Sound propagation through a fluctuating stratified ocean: 

 Theory and observation* 



W. H. Munk 



Institute of Geophysics and Planetary Physics. Scripps Institution of Oceanography. La JoUa. California 92037 



F. Zachariasen 



California Institute of Technology. Pasadena. California 9UQ9 

 (Received 18 November 1975) 



We have derived expressions for the mean-square phase and intensity fluctuations and their spectra for cw 

 sound propagating through a channeled fluctuating ocean. The "supereikonal" approximation reduces to 

 the geometric optics (eikonal) limit for short acoustic wavelengths: X<27rL^/J? and X4Ly/(Rtan^6), where 

 Lh and Ly are horizontal and vertical correlation lengths of the fluctuations, R is range, and tan^ is the 

 ray slope, replacing the traditional (and much more severe) Fresnel condition X<27tL Vi? for a 

 homogeneous isotropic ocean. The results can be expressed in closed form for an exponentially stratified 

 ocean model and associated "canonical sound channel." with superimposed fluctuations from an internal 

 wave model spectrum based on oceanographic observations. The parameters are the stratification scale B, 

 the inertial and buoyancy frequencies W|„ and «(2). the scale /. of internal wave mode numbers, and the 

 internal wave energy per unit area. The results are in reasonable agreement with numerical experiments 

 based on the parabolic wave equation. For the "singlepath" 4-kHz transmission over Cobb Seamount the 

 observed and computed rms fluctuations in phase are 1.6 and 2.5 cycles, respectively; in intensity these are 

 5.5 and 2.2 dB, respectively, with anomalous intensities measured at high frequencies ("sporadic" 

 multipathing?). For the muhipath 406-Hz MIMI transmission, we obtain 4X 10"' and SxlO"' sec"', 

 respectively, for the experimentally determined and the computed rms phase rates. 



Subject Classification: |43| 30.20, |431 30.40; |43] 20.15. 



LIST OF SYMBOLS 



C(x), 6C(x) 

 Ciz); Co, C, C 



rms (60/0)0 = 4.9x10-* 

 >iU)', "o> ^, n 



/5C(x) 5C(y)N 



p{x, y) = 



/5C(x) 5C(y) \ 



V = 2g^5C/C 



a, q; \ = 2Ti/q 



o), k; kl=kl + kl, k^ = k, 



p{x, t)exp[i{gx - at)] 



X,i,<p 



G(x), G(k) 



i, II 



rivf,, 7>) 



T) = (Z- D/^B 



€ = 5.7x10"^ 



ft*, R', ft*" =«*+«■; R 



s', s', s*-=s'+s- 



b=R*-d''S*-/d(R*-f 



e, e 



(R 



sound velocity and fluctuations 



mean velocity profile (z upwards); C at surface z =0 (ignoring mixed layer), 

 at channel axis z=-h, at ray apex z 



fractional surface fluctuation 



buoyancy (Brunt-VaisSIa) frequency 



covariance of fractional fluctuations 



perturbation "potential" 



frequency and wave number of sound signal; wavelength 



frequency and wave number of internal waves (H horizontal, V vertical) 



sound pressure 



log pressure, intensity, and phase 



Green's function and Fourier transform 



perpendicular and parallel to ray path 



dimensions of ray tubes- 



correlation lengths 



dimensionless distance above ray axis F(= - 1 km); B{= 1 km) is stratification 

 scale 



perturbation coefficient [Eq. (84)] 



ranges of upper, lower and combined ray loops; axial ray loop 



arc distances of upper, lower, and combined ray loops 



ray parameter [Eq. (86)1 



ray inclination, axial ray inclination 



radius of curvature of ray 



212 



