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# focusedAnnulusONeil

Compute axial pressure for focused annulus transducer using O'Neil's solution.

## Syntax

```p_axial = focusedAnnulusONeil(radius, diameter, amplitude, phase, frequency, sound_speed, density, axial_position)
```

## Description

`focusedAnnulusONeil` calculates the axial pressure for a focused annulus transducer using O'Neil's solution (O'Neil, H. Theory of focusing radiators. J. Acoust. Soc. Am., 21(5), 516-526, 1949). The annuluar elements are 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).

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.

## Inputs

 `radius` transducer radius of curvature [m] `diameter` 2 x num_elements array containing pairs of inner and outer aperture diameter (diameter of opening) [m] `amplitude` array containing the normal surface velocities for each element [m/s] `phase` array containing the phase for each element [rad] `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]

## Outputs

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

`focusedBowlONeil`, `mendousse`