−4}−3 × 10^{−3} |

f_{a} = 60%a = 0.3 mm | 0.02–0.6 | 3.5–2.5 | 0.6–1 |

f_{a} = 10%a = 0.05 mm | 0.01–0.2 | 9–4.5 | 2–4 |

Foam sublayer | x | m − 1 | x.(m − 1) | Region | Regime |
---|---|---|---|---|---|

f_{a} = 98%a = 10 mm | s to arb | s | s | R, RG-R, RG | Rayleigh-Gans |

f_{a} = 60%a = 0.3 mm | s | arb | s | R | Rayleigh |

f_{a} = 10%a = 0.05 mm | s | arb | s | R | Rayleigh |

## Share and Cite

**MDPI and ACS Style**

Anguelova, M.D.; Gaiser, P.W.
Dielectric and Radiative Properties of Sea Foam at Microwave Frequencies: Conceptual Understanding of Foam Emissivity. *Remote Sens.* **2012**, *4*, 1162-1189.
https://doi.org/10.3390/rs4051162

**AMA Style**

Anguelova MD, Gaiser PW.
Dielectric and Radiative Properties of Sea Foam at Microwave Frequencies: Conceptual Understanding of Foam Emissivity. *Remote Sensing*. 2012; 4(5):1162-1189.
https://doi.org/10.3390/rs4051162

**Chicago/Turabian Style**

Anguelova, Magdalena D., and Peter W. Gaiser.
2012. "Dielectric and Radiative Properties of Sea Foam at Microwave Frequencies: Conceptual Understanding of Foam Emissivity" *Remote Sensing* 4, no. 5: 1162-1189.
https://doi.org/10.3390/rs4051162