^{1}and M. Dale Stokes

^{1}

### Abstract

A model for the underwater noise of whitecaps is presented and compared with the noise measured beneath plunging seawater laboratory waves. The noise from a few hundred hertz up to at least 80 kHz is assumed to be due to the pulses of sound radiated by bubbles formed within a breaking wave crest. The total noise level and its dependence on frequency are a function of bubble creation rate, bubble damping factor and an ‘acoustical skin depth’ associated with scattering and absorption by the bubble plume formed within the crest. Calculation of breaking wavenoise is made using estimates of these factors, which are made independently of the noise itself. The results are in good agreement with wavenoise measured in a laboratory flume when compensated for reverberation. A closed-form, analytical expression for the wavenoise is presented, which shows a −11/6 power-law dependence of noise level on frequency, in good agreement with the −10/6 scaling law commonly observed in the open ocean.

The authors gratefully acknowledge helpful discussions with Dr. David Farmer and Dr. Chris Tindle. This work has been supported by grants from the Office of Naval Research, Ocean Acoustics Division, ONR Grant No. N00014-07-1-1030 and the National Science Foundation, Division of Ocean Sciences, Grant No. OCE 07-27140.

I. INTRODUCTION

II. THE WAVE FLUME EXPERIMENTS

A. General description

B. Reverberation calculations and measurements

1. Model calculations of reverberation in the flume

2. Measurements of reverberation in the flume

3. Reverberation-compensated wavenoise power spectral density estimates

III. WAVENOISEMODEL

A. Creation mechanisms and formation rates of bubbles within the breaking wave

B. The spectrum of the bubble sound pulse

1. The excitation amplitude

2. Bubble damping

C. Bubble plume effects

1. The acoustical skin depth and volume correction factor

2. The volume correction factor for a cylindrical plume

3. The volume correction factor for a disk plume

4. Calculation of the volume correction factor for the experiments

IV. MODEL CALCULATIONS OF WAVENOISE

V. BUBBLE CREATION RATES

VI. DISCUSSION

VII. CONCLUDING REMARKS

### Key Topics

- Bubble dynamics
- 99.0
- Noise propagation
- 54.0
- Acoustic waves
- 49.0
- Acoustic noise
- 34.0
- Ocean waves
- 28.0

## Figures

Contour maps and histograms of the turbulent dissipation rate within the breaking wave crest at 4 successive intervals, determined from an analysis of shear stress-induced dinoflagellate bioluminescence. The exposure time for each image was 0.067 s. Only the four frames with the greatest dissipation are shown.

Contour maps and histograms of the turbulent dissipation rate within the breaking wave crest at 4 successive intervals, determined from an analysis of shear stress-induced dinoflagellate bioluminescence. The exposure time for each image was 0.067 s. Only the four frames with the greatest dissipation are shown.

(A) Geometry for the reverberation calculations. A cross-section through the wave channel and the source image and ray propagation path for the generating sequence ‘SL’ is shown. (B) The phase of the water/glass/air reflection coefficient as a function of angle of incidence for frequencies 1 kHz, 10 kHz and 100 kHz. At low frequencies, the boundary behaves like a pressure-release surface. At high frequencies, the behavior of the boundary depends on the angle of incidence.

(A) Geometry for the reverberation calculations. A cross-section through the wave channel and the source image and ray propagation path for the generating sequence ‘SL’ is shown. (B) The phase of the water/glass/air reflection coefficient as a function of angle of incidence for frequencies 1 kHz, 10 kHz and 100 kHz. At low frequencies, the boundary behaves like a pressure-release surface. At high frequencies, the behavior of the boundary depends on the angle of incidence.

The free space and flume transmission response for the ITC 1007 source and ITC 6050C hydrophone over (A) a 25 kHz band and (B) a 100 kHz band. The black and gray solid lines respectively show the received hydrophone signal in the pool and the flume. The broad peak around 10 kHz is due to a resonance in the source and the smaller peak at 50 kHz (bottom plot) is due to a resonance in the hydrophone. The solid black and broken black lines in the bottom plot show the difference between reverberation from a stationary source versus a source gently moved from side to side across the channel. The black and gray dash-dot lines respectively show the noise floor in the pool and the flume.

The free space and flume transmission response for the ITC 1007 source and ITC 6050C hydrophone over (A) a 25 kHz band and (B) a 100 kHz band. The black and gray solid lines respectively show the received hydrophone signal in the pool and the flume. The broad peak around 10 kHz is due to a resonance in the source and the smaller peak at 50 kHz (bottom plot) is due to a resonance in the hydrophone. The solid black and broken black lines in the bottom plot show the difference between reverberation from a stationary source versus a source gently moved from side to side across the channel. The black and gray dash-dot lines respectively show the noise floor in the pool and the flume.

(A) A comparison of the measured and calculated reverberation in the flume. The solid, gray line shows the calculated reverberation using the source image method. The solid black and broken black lines respectively show the high-frequency and low-frequency measurements of reverberation in the flume. (B) The power spectral density of wave noise from event V40 measured 25 cm below the breaking wave crest. The black dash-dot line shows the wave noise, compensated for the hydrophone response but uncompensated for reverberation. The solid black line shows the wave noise wave noise compensated for both hydrophone response and flume reverberation. The solid gray line shows the noise floor of the flume. (C) The power spectral density of breaking wave packets V68 (gray line) and V27 (black line) plotted as their difference from V40.

(A) A comparison of the measured and calculated reverberation in the flume. The solid, gray line shows the calculated reverberation using the source image method. The solid black and broken black lines respectively show the high-frequency and low-frequency measurements of reverberation in the flume. (B) The power spectral density of wave noise from event V40 measured 25 cm below the breaking wave crest. The black dash-dot line shows the wave noise, compensated for the hydrophone response but uncompensated for reverberation. The solid black line shows the wave noise wave noise compensated for both hydrophone response and flume reverberation. The solid gray line shows the noise floor of the flume. (C) The power spectral density of breaking wave packets V68 (gray line) and V27 (black line) plotted as their difference from V40.

(A) Model calculations of bubble creation rates in event V40. The squares show the bubble size distribution measured at the end of active breaking. The black dash-dot line shows a model calculation of the bubble size distribution based on the fluid turbulent dissipation rates illustrated in Fig. 1 and the binary fragmentation model presented by Martínez-Bazán *et al.* (1999a, 1999b). The short black vertical line denotes the position of the Hinze scale determined from an analysis of the measured bubble size distribution. The solid black curve shows a calculation of the bubble creation rates using the fragmentation model. The solid gray curve shows a hybrid estimate of bubble creation rates based on the model calculations for bubbles larger than the Hinze scale and the observed bubble size distribution for bubbles smaller than the Hinze scale. The black squares show the bubble size distribution at the end of the acoustically active period estimated from the analysis of video images taken through the flume side wall. (B) The bubble size distribution used to calculate the e-folding length for the cylindrical plume of bubbles created by event V40. The black squares show the bubble size distribution at the end of the acoustically active period estimated from the analysis of video images taken through the flume side wall. The black lines show the fitted distribution, extrapolated to radius bubbles at the small end of the bubble spectrum and 10 mm bubbles at the large end of the spectrum.

(A) Model calculations of bubble creation rates in event V40. The squares show the bubble size distribution measured at the end of active breaking. The black dash-dot line shows a model calculation of the bubble size distribution based on the fluid turbulent dissipation rates illustrated in Fig. 1 and the binary fragmentation model presented by Martínez-Bazán *et al.* (1999a, 1999b). The short black vertical line denotes the position of the Hinze scale determined from an analysis of the measured bubble size distribution. The solid black curve shows a calculation of the bubble creation rates using the fragmentation model. The solid gray curve shows a hybrid estimate of bubble creation rates based on the model calculations for bubbles larger than the Hinze scale and the observed bubble size distribution for bubbles smaller than the Hinze scale. The black squares show the bubble size distribution at the end of the acoustically active period estimated from the analysis of video images taken through the flume side wall. (B) The bubble size distribution used to calculate the e-folding length for the cylindrical plume of bubbles created by event V40. The black squares show the bubble size distribution at the end of the acoustically active period estimated from the analysis of video images taken through the flume side wall. The black lines show the fitted distribution, extrapolated to radius bubbles at the small end of the bubble spectrum and 10 mm bubbles at the large end of the spectrum.

The measured (black line) and theoretical (gray line) power spectral density estimates of the sound radiated by a bubble fragmenting into two products.

The measured (black line) and theoretical (gray line) power spectral density estimates of the sound radiated by a bubble fragmenting into two products.

The calculated peak pressure of the pulse of sound radiated by a bubble forced into oscillation by a collapsing neck of air with a 177.6° angle, as a function of bubble radius. The solid line shows the results of an analysis of bubble forcing using 4th order, Runge-Kutta integration of the Rayleigh-Plesset equation for bubbles with radii from to 10 mm. The broken line shows the peak pressure extrapolated from to .

The calculated peak pressure of the pulse of sound radiated by a bubble forced into oscillation by a collapsing neck of air with a 177.6° angle, as a function of bubble radius. The solid line shows the results of an analysis of bubble forcing using 4th order, Runge-Kutta integration of the Rayleigh-Plesset equation for bubbles with radii from to 10 mm. The broken line shows the peak pressure extrapolated from to .

(A) The geometry of a rectangular prism used to illustrate the volume correction factor calculations. (B) The geometry for the cylindrical plume. (C) The geometry for the disk plume.

(A) The geometry of a rectangular prism used to illustrate the volume correction factor calculations. (B) The geometry for the cylindrical plume. (C) The geometry for the disk plume.

(A) Calculations of the scaled volume correction factor for the cylindrical plume geometry. The variation of scaled volume correction factor is shown for various values of the scaled length parameters that define the geometry as a function of scaled distance from the plume center. The circle shows the scaled volume correction factor for the cylindrical plume created beneath the breaking wave packet V40. (B) Calculations of the scaled volume correction factor for the disk plume as a function of distance from the disk plume center.

(A) Calculations of the scaled volume correction factor for the cylindrical plume geometry. The variation of scaled volume correction factor is shown for various values of the scaled length parameters that define the geometry as a function of scaled distance from the plume center. The circle shows the scaled volume correction factor for the cylindrical plume created beneath the breaking wave packet V40. (B) Calculations of the scaled volume correction factor for the disk plume as a function of distance from the disk plume center.

Calculated absorption (A), phase velocity (B) and acoustical skin depth (C) of the bubble plume for event V40 based on the distribution in Fig. 5(B).

Calculated absorption (A), phase velocity (B) and acoustical skin depth (C) of the bubble plume for event V40 based on the distribution in Fig. 5(B).

Model calculations of the radiated wave noise for event V40. The solid black line shows the model results for the wave noise 25 cm below the breaking wave crest. The solid gray line shows the wave noise measured 25 cm below the wave crest and compensated for tank reverberation. The gray dash-dot line shows the noise floor in the flume. The gray dots show the power spectral density of an individual white cap measured in the Pacific ocean during a storm 160 km west of Point Conception in 2000. The level of this curve has been scaled to a distance of 25 cm below the white cap for comparison with the laboratory results.

Model calculations of the radiated wave noise for event V40. The solid black line shows the model results for the wave noise 25 cm below the breaking wave crest. The solid gray line shows the wave noise measured 25 cm below the wave crest and compensated for tank reverberation. The gray dash-dot line shows the noise floor in the flume. The gray dots show the power spectral density of an individual white cap measured in the Pacific ocean during a storm 160 km west of Point Conception in 2000. The level of this curve has been scaled to a distance of 25 cm below the white cap for comparison with the laboratory results.

The bubble creation rates inferred from the wave noise measurements and compared with the modeled rates based on the bubble fragmentation model and the observed bubble size distribution. The solid black line shows the bubble creation rate calculated using the wave noise power spectral density and Eq. (36). The dash-dot line shows the bubble creation rate calculated from the measured bubble spectrum and the MBML fragmentation model. The gray line indicates a power law scaling of −3/2 for comparison.

The bubble creation rates inferred from the wave noise measurements and compared with the modeled rates based on the bubble fragmentation model and the observed bubble size distribution. The solid black line shows the bubble creation rate calculated using the wave noise power spectral density and Eq. (36). The dash-dot line shows the bubble creation rate calculated from the measured bubble spectrum and the MBML fragmentation model. The gray line indicates a power law scaling of −3/2 for comparison.

## Tables

Parameters for the glass wall reflection coefficient.

Parameters for the glass wall reflection coefficient.

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