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about the wave equation

(14 posts) (6 voices)

Started 13 years ago by huangchao
Latest reply from Amita

huangchao

Member
Joined: Nov '10
Posts: 57

Hi,

Thanks for this wonderful toolbox!

The k-Wave solves 3 first order equations (or second order wave equation for homogeneous media), which, at least in my opinion, is valid when the ambient density rho_0 is constant.

Therefore, I am wondering if you can solve the wave equation that models the sound speed, density and absorption heterogeneities. To my knowledge, it should be

where α(r) is the ultrasonic absorption distribution.

Thanks,
Chao

Edit: I don't know why the image does not show properly, but you can find it here:
http://img225.imageshack.us/f/86132079.jpg/ (ADMIN: Fixed - you need the direct image link)

Posted 13 years ago #
Bradley Treeby

Developer
Joined: Mar '10
Posts: 1,643

Hi Chao,

Thanks for your question. The first order simulation functions in k-Wave (kspaceFirstOrder1D, 2D, and 3D) are based on three coupled first equations as you say. Without the absorption term, these are given by:

Here p and rho are acoustic quantities, u and d are the particle velocity and displacement, and rho_0 and c_0 are the background parameters.

These equations are valid for both homogeneous or heterogeneous media (in rho_0 and c_0). If you combine these in the normal fashion including a photoacoustic source, you will get the equation you mention without the absorption term (the absorption model used in k-Wave is based on the fractional Laplacian, see the references for more details).

Note, within the actual k-Wave code, the grad(rho_0) terms are not computed because they exactly cancel. (More subtly, this means the calculated acoustic density is not quite the acoustic density, but that doesn't matter in this case because this parameter is not used for input or output.)

For more information - see the Modelling Wave Propagation section under Getting Started in the k-Wave help files.

I hope that helps. If not, please let us know!

Brad.

Posted 13 years ago #
huangchao

Member
Joined: Nov '10
Posts: 57

Hi, Dr. Treeby,

Thank you for your explanation. It's very helpful.

And I have two other questions:

1. It seems that the absorption parameters, like alpha_power, alpha_coeff, can not be given in the matrix form. For example, when I used matrix to define the absorption parameters in the "2D Time Reversal With Compensation For Acoustic Absorption Example", it will prompt error.

2. Can I define the sensor frequency response in other forms instead of Gaussian shape?

If not, hope the toolbox will support these two functions in the future.

Thanks,
Chao

Posted 13 years ago #
Bradley Treeby

Developer
Joined: Mar '10
Posts: 1,643

Hi Chao,

You are more than welcome. At this stage, the toolbox only supports scalar inputs for the absorption variables. We are planning to modify this to accommodate matrix inputs in the near future.

The sensor frequency response simply takes the FFT of the recorded time domain signals after the simulation is complete, multiplies the frequency domain signals by a zero phase Gaussian window, and then computes the inverse FFT (see kspaceFirstOrder_filterSensorData.m within the private folder).

To implement another type of frequency response, you could manually filter the recorded time domain signals in a similar manner (or in any other way you like) after they are returned from the simulation.

I hope that is of some help,

Brad.

Posted 13 years ago #
huangchao

Member
Joined: Nov '10
Posts: 57

Hi Dr. Treeby,

Thanks for the clarification.

One more question: If the signal is the convolution of the simulated pressure data and the sensor frequency response, and I want to use this signal for reconstruction by time reversal method, other than direct deconvolution before reconstruction (FFT and IFFT), is there any other way to remove the frequency response effect during reconstruction process?

Thanks,
Chao

Posted 13 years ago #
bencox

Developer
Joined: Mar '10
Posts: 292

Dear Chao,

I think deconvolving your frequency response before you do the image reconstruction is probably the safest way, as you can then deal with the problems that the deconvolution presents (noise amplification essentially) before it gets tangled up in an image reconstruction. Deconvolution of a frequency response from a time series will be covered in many standard digital signal processing books.

Kind regards,

Ben

Posted 13 years ago #
xiong

Member
Joined: Aug '10
Posts: 8

Dear Chao and Ben,

I also saw the equation Chao posted. I have a question about it. The unit of alpha(r) in it should be the same as that of time (second) to ensure the unit consistency of the equation. But the acoustic attenuation coefficient (use beta(r) to denote it) usually has the unit of Np/cm or Np/cm/MHz. I don't know the relationship between alpha(r) (ultrasonic absorption distribution) and beta(r) (acoustic attenuation coefficient). If possible, please provide the corresponding equation with beta(r) represents the loss of acoustic wave.

Thank you.

Xiong

Posted 12 years ago #
Bradley Treeby

Developer
Joined: Mar '10
Posts: 1,643

Hi Xiong,

Thanks for your question. You are correct about the units. The alpha(r) in Chao's equation actually represents a relaxation time. This is normally denoted tau and is often called Stokes' relaxation time. In the limit of w*tau << 1, the absorption will be proportional to frequency squared, in other words:

absorption = alpha_0 w^2.

In this case, the relationship between tau and alpha_0 is given by:

alpha_0 = tau/(2*c)

Here c is the sound speed in m/s and w is the frequency in rad/s. Note, this is not the wave equation solved in k-Wave.

I hope that helps!

Brad.

Posted 12 years ago #
xiong

Member
Joined: Aug '10
Posts: 8

Hi Brad,

This is exactly what I want.

Thank you!

Xiong

Posted 12 years ago #
aminzadeh

Member
Joined: Jan '15
Posts: 1

Dear Dr. Treeby

Hello and Thanks for your Toolbox.

As you said k-Wave solve a system of coupled 1st-order equations. why you don't solve the single 2nd-order equation?
I asked this question, because before I know about your Toolbox, I solved the single 2nd order wave eq. and now I see that results are in a good agreement with the k-Wave.

Best regards,
Amin

Posted 9 years ago #
Bradley Treeby

Developer
Joined: Mar '10
Posts: 1,643

Hi Amin,

There are a few reasons why we solve the first-order equations rather than the second-order wave equation in the main simulation functions. First, it allows both mass and force sources to be easily included into the discrete equations. Second, it allows the PML to be easily defined. Third, calculation of the particle velocity allows quantities such as the acoustic intensity to be calculated. This is useful, for example, when modelling how ultrasound heats biological tissue due to acoustic absorption. Finally, there is some evidence that solving the first-order equations gives improved accuracy for heterogeneous media.

If you're interested in homogeneous media, the function kspaceSecondOrder solves the second-order wave equation.

Hope that helps,

Brad.

Posted 9 years ago #
Amita

Member
Joined: Sep '21
Posts: 8

Hi

It is written there in user manual that the tool is solving governing equations for fluid medium, then how k-wave is applicable for solid or viscoelastic medium when the wave equation itself is not for these medium?

or

which governing equations are being solved behind the command pstdElastic2D and pstdElastic3D?

please help.

Posted 2 years ago #
Bradley Treeby

Developer
Joined: Mar '10
Posts: 1,643

See here.

Posted 2 years ago #
Amita

Member
Joined: Sep '21
Posts: 8

ok
Thank you

Posted 2 years ago #

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A MATLAB toolbox for the time-domain simulation of acoustic wave fields

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A MATLAB toolbox for the time-domain
simulation of acoustic wave fields