Abstract
When modeling sound propagation, the use of fractional derivatives leads to models that better describe observations of attenuation and dispersion. The wave equation for viscous losses involving integer-order derivatives only leads to an attenuation which is proportional to the square of the frequency. This does not always reflect reality. The acoustic wave equation with loss operator is generalized to the concept of variable-order derivatives in this work. The generalized equation is solved via the Crank-Nicholson scheme. The stability and the convergence of this case are examined in detail.
Highlights
Fractional derivatives are well suited to describe wave propagation in complex media
The wave equation for viscous losses involving integer-order derivatives only leads to an attenuation which is proportional to the square of the frequency
6 Numerical simulation we show the numerical simulation of the acoustic wave equation with variational-order derivative loss operator
Summary
Fractional derivatives are well suited to describe wave propagation in complex media. Several simulators take a modified nonlinear wave equation as a starting point by replacing the traditional loss operator by fractional derivatives [ , , ] or a convolution in time [ , ]. This present work is devoted to the discussion underpinning the description of the extension of an acoustic wave equation to the concept of the variational-order derivative and the solution of the generalized equation using the Crank-Nicholson scheme. Under these circumstances, the loss operator was proposed as [ ]. For constant-order time fractional diffusion equations, the implicit difference approximation scheme was proposed in [ ].
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