Hi
Thank you for your great toolbox.
I was reading about the implementation of absorption modell which is implemented as a integro-differential operator in the equation of state, and I think there was a very little error. As stated in the manual "To avoid needing to explicitly calculate the time derivative of the acoustic density (which would require storing a copy of at least ρ n and ρ n−1 in memory), the temporal derivative
of the acoustic density is replaced using the linearized mass conservation equation", but if we derive the equation from Navier equation for viscous fluid (also considering thermal conductivity which lead to Kuznetsov equation) the density should be outside of the Fourier transform.
i think this error happened because, equation of state supposed to relate density and pressure and we tried to replace time derivative of density with divergence of flow.
k-Wave
A MATLAB toolbox for the time-domain
simulation of acoustic wave fields
Acoustic absorption modell for inhomogeneous medium
(4 posts) (2 voices)-
Posted 5 years ago #
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Hi 1369mohammadi,
Thanks for your comment. I'm not sure I follow what you mean by outside the Fourier transform? If you add a classical viscous absorption term to the momentum equation, I agree that there will be no time derivative. However, if you add a classical thermal loss term to the equation of state, you will still have a time derivative of the density. When you solve this numerically, this will need to be discretised somehow (e.g., as the manual mentions). In some sense, the fractional loss term in k-Wave replaces the thermal loss term.
Brad.
Posted 5 years ago # -
Thank you for your response.
I'm sorry for my bad explanations. I was concerned about inhomogeneous medium (not the fractional loss). I think rho_0 in the first term in RHS of equation 2.24 (in the manual) should be outside of the spatial Fourier transform. If the medium were homogeneous it would be a constant and won't matter if we place it inside or outside of spatial Fourier transform, but for a medium with inhomogeneous density it would be different. A simply viscous medium was a supporting example for my logic (a counterexample for current formula).Posted 5 years ago # -
If you look at Eqs. 2.4 and 2.5 in the manual, the absorption term has two operators that act on rho (the acoustic density). These are a (fractional) spatial derivative and a temporal derivative. We are free to apply these operators in whichever order we chose, but the second operation must apply to the output of the first operation. In other words, we can either take a time derivative, and then do a space derivative of the output, or vice versa.
Now, in k-Wave we re-write the time derivative of rho using the mass conservation equation. This equation introduces the mass density. The second operator (the fractional Laplacian) must then act on this. In other words, the mass density stays inside the Fourier transform.
Hope that makes sense,
Brad.
Posted 5 years ago #
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