Speaker Driver Simulation With Room Response
For high frequency analysis.
July 24, 2023
Extended version of Speaker Directivity Modeler with room response calculation from radiated sound power and free-field response (direct sound).
Simplifications in the model
The simulation makes use of the following simplifications:
- Speaker driver has a flat, completely rigid diaphragm (all parts of the speaker vibrate in phase with the same amplitude, no cone resonances).
- Driver is mounted in a wall (hemispherical radiation, half-space simulation, no diffraction from the baffle edge).
- High-pass filter behavior, low-pass filter behavior are not part of the simulation.
- Reflections create a diffuse sound field, absorption is independent from frequency.
Sound power is the total radiated sound into full space (in our case, into half space due to the infinite baffle). The radiated sound power of a speaker driver mounted in a wall is flat up to a certain frequency. Above that frequency the radiated sound power decreases with increasing frequency. The cut-off frequency depends on the diameter of the diaphragm and is independent of angle.
Sound power curve is important in closed spaces. On-axis response room response of a speaker driver is a combination of the sound power response (scaled according to room gain) and the on-axis response (a horizontal line for an ideal loudspeaker).
Diffuse sound field
Diffuse sound field curve represents the reflections and it is calculated from the sound power response and (diffuse) room gain. It is generated by scaling the sound power response with the room gain.
Room gain = speaker level relative to free-field, reinforcement from diffuse sound field
Level of a point source in a sound reflective space relative to free-field, assuming that reflections create a diffuse sound field (this assumption is valid above 1 kHz). In other words, this is the gain from diffuse sound field. Setting the room gain to 0 dB means that the speaker is in free-field (no reflections).
Room gain is a function of:
- room size (height is the most important parameter),
- room absorption (walls, furnishing, windows, etc...),
- distance from the source,
- directivity of the source.
Though room gain - and hence the room response - is a function of room absorption, in a typical room the deviation from the on-axis free field response can be attributed to the non-flat directivity response.
The following graph shows the room gain as a function of distance. Room height is 3 meters, parameter is the floor area, source DI = 3 dB (sound source on a large baffle with half space radiation, wavelength is larger than the diameter of the source).
- Normalized room response is actually a room gain vs. frequency curve.
- Simulated room response has two different meanings. It can be considered the frequency response of an ideal system that meets the above criteria (see "Simplifications in the model"). The response can also be used as a difference curve to predict the room response from a free-field on-axis speaker driver measurement.
(room_curve_dB = normalized_room_curve_dB + on_axis_free_field_curve_dB)
Measurement vs model
On-axis measurement of a small full-range loudspeaker in a 20 m2 room. The cone diameter of the speaker is 6 cm and the microphone distance from the loudspeaker is two meters.
The blue curve is the difference between the 50 ms gated response (room response) and 1.6 ms gated response (free-field response, valid above approx. 500 Hz). In other words, the room response is normalized to the free-field response. The red curve is the simulated on-axis response with 6 dB room gain.
Simulation of pressure microphones
The software is also suitable for modeling directivity (between ± 90°) and room responses of pressure microphones.
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