The nonlinear progressive wave equation (NPE) is a time-domain formulation of Euler’s fluid equations designed to model low-angle wave propagation using a wave-following computational domain [ McDonald et al., J. Acoust. Soc. Am. 81]. The wave-following frame of reference permits the simulation of long-range propagation that is useful in modeling the effects of blast waves in the ocean waveguide. However, the current model does not take into account sediment attenuation, a feature necessary for accurately describing sound propagation into and out of the ocean sediment. These attenuating, dispersive sediments are naturally captured with linear, frequency-domain solutions through use of complex wavespeeds, but a comparable treatment is nontrivial in the time-domain. Recent developments in fractional loss operator methods allow for frequency-dependent loss mechanisms to be applied in the time-domain providing physically realistic results [Prieur et al., J. Acoust. Soc. Am. 130]. Using these approaches, the governing equations used to describe the NPE are modified to use fractional derivatives in order to develop a fractional NPE. The updated model is then benchmarked against a Fourier-transformed parabolic equation solution for the linear case using various sediment attenuation factors.

The nonlinear progressive wave equation (NPE) is a time-domain model used to calculate long-range shock propagation using a wave-following computational domain. Current models are capable of treating smoothly spatially varying medium properties, and fluid-fluid interfaces that align horizontally with a computational grid that can be handled by enforcing appropriate interface conditions. However, sloping interfaces that do not align with a horizontal grid present a computational challenge as application of interface conditions to vertical contacts is non-trivial. In this work, range-dependent environments, characterized by sloping bathymetry, are treated using a rotated coordinate system approach where the irregular interface is aligned with the coordinate axes. The coordinate rotation does not change the governing equation due to the narrow-angle assumption adopted in its derivation, but care is taken with applying initial, interface, and boundary conditions. Additionally, sound pressure level influences on nonlinear steepening for range-independent and range-dependent domains are used to quantify the pressures for which linear acoustic models suffice. A study is also performed to investigate the effects of thin sediment layers on the propagation of blast waves generated by explosives buried beneath mud line.

The nonlinear progressive wave equation (NPE) is a time-domain formulation of the Euler fluid equations designed to model low-angle wave propagation using a wave-following computational domain. The wave-following frame of reference permits the simulation of long-range propagation and is useful in modeling blast wave effects in the ocean waveguide. Existing models do not take into account frequency-dependent sediment attenuation, a feature necessary for accurately describing sound propagation over, into, and out of the ocean sediment. Sediment attenuation is addressed in this work by applying lossy operators to the governing equation that are based on a fractional Laplacian. These operators accurately describe frequency-dependent attenuation and dispersion in typical ocean sediments. However, dispersion within the sediment is found to be a secondary process to absorption and effectively negligible for ranges of interest. The resulting fractional NPE is benchmarked against a Fourier-transformed parabolic equation solution for a linear case, and against the analytical Mendousse solution to Burgers' equation for the nonlinear case. The fractional NPE is then used to investigate the effects of attenuation on shock wave propagation.

High intensity underwater sources, such as explosions, generate nonlinear finite-amplitude pulses that behave differently than linear acoustic pulses within shallow water waveguides. The nonlinearity is known to decrease the critical angle for total internal reflection from that of the linear case when the seafloor is approximated as a fluid. However, this result has not been extensively studied for elastic seafloors where shear waves are present. In this work, a time-domain model is developed assuming an isotropic linear elastic bottom, inviscid water column, and allowing for nonlinear advective acceleration and a nonlinear equation of state. The model is numerically implemented using a high-order Godunov scheme and then benchmarked against tank experiment data for the linear case. The nonlinear model is used to study the critical grazing angle for ocean bottoms of varying shear speeds to determine the combined effect of nonlinearity and elasticity on bottom penetration.

The nonlinear progressive wave equation (NPE) is a time-domain formulation of Eulers fluid equations designed to model low angle wave propagation using a wave-following computational domain. The standard formulation consists of four separate mathematical quantities that physically represent refraction, nonlinear steepening, radial spreading, and diffraction. The latter two of these effects are linear whereas the steepening and refraction are nonlinear. This formulation recasts pressure, density, and velocity into a single variable a dimensionless pressure perturbation which allows for greater efficiency in calculations.The wave-following frame of reference permits the simulation of long-range propagation that is useful in modeling the effects of blast waves in the ocean waveguide. Nonlinear effects such as weak shock formation are accurately captured with the NPE. The numerical implementation is a combination of two numerical schemes: a finite-difference Crank-Nicholson algorithm for the linear terms of the NPE and a flux-corrected transport algorithm for the nonlinear terms. In this work, an existing implementation is extended to allow for a penetrable fluid bottom. Range-dependent environments, characterized by sloping bathymetry, are investigated and benchmarked using a rotated coordinate system approach.