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 an infinite baffle (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). Above a certain frequency the radiated sound power of speaker drivers 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 at HF
Room gain at HF = speaker level with reflections, reinforcement from diffuse sound field, diffuse room gain, high-frequency room gain
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).
The sound field created by reflections is diffuse, but the total sound field is not necessarily diffuse (depends on room gain). This gain is not the same as the low-frequency room gain, which increases with decreasing frequency (reflections become more and more coherent).
Diffuse room gain values (well furnished room, without special acoustical treatment, absorption coefficient doesn't change with frequency, room height: 3 meters, room size: max 30 m2):
- Near-field listening (e.g.: sitting close to computer speakers): 1-3 decibels.
- "Normal" listening: 4-7 dB.
- Speaker is close to a wall, listening position is close to the opposite wall: 8-10 dB.
Diffuse room gain values higher than 10 dB are unrealistic. Values around 5 dB seems typical for most rooms and listening distances.
The following graph shows the room gain as a function of distance from the source for a small (20 m2) and large rooms (large room: width and height of the room is much larger than its height). Room height is 3 meters, source DI = 3 dB (sound source on a large baffle with half space radiation, wavelength is larger than the diameter of the source).
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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