AcousticSolver
Package: kwave.toolbox Superclasses: kwave.toolbox.TimeDomainSolver
Time domain acoustic wave solver.
Description
AcousticSolver implements a time-domain solution to the wave equation written as three coupled equations in the form:
$$ \frac{\partial \vec{u} }{\partial t}=-\frac{1}{\rho_0 }\nabla (p) $$
$$ \frac{\partial \rho }{\partial t}=-\rho_0 \nabla \cdot (\vec{u} ) $$
$$ p=c_0^2 \rho $$
When absorption y is declared as a medium property
$$ p=c_0^2 \rho +\tau (-\nabla^2 )^( y/2-1)\frac{\partial \rho }{\partial t}+\eta (-\nabla^2 )^( (y-1)/2)\rho $$
where \(p\) is the acoustic pressure, \(\vec{u}\) is the acoustic particle velocity, \(\rho\) is the acoustic density, \(\rho_0\) is the ambient density, \(c_0\) is the sound speed, and \(t\) is time.
The computation is based on a k-space pseudospectral scheme in which spatial gradients are calculated using the Fourier collocation spectral method, and temporal gradients are calculated using a k-space corrected finite difference scheme. For a homogeneous medium, the formulation is exact and unconditionally stable. For a heterogeneous medium, the time scheme allows larger time-steps to be taken for the same level of accuracy compared to conventional pseudospectral time-domain methods.
The simulation is defined by five input objects which define the computational grid, medium properties, sources, sensors, and settings.
After an object of the AcousticSolver class is created, the simulation is run by calling solver.run. During the simulation, a visualisation of the pressure field is displayed. The current pressure can be queried (or modified) at any point using the property solver.pressure.
Examples
A simple of example of solving a initial value problem in 1D is shown below.
clearvars;
import kwave.toolbox.*
% Grid.
kgrid = Grid(128, 1e-3, 20);
% Medium.
medium = AcousticMedium(kgrid);
medium.soundSpeed = 1500;
medium.density = 1000;
% Source.
source = AcousticSource(kgrid);
source.initialPressure = exp( -kgrid.xVec.^2 ./ (10 * kgrid.dx).^2 );
% Settings.
settings = Settings;
settings.plotFrequency = 1;
% Solve.
solver = AcousticSolver(kgrid, medium, source, [], settings);
CFL = 0.5;
dt = CFL * kgrid.dx / medium.soundSpeed;
solver.run(Nt=80, dt=dt);
% Plot.
figure;
plot(1e3 * kgrid.xVec, source.initialPressure);
hold on;
plot(1e3 * kgrid.xVec, solver.pressure);
set(gca, 'YLim', [0, 1]);
grid on;
legend('p_0', 'p_{final}');
xlabel('Position [mm]');
ylabel('Pressure [Pa]');
Input Arguments
kgrid- (kwave.toolbox.Grid) Object which defines the simulation grid size.medium- (kwave.toolbox.AcousticMedium) Object which defines the medium properties.source- (kwave.toolbox.AcousticSource) Object which defines the source properties.sensor- (kwave.toolbox.AcousticSensor) Object which defines the sensor properties.settings- (kwave.toolbox.Settings) Object which defines the simulation settings.
Properties
pressure- (numeric) Pressure field [Pa].densitySplit- (numeric) Vector field of the split-field components of the acoustic density [kg/m^3].velocity- (numeric) Vector field of the acoustic particle velocity [m/s]absorptionType- (string) Must be a member of { 'on', 'off', 'noAbsorption', 'noDispersion'} defaults to 'off'. Switch to include to exclude each of the absorption terms within the equation of state.