Acoustic Fields, Microphones & Loudspeakers

Sound is a coupled pressure-and-motion field in a medium. A microphone converts one aspect of that field into an electrical signal; a loudspeaker converts electrical drive into mechanical motion and then into a field. Neither has one universal “frequency response”: geometry, direction, acoustic loading, distance, boundary conditions, and electrical loading are part of the measurement.

The durable mental model is an energy chain with reference planes. At each boundary, identify the quantity being conserved or transformed, its impedance, and the calibration that connects the number to pascals, metres per second, watts, volts, or digital full scale.

Pressure, velocity, impedance, and intensity #

Small-signal acoustics linearizes fluctuations around ambient pressure. Sound pressure $p(t)$ is a scalar in pascals; particle velocity $\mathbf u(t)$ is the local oscillatory velocity of the medium, not the propagation speed.

For a plane progressive wave in a lossless fluid,

$$Z_0=\frac{p}{u}=\rho c,$$

where $\rho$ is density and $c$ is sound speed. Instantaneous intensity is

$$\mathbf I(t)=p(t)\mathbf u(t),$$

and average intensity for in-phase RMS quantities is $I=p_{rms}^2/(\rho c)$. In reactive near fields or standing waves, pressure and velocity are not generally in phase; high pressure need not mean outward net power.

Sound-pressure level in air is

$$L_p=20\log_{10}\left(\frac{p_{rms}}{20\ \mu\mathrm{Pa}}\right)\ \mathrm{dB}.$$

Because acoustic intensity is proportional to pressure squared under plane-wave conditions, the pressure ratio uses $20\log_{10}$ while a power ratio uses $10\log_{10}$. A decibel value without reference quantity, bandwidth, weighting, and detector time behavior is incomplete.

Spreading, boundaries, and fields #

An ideal point source in a homogeneous free field spreads power over a sphere. Far enough from the source,

$$p_{rms}(r)\propto\frac1r, \qquad I(r)\propto\frac1{r^2},$$

so doubling distance reduces level by about 6 dB. This is not a universal room rule. Ground, walls, ducts, arrays, atmospheric gradients, absorption, and coherent reflections change spreading and interference.

flowchart TB
  subgraph FIELD["Radiation and field"]
    direction LR
    E["Electrical drive"] --> M["Diaphragm motion"] --> F["Pressure + particle velocity"] --> R["Room / boundaries"]
  end
  subgraph CAP["Capture"]
    direction LR
    P["Microphone"] --> A["Preamp + ADC"] --> D["Calibrated samples"]
  end
  R --> P

The near field contains reactive energy and geometry-dependent interference; pressure/velocity ratio differs from $\rho c$. In the far field, angular pattern stabilizes and outgoing-wave approximations become useful. Far-field distance depends on aperture $D$ and wavelength $\lambda$; a common scale is $2D^2/\lambda$, but the actual criterion belongs to the source and required error.

Boundaries create reflected images and standing waves. In a room, the transfer function changes dramatically over centimetres at high frequencies and across seats at low-frequency modes. A single point response is not “the room.” Spatial averaging answers a different question than precise point calibration.

Directivity is frequency dependent #

A source or receiver small compared with wavelength is approximately omnidirectional. As aperture becomes comparable to wavelength, phase varies across it and the pattern narrows. Arrays deliberately control those phases to steer beams and nulls.

Directivity factor $Q$ is radiated intensity in a direction relative to an omnidirectional source with the same total acoustic power. Directivity index is

$$DI=10\log_{10}Q.$$

A polar plot must name frequency; one polar curve cannot characterize a broadband loudspeaker or microphone. Off-axis response matters because reflected sound reaches the listener from many angles and because a nominally flat on-axis transducer can radiate uneven total power.

Microphones sense different field quantities #

A pressure microphone exposes one side of a diaphragm to the field and the other to a reference cavity; it is approximately omnidirectional at low frequency. A pressure-gradient microphone responds to pressure difference across separated ports, producing directional patterns but also proximity effect and sensitivity to port spacing.

Measurement microphones are corrected for a specified field:

• pressure response: pressure at a closed boundary or coupler;
• free-field response: compensates the microphone's disturbance of a wave arriving from a specified direction, commonly on axis;
• diffuse-field response: averages incidence from many directions.

The Brüel & Kjær handbook stresses that using the wrong correction produces a frequency-dependent error. A microphone calibrated flat in a free field is not automatically flat in a diffuse reverberant field.

Sensitivity may be given in mV/Pa or dBV/Pa. For sensitivity $S$ V/Pa and measured RMS voltage $V$,

$$p=\frac{V}{S}$$

only if the preamplifier gain, loading, frequency correction, and cable/interface behavior are included. Self-noise sets the low end; diaphragm nonlinearity, electronics, and clipping set the high end. Dynamic range must be specified over bandwidth.

Loudspeakers are coupled electromechanical systems #

A moving-coil driver combines electrical impedance, force factor, moving mass/compliance/damping, diaphragm radiation, enclosure, and crossover. The input impedance varies with frequency; “8 Ω” is a nominal class, not a resistor. Near resonance, cone motion and electrical impedance are controlled by stored mechanical/acoustic energy; above the piston region, breakup and beaming invalidate a rigid-diaphragm model.

Sensitivity is often stated as SPL at 1 m for 2.83 V or 1 W under defined conditions. Those are equivalent only for an 8 Ω resistive load. Maximum SPL also depends on excursion, thermal compression, distortion, bandwidth, and duration. A high sensitivity number at one frequency is not broadband output capability.

Enclosures change acoustic loading:

Crossovers divide bands while their phase, acoustic centres, and directivity overlap. Flat electrical branches do not guarantee flat acoustic sum.

Rooms add decay and intelligibility constraints #

Reverberation time $T_{60}$ is the time for a diffuse-field decay to fall 60 dB. Sabine's approximation,

$$T_{60}\approx0.161\frac{V}{A},$$

relates room volume $V$ in m$^3$ to equivalent absorption area $A$ in m$^2$ under assumptions of a sufficiently diffuse field and modest absorption. Real rooms violate those assumptions through localized absorption, coupled volumes, modes, and non-exponential decay; report frequency bands and measurement method.

Speech intelligibility depends on direct-to-reverberant ratio, background noise, bandwidth, distortion, and temporal smearing. Raising level can worsen reverberant masking or trigger limiting; equalization cannot remove late reflections everywhere. Geometry and acoustic treatment precede DSP when the defect is spatial.

Calibrate the complete chain #

1. Define the acoustic quantity, location, direction, bandwidth, and field type.
2. Apply a traceable acoustic calibrator or comparison at the microphone reference plane.
3. Record microphone sensitivity and correction, preamp gain, ADC full-scale mapping, sample rate, and environmental conditions.
4. Check noise floor and overload with the intended bandwidth and weighting.
5. Measure repeatability after repositioning; a suspiciously perfect repeat can mean geometry was never challenged.
6. For transfer response, use known excitation and synchronous/correlation methods to separate response from unrelated noise.
7. Gate reflections only when the resulting time window and low-frequency resolution are reported.
8. Preserve raw data so later calibration or weighting corrections remain possible.

The essential discipline is to name the field and reference plane. “Flat from 20 Hz to 20 kHz” is not a physical claim until distance, angle, environment, level, tolerance, and calibration are attached.