Analysis Strategy

Numerical methods are ideal for investigating abstract concepts alongside real-world examples. It is a simple matter to calculate steady-state sound pressure levels from radiating surface in half-space using the Boundary Element Method (BEM). However, BEM, based on the free-space Green's function, will not cater for rigid boundaries other than flat rigid planes, hence we cannot use BEM to model an infinite horn.

While a finite conical horn may be readily modeled using FEM it is difficult to terminate it with the correct acoustic impedance and avoid mouth reflections. The strategy adopted here was to carry out the analysis in the time domain and terminate the calculation before waves reflections reach the sampling points. Frequency responses may then be obtained using a Fourier transform.

In this paper the regions of fluid are modeled with quadratic axisymmetric acoustic finite elements allowing the use of a implicit transient solver , as described in reference [2]. Shell elements with enforced axial motion represent a diaphragm in rigid body motion. [3]

Transient FEM

The natural boundary condition for pressure-based acoustic finite elements is the Neumann condition, corresponding to a rigid boundary. The fluid region is infinite but must be truncated so the number of elements is not too high, avoiding excessive calculation times.

Figure 1 illustrates the geometry used for a finite conical horn with a spherical cap radiator. This figure is not to scale and the element size is much bigger than in the actual models to allow graphical reproduction. The pressure is sampled at the arc of points. Since the arc is approximately coincident with apparent source of the radiation, the impulse reaches all points at the same time. The impulse then travels to the end of the mesh where it is reflected back towards the points. The analysis is terminated before the reflected impulse reaches the sampling points.

For all of the models presented, the impulse responses were extracted at on an arc of points 0.12m from the source center. It was found that a model radius of 0.16m was sufficient for the impulse to have decayed prior to reflections from the end of the mesh reaching the sampling points.

The mesh size required for this type of analysis is related to the time step and excitation signal as described in [3]. The models presented have a nominal element size of 0.7mm with a time step of 0.5e-6 seconds. A smoothed step function defines diaphragm displacement to provide a velocity “impulse”. To allow direct comparison all modeled radiators in this paper have a one inch outside diameter.

Result post processing

To facilitate straightforward interpretation of the data the results have been post-processed as described below.

Response curve

The impulse responses were Fourier transformed and normalized, by dividing with the excitation spectra. The resulting constant velocity spectra are then divided by jω converting them to constant acceleration spectra. Constant acceleration corresponds to the mass controlled region of a loudspeaker. Levels have been normalized to give 0dB from the planar piston and a 50dB range is displayed.

Directivity Contour plot

Since it is usual for loudspeakers to be balanced to give a compromise between flat energy and frequency response the directivity contours are normalized to the RMS response averaged over –30 to +30 degrees. A range of 10dB with 1.5dB steps has been used.

Impulse Contour plot

The initial results have not been illustrated here since they are dominated by the overall response. Since in practice loudspeakers are equalized to have a substantially flat response on axis it was decided to normalize the on axis response This was achieved by applying an inverse filter to the FEM results in the time domain.

The results shown here have a range of 35dB with 5dB steps.

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