Acoustic simulation tools to study speaker directivity

One of the most important features of loudspeakers and drivers is how their directivity (radiation pattern) varies as a function of frequency. For example, a 10 cm diameter speaker built into a wall radiates in all directions with the same intensity up to 1 kHz (constant directivity and sound power), on the other hand the intensity of the radiation gradually decreases off-axis from 1 kHz (increasing directivity, decreasing sound power). The reason for that the speaker is not an ideal point source and becomes directive at high frequencies due to destructive interference.

The simplest directivity model for cone speakers is the circular rigid piston mounted in an infinite baffle. While this is not the most accurate model, still a fair good approximation of real speakers up to a certain frequency. This frequency depends on the cone shape, material etc...

There are plenty of tools to display directivity: polar diagram (polar chart), DI (directivity index) vs. frequency, sound power vs. frequency, sonogram, parametric frequency response graphs.

Speaker Polar Chart Display 1.02

Software platforms: Excel | OpenOffice | LibreOffice

This Excel workbook displays the far field polar response of a theoretical cone speaker mounted in an infinitely large baffle. In the simulation all parts of the speaker vibrate in phase and with the same amplitude, and the speaker is completely flat.

The input parameters are the diameter of the cone and the frequency. In the polar graph the normalized sound pressure level is shown as a function of the angle at the selected frequency.

Speaker Polar Chart Display

Version: 1.02
Format: xls (Microsoft Excel 2003 Workbook)
Required software: LibreOffice Calc | Microsoft Excel | OpenOffice Calc
License: freeware

Speaker Directivity Simulator 1.03

Software platforms: Excel | OpenOffice1 | LibreOffice1

This Excel workbook displays the far field directivity of a theoretical cone speaker mounted in an infinite baffle. The user can select from frequency response graph, 2D color map (sonogram) and waterfall plot. In version 1.03 the simulation is extended with sound power and directivity index calculation.

Note: above a certain frequency the directivity calculated from the rigid piston model is different from the directivity of real cone speakers due to cone break-up modes, propagation delay in the cone and cone geometry.

Cone speaker (circular piston) directivity simulator

Version: 1.03
Format: xls (Microsoft Excel 2003 Workbook)
Required software: LibreOffice Calc1 | Microsoft Excel | OpenOffice Calc1
License: freeware

Note: In OpenOffice Calc and LibreOffice Calc waterfall plot and 2D color map are not available.

About the accuracy of the model

As the speaker becomes more directional (starts to 'beaming') with increasing frequency, the difference between the output of the rigid piston model and the normalized off-axis measurements of cone speakers becomes greater. If the maximum allowed deviation from the measured response is 2 dB (the measured off-axis response is normalized to 0 deg), then the high frequency limit of the flat piston model for cone speakers:

(The value of ka is calculated and displayed.)

Simulation vs. measurement with an example: the graph below shows the simulated response of a 6.2 cm speaker (D = 6.2 cm, Sd = 30.2 cm2) from 30° off-axis and the normalized off-axis response of the Dayton Audio PC83-4 3" full range speaker from 30°. The speakers have the same effective cone area.

Dayton Audio PC83-4 normalized 30 deg off-axis response, simulation vs. measurement

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