AXIAL WAVE IN LONG-RANGE PROPAGATION IN A RANGE-INDEPENDENT OCEAN

2004 ◽  
Vol 12 (02) ◽  
pp. 127-147 ◽  
Author(s):  
NATALIE S. GRIGORIEVA ◽  
GREGORY M. FRIDMAN
Keyword(s):  
Integral Representation ◽  
Long Range ◽  
Sound Speed ◽  
Parabolic Cylinder ◽  
Time Of Arrival ◽  
Channel Axis ◽  
Sound Channel ◽  
Cyclic Frequency ◽  
Geometrical Acoustics ◽  
Parabolic Cylinder Functions

In many long-range propagation experiments the source and receiver are placed close to the depth of the waveguide (SOFAR) axis to minimize the interaction of the acoustic field with the ocean's surface and bottom. The time-of-arrival patterns of these experiments consist of resolvable, geometrical-like arrivals followed by an axial crescendo of unresolved energy. It is impossible to explain this late-arriving energy using the geometrical acoustics because of the presence of cusp caustics repeatedly along the waveguide axis. The interference of the wave fields corresponding to the rays located in the vicinity of the caustics near the waveguide axis produces a special "axial wave" that propagates along this axis. The purpose of the paper is to obtain the integral representation of the axial wave for an arbitrary deep-water waveguide in a range-independent medium in long-range acoustic propagation in the ocean when the source and receiver are located close to the depth of the sound-channel axis. The integral representation for the axial wave is derived with the use of solutions of the Helmholtz equation concentrated near the sound-channel axis and which decrease exponentially outside a strip containing the axis. These solutions have the form of the exponentials multiplied by parabolic cylinder functions whose arguments are sections of series in powers of ω-1/2, where ω is a cyclic frequency. Numerical results are obtained for the Munk canonical sound-speed profile.

EFFECT OF HORIZONTAL INHOMOGENEITY OF THE OCEAN ON INTERFERENCE OF NEAR-AXIAL WAVES IN LONG-RANGE ACOUSTIC PROPAGATION

2006 ◽  
Vol 14 (04) ◽  
pp. 415-443 ◽  
Author(s):  
NATALIE S. GRIGORIEVA ◽  
GREGORY M. FRIDMAN
Keyword(s):  
Integral Representation ◽  
Long Range ◽  
Deterministic Model ◽  
Sound Field ◽  
Acoustic Propagation ◽  
Time Of Arrival ◽  
Channel Axis ◽  
Sound Channel ◽  
Narrow Strip ◽  
Geometrical Acoustics

When the source and receiver are located close to the depth of the waveguide axis, there exist cusped caustics repeatedly along the axis. A description of the propagation of energy along the waveguide axis in terms of geometrical acoustics is not valid in neighborhoods of cusped caustics, because in these neighborhoods the waves associated with individual ray paths interfere with one another. Neighborhoods of interference grow with range, and at long distances they overlap. This results in the formation of a diffractive (as opposed to ray, i.e., geometrical acoustics) component of the field — the axial wave — that propagates along the sound-channel axis. In this paper, the integral representation of the axial wave obtained before for an arbitrary deep-water waveguide in a three-dimensional range-independent medium is generalized to a range-dependent ocean. The integral representation of the axial wave is derived with the use of solutions of the homogeneous Helmholtz equation concentrated near the sound-channel axis and which decrease exponentially outside a narrow strip containing the axis. The observed time-of-arrival patterns from a number of long-range ocean acoustic propagation experiments show early geometrical-like arrivals followed by a crescendo of energy that propagates along the sound-channel axis and is not resolved into individual arrivals. The practical application of the developed analytic expression for the sound field near the axis of an ocean type waveguide is the discrimination of noninterfering (resolved) and interfering (nonresolved) arrivals. In this paper, the axial wave is simulated for a deterministic model of a range-dependent medium, where the range-dependence results for such things as change in geographic location. The model is based on the information about sound-speed profiles as a function of range between the source and receiving array for the AET experiment. The sound source frequency is taken equal to 75Hz. The propagation range is 3250 km.

INVESTIGATION OF NEAR-AXIAL INTERFERENCE EFFECTS FOR PROPAGATION IN A DUCTED WAVEGUIDE

2004 ◽  
Vol 12 (03) ◽  
pp. 355-386 ◽  
Author(s):  
NATALIE S. GRIGORIEVA ◽  
GREGORY M. FRIDMAN ◽  
DAVID R. PALMER
Keyword(s):  
Long Range ◽  
Acoustic Propagation ◽  
Index Of Refraction ◽  
Time Of Arrival ◽  
Two Dimensional ◽  
Interference Effects ◽  
Channel Axis ◽  
Model Case ◽  
Sound Channel ◽  
Axial Waves

The observed time-of-arrival patterns from a number of long-range ocean acoustic propagation experiments show early geometrical-like arrivals followed by a crescendo of energy that propagates along the sound channel axis and is not resolved into individual arrivals. To describe in a simple model case the interference of near-axial waves which results in forming the so-called axial wave and propose formulas for the axial wave in more general cases, the two-dimensional reference point source problem for the parabolic index of refraction squared is investigated. Using the method proposed by V. Buldyrev, the integral representation for the exact solution is transformed in such a way to extract ray summands corresponding to rays radiated from the source at angles less than a certain angle, the axial wave, and a term corresponding to the sum of all the rays having launch angles greater than the indicated angle. Numerical results for the axial wave and the last term are obtained for parameters corresponding to long-range ocean acoustic propagation experiments.

scholarly journals Parabolic Cylinder Functions: Examples of Error Bounds for Asymptotic Expansions

2003 ◽  
Vol 01 (03) ◽  
pp. 265-288
Author(s):  
Nico M. Temme ◽  
Raimundas Vidunas
Keyword(s):  
Differential Equation ◽  
Integral Representation ◽  
Computer Algebra ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Parabolic Cylinder ◽  
Numerical Computations ◽  
Parabolic Cylinder Functions ◽  
Uniform Expansions ◽  
Special Case

Several asymptotic expansions of parabolic cylinder functions are discussed and error bounds for remainders in the expansions are presented. In particular, Poincaré-type expansions for large values of the argument z and uniform expansions for large values of the parameter are considered. It is shown how expansions can be derived by using the differential equation, and, for a special case, how an integral representation can be used. The expansions are based on those given in Olver (1959) and on modifications of these expansions given in Temme (2000). Computer algebra techniques are used for obtaining representations of the bounds and for numerical computations.

scholarly journals On certain infinite integrals involving Struve functions and parabolic cylinder functions

1946 ◽  
Vol 7 (4) ◽  
pp. 171-173 ◽  
Author(s):  
S. C. Mitra
Keyword(s):  
Present Note ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Struve Functions ◽  
Image Position

The object of the present note is to obtain a number of infinite integrals involving Struve functions and parabolic cylinder functions. 1. G. N. Watson(1) has proved thatFrom (1)follows provided that the integral is convergent and term-by-term integration is permissible. A great many interesting particular cases of (2) are easily deducible: the following will be used in this paper.

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