k-Wave Toolbox


Compute O'Neil's solution for focused bowl transducer.


[p_axial, p_lateral] = focusedBowlONeil(radius, diameter, velocity, ...
    axial_position, lateral_position, frequency, sound_speed, density)


focusedBowlONeil calculates O'Neil's solution (O'Neil, H. Theory of focusing radiators. J. Acoust. Soc. Am., 21(5), 516-526, 1949) for the axial and lateral pressure amplitude generated by a focused bowl transducer when uniformly driven by a continuous wave sinusoid at a given frequency and normal surface velocity.

The solution is evaluated at the positions along the beam axis given by axial_position (where 0 corresponds to the transducer surface), and lateral positions through the geometric focus given by lateral_position (where 0 corresponds to the beam axis).

Note, O'Neil's formulae are derived under the assumptions of the Rayleigh integral, which are valid when the transducer diameter is large compared to both the transducer height and the acoustic wavelength.


% define transducer parameters
radius      = 140e-3;   % [m]
diameter    = 120e-3;   % [m]
velocity    = 100e-3;   % [m/s]
frequency   = 1e6;      % [Hz]
sound_speed = 1500;     % [m/s]
density     = 1000;     % [kg/m^3]

% define position vectors
axial_position   = 0:1e-4:250e-3;       % [m]
lateral_position = -15e-3:1e-4:15e-3;   % [m]

% evaluate pressure
[p_axial, p_lateral] = focusedBowlONeil(radius, diameter, velocity, ...
    frequency, sound_speed, density, axial_position, lateral_position);

% plot
subplot(2, 1, 1);
plot(axial_position .* 1e3, p_axial .* 1e-6, 'k-');
xlabel('Axial Position [mm]');
ylabel('Pressure [MPa]');
subplot(2, 1, 2);
plot(lateral_position .* 1e3, p_lateral .* 1e-6, 'k-');
xlabel('Lateral Position [mm]');
ylabel('Pressure [MPa]');


radius transducer radius of curvature [m]
diameter diameter of the circular transducer aperture [m]
velocity normal surface velocity [m/s]
frequency driving frequency [Hz]
sound_speed speed of sound in the propagating medium [m/s]
density density in the propagating medium [kg/m^3]
axial_position vector of positions along the beam axis where the pressure amplitude is calculated [m]
lateral_position vector of positions along the lateral direction where the pressure amplitude is calculated [m]


p_axial pressure amplitude at the positions specified by axial_position [Pa]
p_lateral pressure amplitude at the positions specified by lateral_position [Pa]

See Also