k-Wave User Forum » Topic: equivalence of wave equations

k-Wave User Forum » Topic: equivalence of wave equations http://www.k-wave.org/forum/topic/equivalence-of-wave-equations Support for the k-Wave MATLAB toolbox en-US Wed, 13 May 2026 00:22:33 +0000 http://bbpress.org/?v=1.0.2 <![CDATA[Search]]> q http://www.k-wave.org/forum/search.php bencox on "equivalence of wave equations" http://www.k-wave.org/forum/topic/equivalence-of-wave-equations#post-4757 Sun, 05 Oct 2014 10:13:43 +0000 bencox 4757@http://www.k-wave.org/forum/ HI Hamid, Define your medium properties so that the matrices for sound_speed and density are the same size as the domain: <code>% define the medium properties</code> <code>medium.sound_speed = 1500*ones(Nx, Nz);</code> <code>medium.sound_speed(1:50,:) = 1600;</code> <code>medium.density = 1000*ones(Nx, Nz);</code> <code>medium.density(1:50,:) = 1040;</code> Ben Hamid on "equivalence of wave equations" http://www.k-wave.org/forum/topic/equivalence-of-wave-equations#post-4756 Fri, 03 Oct 2014 21:38:20 +0000 Hamid 4756@http://www.k-wave.org/forum/ Hello, Could you please give me some advice on the following paper of yours? <a href="http://www.k-wave.org/papers/2010-Treeby-JBO.pdf" rel="nofollow">http://www.k-wave.org/papers/2010-Treeby-JBO.pdf</a> 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, bencox on "equivalence of wave equations" http://www.k-wave.org/forum/topic/equivalence-of-wave-equations#post-1148 Mon, 31 Dec 2012 01:08:13 +0000 bencox 1148@http://www.k-wave.org/forum/ 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 huangchao on "equivalence of wave equations" http://www.k-wave.org/forum/topic/equivalence-of-wave-equations#post-1147 Sun, 30 Dec 2012 20:50:08 +0000 huangchao 1147@http://www.k-wave.org/forum/ 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 huangchao on "equivalence of wave equations" http://www.k-wave.org/forum/topic/equivalence-of-wave-equations#post-1115 Mon, 17 Dec 2012 00:51:41 +0000 huangchao 1115@http://www.k-wave.org/forum/ I see, so the inhomogeneous wave equation + the Sommerfeld radiation condition is equivalent to the initial value problem. Thanks for your help. Chao bencox on "equivalence of wave equations" http://www.k-wave.org/forum/topic/equivalence-of-wave-equations#post-1114 Mon, 17 Dec 2012 00:39:00 +0000 bencox 1114@http://www.k-wave.org/forum/ 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 huangchao on "equivalence of wave equations" http://www.k-wave.org/forum/topic/equivalence-of-wave-equations#post-1113 Mon, 17 Dec 2012 00:05:59 +0000 huangchao 1113@http://www.k-wave.org/forum/ 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