Abstract

A simulated sea surface with a Pierson-Neumann power spectrum was generated by a numerical model. The image was recorded on photographic film by means of a microdensitometer with a writing mode. To obtain the bidimensional power spectrum of this simulated image of the sea surface, a coherent optical system was used. This power spectrum has information about frequencies in the highest energy peak and the direction that the waves have at a specific time. The Pierson-Neumann power spectrum used to generate the simulated sea surface was compared with the bidimensional power spectrum obtained with the coherent optical system. Attenuation of the high frequencies in the measured spectrum was observed. This attenuation was probably caused by distribution of density values in the film or by the aperture of the detector used in the coherent optical system. Optical autocorrelations of the simulated sea surface were obtained, and a high degree of correlation in the direction perpendicular to the wind was found.

© 1985 Optical Society of America

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References

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  1. N. G. Jerlov, Marine Optics, Vol. 14 (American Elsevier, New York, 1976).
    [CrossRef]
  2. N. E. Barber, “Finding the Direction of Travel of Sea Waves,” Nature London 154, 1048 (1954).
    [CrossRef]
  3. Y. Sugimori, “A Study of the Application of the Holographic Method of the Determination of the Directional Spectrum of Ocean Waves,” Deep Sea Research and Oceanographic Abstracts, 22(5), May1975.
    [CrossRef]
  4. D. Bruno, J. Novarini, “Análisis comparativo de superficies modeladas numéricamente con distintos espectros de potencia,” Junio 1976, Republka Argentina, Servicio de Hidrografia Naval, DOF-ITIO-76 (1976).
  5. W. Caruthers, J. Novarini, “Numerical Modeling of Randomly Rough Surfaces with Application to Sea Surfaces,” Texas A & M Univ., Dept Oceanography Tech. Rep. Ref. 71-13-T (1971).
  6. D. Casasent, Optical Data Processing (Springer, Berlin, 1978).
    [CrossRef]
  7. B. Kinsman, Wind Waves (Prentice-Hall, Englewood Cliffs, N.J., 1965).
  8. S. Denzil, “Directional Energy Spectra of the Sea from Photographs,” J. Geophysical Research, 74, No. 8 (April15, 1969).
  9. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), pp. 14–16.
  10. C. S. Cox, Optical Aspects of Oceanography (Academic, New York, 1974), Chap. 3.

1976 (1)

D. Bruno, J. Novarini, “Análisis comparativo de superficies modeladas numéricamente con distintos espectros de potencia,” Junio 1976, Republka Argentina, Servicio de Hidrografia Naval, DOF-ITIO-76 (1976).

1975 (1)

Y. Sugimori, “A Study of the Application of the Holographic Method of the Determination of the Directional Spectrum of Ocean Waves,” Deep Sea Research and Oceanographic Abstracts, 22(5), May1975.
[CrossRef]

1969 (1)

S. Denzil, “Directional Energy Spectra of the Sea from Photographs,” J. Geophysical Research, 74, No. 8 (April15, 1969).

1954 (1)

N. E. Barber, “Finding the Direction of Travel of Sea Waves,” Nature London 154, 1048 (1954).
[CrossRef]

Barber, N. E.

N. E. Barber, “Finding the Direction of Travel of Sea Waves,” Nature London 154, 1048 (1954).
[CrossRef]

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), pp. 14–16.

Bruno, D.

D. Bruno, J. Novarini, “Análisis comparativo de superficies modeladas numéricamente con distintos espectros de potencia,” Junio 1976, Republka Argentina, Servicio de Hidrografia Naval, DOF-ITIO-76 (1976).

Caruthers, W.

W. Caruthers, J. Novarini, “Numerical Modeling of Randomly Rough Surfaces with Application to Sea Surfaces,” Texas A & M Univ., Dept Oceanography Tech. Rep. Ref. 71-13-T (1971).

Casasent, D.

D. Casasent, Optical Data Processing (Springer, Berlin, 1978).
[CrossRef]

Cox, C. S.

C. S. Cox, Optical Aspects of Oceanography (Academic, New York, 1974), Chap. 3.

Denzil, S.

S. Denzil, “Directional Energy Spectra of the Sea from Photographs,” J. Geophysical Research, 74, No. 8 (April15, 1969).

Jerlov, N. G.

N. G. Jerlov, Marine Optics, Vol. 14 (American Elsevier, New York, 1976).
[CrossRef]

Kinsman, B.

B. Kinsman, Wind Waves (Prentice-Hall, Englewood Cliffs, N.J., 1965).

Novarini, J.

D. Bruno, J. Novarini, “Análisis comparativo de superficies modeladas numéricamente con distintos espectros de potencia,” Junio 1976, Republka Argentina, Servicio de Hidrografia Naval, DOF-ITIO-76 (1976).

W. Caruthers, J. Novarini, “Numerical Modeling of Randomly Rough Surfaces with Application to Sea Surfaces,” Texas A & M Univ., Dept Oceanography Tech. Rep. Ref. 71-13-T (1971).

Sugimori, Y.

Y. Sugimori, “A Study of the Application of the Holographic Method of the Determination of the Directional Spectrum of Ocean Waves,” Deep Sea Research and Oceanographic Abstracts, 22(5), May1975.
[CrossRef]

Deep Sea Research and Oceanographic Abstracts (1)

Y. Sugimori, “A Study of the Application of the Holographic Method of the Determination of the Directional Spectrum of Ocean Waves,” Deep Sea Research and Oceanographic Abstracts, 22(5), May1975.
[CrossRef]

J. Geophysical Research (1)

S. Denzil, “Directional Energy Spectra of the Sea from Photographs,” J. Geophysical Research, 74, No. 8 (April15, 1969).

Junio 1976, Republka Argentina, Servicio de Hidrografia Naval (1)

D. Bruno, J. Novarini, “Análisis comparativo de superficies modeladas numéricamente con distintos espectros de potencia,” Junio 1976, Republka Argentina, Servicio de Hidrografia Naval, DOF-ITIO-76 (1976).

Nature London (1)

N. E. Barber, “Finding the Direction of Travel of Sea Waves,” Nature London 154, 1048 (1954).
[CrossRef]

Other (6)

N. G. Jerlov, Marine Optics, Vol. 14 (American Elsevier, New York, 1976).
[CrossRef]

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), pp. 14–16.

C. S. Cox, Optical Aspects of Oceanography (Academic, New York, 1974), Chap. 3.

W. Caruthers, J. Novarini, “Numerical Modeling of Randomly Rough Surfaces with Application to Sea Surfaces,” Texas A & M Univ., Dept Oceanography Tech. Rep. Ref. 71-13-T (1971).

D. Casasent, Optical Data Processing (Springer, Berlin, 1978).
[CrossRef]

B. Kinsman, Wind Waves (Prentice-Hall, Englewood Cliffs, N.J., 1965).

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Figures (13)

Fig. 1
Fig. 1

Four simulated sea surfaces with area of 500,000 m2 each. The repetition of the surfaces increases the SNR in the optical system.

Fig. 2
Fig. 2

Directional power spectrum of Pierson-Neumann for a wind velocity of 5.0 m/sec.

Fig. 3
Fig. 3

Film Kodak 2476 response to values of density written with the microdensitometer.

Fig. 4
Fig. 4

Coherent optical system for obtaining the power spectrum of the sea images contained in the transparency.

Fig. 5
Fig. 5

Coherent optical system for obtaining the correlation between two images.

Fig. 6
Fig. 6

Simulated sea surface for a Pierson-Neumann power spectrum.

Fig. 7
Fig. 7

Power spectrum of the simulated sea surface (Fig. 1). Scale 12:1.

Fig. 8
Fig. 8

Result of the measurement of the power spectrum (Fig. 7) obtained in the coherent optical system (Fig. 4).

Fig. 9
Fig. 9

Comparison between the power spectrum obtained experimentally and the theoretical Pierson-Neumann power spectrum (continue line).

Fig. 10
Fig. 10

Autocorrelations.

Fig. 11
Fig. 11

Real sea surface. Area, 68,900 m2.

Fig. 12
Fig. 12

Power spectrum of a real sea surface (Fig. 11).

Fig. 13
Fig. 13

Result of the measurement of the power spectrum of the real sea surface (Fig. 12).

Tables (2)

Tables Icon

Table 1 Wavelength of the Waves

Tables Icon

Table II Localization of Sea Wave Numbers on the Fourier Plane of the Optical System

Equations (13)

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A 2 ( w , θ ) = ( C / w 6 ) exp ( 2 g 2 / w 2 u 2 ) cos 2 θ for w 1 < w < π / 2 < θ < π / 2 0 otherwise ,
G = 3 C ( π / 2 ) 3 / 2 ( u / 2 g ) 5 ,
w = ( g K ) 1 / 2 , K = ( K x 2 + K y 2 ) 1 / 2 , θ = tan 1 ( K y / K x ) ,
J [ ( w , θ ) / ( K x , K y ) ] = [ ( g ) 1 / 2 / 2 ] K 3 / 2 .
E z ( K x , K y ) = J [ ( w , θ ) / ( K x , K y ) ] A 2 [ w ( K x , K y ) , θ ( K x , K y ) ]
Ez ( K x , K y ) = ( C / 2 g 5 / 2 K 9 / 2 ) exp ( 2 g / u 2 K ) cos 2 [ tan 1 ( K y / K x ) ] .
U f ( x f , y f ) = + t 0 ( x 0 , y 0 ) × exp [ ( 2 π j / λ f ) ( x 0 x f + y 0 y f ) ] d x 0 d y 0 ,
a 1 ( x 1 , y 1 ) = t 01 + t 11 ( x , y ) ,
a 2 ( x 2 , y 2 ) = A 1 ( x 2 / λ f , y 2 / λ f ) ,
a 3 ( x 3 , y 3 ) = t 11 ( x 3 , y 3 ) [ t 02 + t 12 ( x 3 , y 3 ) ] .
a 4 ( x 4 , y 4 ) = A 3 ( x 4 / λ f , y 4 / λ f ) ,
a 4 ( x 4 , y 4 ) = T 02 T 11 ( 0 , 0 ) + T 11 ( x 4 / λ f , y 4 / λ f ) * T 12 ( x 4 / λ f , y 4 / λ f ) ,
C ff ( x , y ) = + T 11 ( x + Δ x , y + Δ y ) dxdy .

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