k-Wave

1. huangchao
   

   Member
   Joined: Nov '10
   Posts: 57

   Hi Dr. Cox and Dr. Treeby,

   In your papers, you said that when the source term can be written as H(x,t)=A(x)δ(t), the inhomogeneous wave equation with source terms was equivalent to the initial value problem, where the initial conditions were given.

   But in order to show that the solutions of the two PDEs are the same, additional conditions are also required when solving the inhomogeneous wave equation so that its solution can be specified, so I wonder what are those additional conditions? Are they the boundary conditions that the acoustic pressure and its spatial derivative equal zeros at infinity, or the initial conditions that the acoustic pressure and its temporal derivative equal zeros at t=0.

   Thanks,
   Chao

   Posted 11 years ago #

2. bencox
   

   Developer
   Joined: Mar '10
   Posts: 292

   Hi Chao,

   In the initial value problem the initial conditions are taken as p(t=0) = G*H, where G is the Grueneisen parameter, and dp/dt = 0, which is satisfied if the particle velocity is initially zero. The boundary conditions could be the same in both cases; we usually assume free space conditions, ie. satisfying the Sommerfeld radiation condition. Does that help?

   Ben

   Posted 11 years ago #

3. huangchao
   

   Member
   Joined: Nov '10
   Posts: 57

   I see, so the inhomogeneous wave equation + the Sommerfeld radiation condition is equivalent to the initial value problem.

   Thanks for your help.

   Chao

   Posted 11 years ago #

4. huangchao
   

   Member
   Joined: Nov '10
   Posts: 57

   Hi Dr. Cox,

   I have another related quick question:

   In time reversal reconstruction, the homogeneous wave equation □p(x,t)=0 subjects to initial and boundary conditions: p(x,t=0)=0, dp/dt = 0, p(x_s,t)=p_m(x_s,T-t), where □ is the wave operator, x_s is the sensor location, and p_m is the measurement.

   I wonder if the above homogeneous wave equation with initial and boundary conditions is equivalent to the inhomogeneous wave equation □p(x,t)=p_m(x_s,T-t) subjecting to only initial conditions: p(x,t=0)=0, dp/dt = 0, where I just move the boundary condition to the source term and leave the initial conditions unchanged.

   Thanks,
   Chao

   Posted 11 years ago #

5. bencox
   

   Developer
   Joined: Mar '10
   Posts: 292

   Hi Chao,

   Good question. Putting a source at the boundary is not the same as imposing a time-varying Dirichlet boundary condition. In the former you are adding the pressure to the existing acoustic field, in the latter you replace whatever the value of the existing field is with the new value. Imposing the time reversed pressure as a boundary condition will give an exact reconstruction (if you have a closed measurement surface) but adding it as a source will not. (It will probably give something resembling the correct image but it won't be exact.)

   Ben

   Posted 11 years ago #

6. Hamid
   

   Member
   Joined: Oct '13
   Posts: 3

   Hello,

   Could you please give me some advice on the following paper of yours?
   http://www.k-wave.org/papers/2010-Treeby-JBO.pdf

   I would like to generate some simulation data from lossless medium, as you mentioned in the paper the combination of the linear first order equations would be a single second order wave equation. In such a medium I want to put a source and detect the signals. I tried the code you brought in the paper

   % create the computational grid
   Nx = 256;
   Nz = 128;
   dx = 50e-6;
   dz = 50e-6;
   kgrid = makeGrid(Nx, dx, Nz, dz);
   % define the medium properties
   medium.sound_speed = 1500
   ones(Nz, Nx);
   medium.sound_speed(1:50,:) = 1600;
   medium.density = 1000
   ones(Nz, Nx);
   medium.density(1:50,:) = 1040;
   % define the initial pressure
   disc_x_pos = 120;
   disc_z_pos = 75;
   disc_radius = 8;
   disc_mag = 3;
   source.p0 = disc_mag*makeDisc(Nx, Nz,disc_x_pos, disc_z_pos, disc_radius);
   % define a centered circular sensor
   sensor_radius = 2.5e-3;
   num_sensor_points = 50;
   sensor.mask = makeCartCircle(sensor_radius, num_sensor_points);
   % run the simulation
   sensor_data = kspaceFirstOrder2D(kgrid, medium, source, sensor);

   But it didn’t work and I got the error message of

   Running k-Wave simulation...
   start time: 03-Oct-2014 13:26:20
   reference sound speed: 1600m/s
   dt: 9.375ns, t_end: 8.9437us, time steps: 955
   input grid size: 256 by 128 grid points (12.8 by 6.4mm)
   maximum supported frequency: 16MHz
   smoothing p0 distribution...
   calculating Delaunay triangulation...
   precomputation completed in 0.20481s
   starting time loop...
   Error using .*
   Matrix dimensions must agree.

   Error in kspaceFirstOrder2D (line 762)
   ux_sgx = bsxfun(@times, pml_x_sgx, ...

   Error in Untitled (line 25)
   sensor_data = kspaceFirstOrder2D(kgrid, medium, source, sensor);

   Thanks,

   Posted 10 years ago #

7. bencox
   

   Developer
   Joined: Mar '10
   Posts: 292

   HI Hamid,

   Define your medium properties so that the matrices for sound_speed and density are the same size as the domain:

   % define the medium properties
medium.sound_speed = 1500*ones(Nx, Nz);
medium.sound_speed(1:50,:) = 1600;
medium.density = 1000*ones(Nx, Nz);
medium.density(1:50,:) = 1040;

   Ben

   Posted 10 years ago #

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