Kenneth E. Hawker

A normal mode theory of acoustic Doppler effects in the oceanic waveguide

This paper considers the problem of computing the acoustic field generated by a moving point source. In particular, the acoustic field is obtained in terms of the normal modes of a horizontally stratified ocean. The source motion is assumed to be uniform (unaccelerated), but is not restricted to a path radial to the receiver. The structure of the Fourier inversion integral is carefully analyzed and an evaluation is carried out by the method of stationary phase. The stationary phase point is explicitly computed as an expansion in powers of the ratio of the source speed to the mode group velocity. The resulting expression for the velocity potential is examined for Doppler effects for both instantaneous (modal) Doppler as well as Doppler determined by a finite bandwidth Fourier transform.

Consistent coupled mode theory of sound propagation for a class of nonseparable problems

This article examines the effects of boundary condition approximations that arise whenever the coupled mode theories of A. D. Pierce [J. Acoust. Soc. Am. 37, 19–27 (1965)] and D. M. Milder [J. Acoust. Soc. Am. 46, 1259–1263 (1969)], are applied to propagation problems involving range dependent boundaries. This boundary condition approximation requires that the depth derivative, rather than the normal derivative of the field, be continuous across a sloping interface. The approximation is necessary in order to carry out the partial separation of depth and range variables effected in the mathematical formulation of the theory. This article will show that a consequence of this approximation is that conventional coupled mode theory applied to dissipationless media with nonhorizontal boundaries does not conserve energy. It is shown that a correction to coupled mode theory can be derived such that the proper boundary conditions are satisfied and energy is conserved to first order in the local bottom slope. Moreo...

On the calculation of normal mode group velocity and attenuation

The group velocity for a normal mode can be calculated without invoking a finite difference approximation requiring a second eigenmode calculation. The reciprocity relation is employed in a derivation of the normal mode group velocity and attenuation coefficient. The group velocity thus calculated is more accurate than a comparable finite difference approximation. Arbitrarily arranged layers of solid and fluid media are considered.

Effects of density gradients on bottom reflection loss for a class of marine sediments

The sensitivity of bottom reflection loss to a density gradient is examined within the context of a physically meaningful model consisting of a fluid sediment layer having depth‐dependent density and sound speed overlying a substrate having solid properties. The bottom grazing‐angle range is divided into two regions, the low‐angle and the high‐angle regions. In the low‐angle region sediment sound‐speed gradients refract incident energy upward before it encounters the substrate. In the high‐grazing‐angle region, energy impinges on the substrate. Through the use of a numerical plane‐wave reflection coefficient model it is shown that the effect of a density gradient on low‐angle bottom loss is small. An expression indicating this effect is derived and is shown to agree with numerical bottom‐loss calculations. The high‐angle bottom loss is more influenced by a density gradient because of the impedance changes at the sediment–substrate interface which accompany a density gradient. It is shown that introduction...

The existence of Stoneley waves as a loss mechanism in plane wave reflection problems

In a previous paper [’’Influence of Stoneley waves on plane‐wave reflection coefficients: Characteristics of bottom reflection loss,’’ K. E. Hawker, J. Acoust. Soc. Am. 64, 548 (1978)] it was shown that for certain configurations of layered fluids overlying a rigid (solid) substrate, the reflection loss would display narrow peaks of significant amplitude. It was suggested in that paper that these peaks were due to excitation of Stoneley waves at the fluid–solid interface. In the present paper this association is made more precise and definite. Through an extension of the classical theory of Stoneley waves to the case of inhomogeneous media, it is shown that the angles at which the reflection‐loss peaks occur are precisely those angles for which the horizontal component of phase velocity in the fluid equals the Stoneley wave phase velocity. In addition, the near independence of the reflection loss peaks on frequency is explained quantitatively.

An examination of the influence of the range dependence of the ocean bottom on the adiabatic approximation

The effects of radial sediment sound‐speed derivative and sloping water–sediment interface on the adiabatic approximation to the set of coupled radial equations of coupled‐mode theory are examined. The investigation was carried out using the adiabatic criterion of Milder. This criterion requires that the coupling coefficient between adjacent modes be small compared to the mode‐cycle distance. It is determined that sediments having large characteristic acoustic impedances can have larger radial sediment sound‐speed gradients within the adiabatic approximation than can sediments with smaller characteristic impedances. It is also determined that this trend is reversed for the case of sloping water–sediment interface with sediments of small characteristic impedance being able to have more bottom slope within the adiabatic approximation than sediments of higher impedance.

A plane wave reflection loss model based on numerical integration

A method is developed for computation of the complex plane‐wave reflection coefficient for a class of geoacoustic models of the ocean bottom, consisting of an arbitarary number of inhomogeneous (fluid) sediment layers overlying a a semi‐infinite homogeneous solid substrate (basement). Within the sediment layers the density, sound speed, and attenuation are permitted to vary arbitrarily with depth. An appropriate formulation of the problem permits the reflection coefficient to be constructed using the results obtained from the direct numerical integration of the depth separated wave equation. The entire computation is thus reduced to that of an initial value problem to which established methods can be applied. The Runge–Kutta method which was chosen permits user selected local error tolerances. Extensive application of the resulting computer code has shown that for a wide class of physically reasonable models global error is bounded, and accuracy far in excess of the minimum required is easily obtained. Requirements imposed on computer word size and available core are modest, and the model can be implemented on any modern general purpose computer and perhaps on a minicomputer.

Influence of Stoneley waves on plane‐wave reflection coefficients: Characteristics of bottom reflection loss

This paper considers characteristics of plane‐wave reflection loss for a model of the ocean’s sub‐bottom consisting of a single inhomogeneous fluid layer (sediment) overlying a semi‐infinite solid (basement). It is found by direct computation that, under certain conditions of sediment sound speed gradient, density, and compressional‐wave speed ratios and attenuations, a large peak in bottom loss can occur over a narrow range of grazing angles. These angles occur below the shear and compressional wave critical angles; hence, they are not directly related to transport of energy by these wave fields. It is found, however, that the presence of shear waves in the substrate is necessary for such a peak to occur; that is, if the shear‐wave speed is set to zero in the substrate, the anomalous loss peak vanishes. Furthermore, these loss peaks are due to absorption rather than the transport of energy to infinity as evidence by the fact that, when all absorptions vanish, so do the anomalous peaks in bottom loss. The...

A study of the effects of ocean bottom roughness on low‐frequency sound propagation

The effects of bottom (water–sediment interface) roughness on low‐frequency sound propagation is examined using a theory of scattering based on boundary conditions applied at the mean boundary. Boundary roughness effects on normal mode attenuation coefficients are examined for two sediment types and two frequencies. For the cases studied it was found that, for fixed bottom roughness, the attenuation coefficients increased with frequency as expected, but showed insensitivity to the correlation length of the bottom roughness. It was also determined that, for fixed bottom roughness, a harder sand‐type bottom produced more scattering attenuation than a clay‐type bottom.

A study of the acoustical effects of sub‐bottom absorption profiles

This paper considers the acoustical effects of variations of sediment absorption with depth for several sediment types. A simple measure of absorption effects is given by the total absorption along the sub‐bottom ray paths. This total absorption is considered for a variety of absorption and sound‐speed gradients for both semi‐infinite and finite layers. The plane wave reflection loss (bottom loss) is then considered for several treatments of attenuation, layering, and sediment type. It is found that, for clay sediments, plausible variations in absorption gradients can result in variations in bottom loss of several decibels. These conclusions are reinforced by consideration of the absorption gradient effects on normal mode attenuation coefficients.