Why is mathematics important in microphone use? A microphone converts changing air pressure into an electrical signal, but placement and recording choices determine what enters that signal. Distance changes the balance of direct sound and room sound. Direction changes response through a polar pattern. Gain changes level but not the acoustic signal-to-noise ratio already captured. Decibels compress huge ratios into manageable numbers.
This article develops educational models for sound pressure, inverse-distance change, polar patterns, signal-to-noise ratio, sampling and stereo timing. It does not prescribe hearing exposure limits, rig suspended equipment or replace manufacturer instructions. Sound levels can damage hearing, stands can fall, and powered systems have electrical risks. Use current workplace guidance and qualified audio practice where applicable.
Choose Your Reading Route
- Start with level: convert pressure and voltage ratios to decibels.
- Move the microphone: model distance, direct-to-reverberant balance and proximity.
- Turn the microphone: read polar plots and off-axis response.
- Build a recording chain: track sensitivity, gain, noise and headroom.
- Work examples: plan speech capture and a stereo pair.
Sound Pressure and Decibels
Sound pressure p is a changing pressure around atmospheric pressure, often represented by root-mean-square value for a steady signal. Sound pressure level is
Lp = 20 log10(p/p0) dB,
where reference pressure in air is commonly p0 = 20 micropascals.
If p = 0.20 Pa,
Lp = 20 log10(0.20/20×10⁻6) = 80 dB.
The decibel value is a ratio, not a standalone physical unit. The reference and weighting must be stated when needed.
Why 20 for pressure and voltage
Power in the same impedance is proportional to squared pressure or voltage. Since 10 log10(P2/P1) and P ratio equals amplitude ratio squared,
10 log10(A²) = 20 log10(A).
For power ratios directly, use 10 log10. Confusing the two can double a result.
Useful ratios
- Doubling amplitude: 20 log10 2 = +6.02 dB.
- Halving amplitude: −6.02 dB.
- Ten times amplitude: +20 dB.
- Twice power: 10 log10 2 = +3.01 dB.
- Ten times power: +10 dB.
These are mathematical ratios, not exact statements about perceived loudness. Hearing response depends on frequency, duration, level and listener.
Adding independent sound levels
Decibel values cannot be arithmetically added as though they were linear pressures. For independent sources with levels L1 and L2,
Ltotal = 10 log10(10^(L1/10) + 10^(L2/10)).
Two independent 70 dB sources combine to about 73.01 dB, not 140 dB. Adding ten equal independent sources adds 10 dB.
Coherent signals can interfere by phase, so pressure addition may reinforce or cancel. The independent-energy formula has assumptions.
Distance and the Inverse-Distance Model
For a small source radiating into free space, pressure amplitude approximately decreases in proportion to 1/r in the far field. The level difference between distances r1 and r2 is
Delta L = 20 log10(r1/r2).
Doubling distance gives 20 log10(1/2) = −6.02 dB. Halving distance gives +6.02 dB.
Worked distance table
If direct sound is 84 dB at 0.50 m under the ideal model:
| Distance | Change from 0.50 m | Predicted direct level |
|---|---|---|
| 0.25 m | +6.02 dB | 90.02 dB |
| 0.50 m | 0 dB | 84.00 dB |
| 1.00 m | −6.02 dB | 77.98 dB |
| 2.00 m | −12.04 dB | 71.96 dB |
Real rooms reflect sound, sources are not perfect points, and near-field behaviour differs. The table predicts direct-field change, not total room level.
Direct and reverberant sound
Direct sound generally falls with distance. Reverberant sound can be more nearly uniform over part of a room. Moving farther away therefore reduces the direct-to-reverberant ratio even if total meter reading changes less than 6 dB per doubling.
If direct level at one position is 78 dB and reverberant background is 68 dB, their independent total is
10 log10(10^7.8 + 10^6.8) = 78.41 dB.
Move so direct sound falls 6 dB to 72 dB while reverberant remains 68 dB. Total becomes 73.46 dB. The meter falls only 4.95 dB, but clarity can change substantially because the direct-reverberant difference shrank from 10 to 4 dB.
Critical distance
Critical distance is where direct and reverberant energy are equal under a room model. It depends on source directivity and room absorption. Staying within it often improves directness, but placement also must suit tone, movement and practical framing.
Did You Know?
Moving a microphone closer can improve the desired-to-room ratio without increasing the source’s acoustic output. Electronic gain raises desired signal and captured room together; distance changes their acoustic balance before conversion.
Polar Patterns as Functions of Angle
A polar pattern shows sensitivity versus arrival angle, usually at a stated frequency. It is not a map of where the microphone “listens” in metres. Radius represents relative response, and angle represents direction.
Omnidirectional model
An ideal omni has
R(theta) = 1.
at every angle. Real microphones become directional at high frequencies because their bodies are not infinitesimal and ports interact with wavelength.
Figure-eight model
An ideal pressure-gradient figure-eight has signed response
R(theta) = cos theta.
Magnitude is maximum front and rear, zero at 90° and 270°, with polarity reversal at the back.
Cardioid model
An ideal cardioid combines omni and figure-eight components:
R(theta) = 0.5(1 + cos theta).
At 0°, R = 1. At 90°, R = 0.5, which is 20 log10 0.5 = −6.02 dB. At 180°, the ideal response is zero.
Real rear rejection is finite and frequency-dependent. A polar plot measured at 1 kHz cannot describe 100 Hz or 10 kHz automatically.
General first-order family
Write
R(theta) = a + (1−a)cos theta.
a = 1 gives omni; a = 0.5 cardioid; a = 0 figure-eight. Other a values produce subcardioid, supercardioid or hypercardioid-like patterns with different rear lobes and nulls.
Find a null by setting R = 0:
cos theta_null = −a/(1−a).
For a = 0.25, cos theta = −1/3, giving nulls near 109.47° and 250.53°.
Signed versus magnitude response
Polar plots usually display magnitude, so negative response appears as a positive radial lobe. The negative sign denotes phase reversal, not negative loudness. Stereo and multi-microphone work must retain phase information.
Frequency Changes Directionality
Wavelength is
lambda = c/f,
where c is sound speed and f frequency. At c = 343 m/s:
- 100 Hz wavelength is 3.43 m.
- 1 kHz wavelength is 0.343 m.
- 10 kHz wavelength is 0.0343 m.
A microphone body or port spacing becomes acoustically large at shorter wavelengths. Diffraction and interference therefore change the pattern with frequency.
Off-axis colour
A source 90° off a nominal cardioid may be 6 dB lower at one frequency but far more or less at another. The resulting frequency response changes tone, called off-axis coloration.
Pointing a null at unwanted sound is useful only if the null exists across relevant frequencies and placement does not worsen desired sound. Manufacturer polar plots should be read at multiple frequencies.
Orientation conventions
End-address and side-address microphones have different acoustic axes. A visually obvious “front” may not be the active direction. Always use product markings and documentation.
Sensitivity, Gain and Voltage
Microphone sensitivity may be specified as volts per pascal or dBV per pascal at a frequency such as 1 kHz.
If sensitivity is 20 mV/Pa and incident pressure is 0.10 Pa, ideal output is
V = 0.020×0.10 = 0.0020 V = 2.0 mV.
In dBV, 2 mV is
20 log10(0.002/1) = −53.98 dBV.
If a preamplifier adds 40 dB voltage gain, gain factor is 10^(40/20) = 100. Output becomes 0.20 V or −13.98 dBV, before headroom and loading effects.
dBu and dBV differ
dBV references 1 V. dBu references approximately 0.775 V. A value of 0 dBu is not 0 dBV. Always keep the reference.
Gain does not repair acoustic signal-to-noise ratio
If speech and room noise reach the capsule with a 12 dB difference, ideal linear gain raises both by the same amount. Later gain cannot recreate separation absent at capture.
Low electronic noise matters when microphone output is small. A sensitive microphone may need less preamp gain, but maximum level, self-noise, source distance and interface performance all matter.
Noise, Dynamic Range and Headroom
Signal-to-noise ratio is a level difference:
SNR = Lsignal − Lnoise
when quantities use compatible definitions.
If a desired RMS signal is −18 dBFS and noise is −60 dBFS, SNR is 42 dB. Digital full scale, dBFS, has zero as maximum representable peak convention; analogue references differ.
Equivalent input noise
Preamplifier noise can be referred to its input so it can be compared with microphone signal. Combining independent noise sources requires energy addition, not arithmetic decibel addition.
If microphone-equivalent noise is −120 dBu and preamp EIN is −128 dBu, combined level is
10 log10(10^(−12) + 10^(−12.8)) dBu ≈ −119.36 dBu.
The louder noise source dominates.
Headroom
Average speech level may be far below peaks. Recording at −18 dBFS average with peaks 12 dB higher leaves about 6 dB before 0 dBFS. A surprise 8 dB peak would clip.
Peak and RMS are different. Crest factor is peak-to-RMS ratio in dB. Percussion can have larger crest factor than compressed speech.
Quantisation and bit depth
An ideal N-bit converter has 2^N codes. A commonly cited ideal sine-wave quantisation SNR is approximately
6.02N + 1.76 dB.
For 16 bits, about 98.08 dB. Real converters have analogue noise, distortion and nonideal behaviour, and recordings rarely use every code. Bit depth does not determine microphone self-noise or room noise.
Sampling and Aliasing
Sampling rate fs must exceed twice the highest represented frequency in an ideal band-limited signal:
fs > 2fmax.
An input above Nyquist can alias into the baseband. If fs = 48 kHz and a 30 kHz component reaches sampling without filtering, one alias appears at |30−48| = 18 kHz.
Real converters use anti-alias filtering, oversampling and transition bands. The theorem is a condition, not an instruction to omit filters.
Why Mathematics? | Digital Audio Sampling, Nyquist Rate and Aliasing develops sampling in depth.
Worked Example 1: Recording a Speaker
A cardioid microphone is 0.40 m from a speaker’s mouth. Desired direct speech at the capsule is 82 dB SPL. A ventilation source produces 62 dB SPL at the capsule, and room reverberation attributable to the speech is 68 dB SPL. Assume independent components for this simplified energy calculation.
Desired-to-ventilation ratio
82 − 62 = 20 dB.
The desired speech energy is 100 times the ventilation energy because 10^(20/10) = 100.
Combine unwanted components
Ventilation and reverberant components combine:
Lunwanted = 10 log10(10^6.2 + 10^6.8) = 68.97 dB.
Desired-to-unwanted difference is 82 − 68.97 = 13.03 dB.
Move to 0.20 m
Under inverse-distance direct-field behaviour, direct speech rises 6.02 dB to 88.02 dB. Suppose ventilation and reverberant field remain approximately unchanged. Desired-to-unwanted difference becomes 19.05 dB.
Electronic gain could restore the original meter level, but the closer placement has captured a better acoustic ratio by about 6 dB.
Include cardioid angle
Suppose ventilation arrives at 120° in the ideal cardioid model:
R = 0.5(1 + cos 120°) = 0.25.
Amplitude attenuation is 20 log10 0.25 = −12.04 dB, reducing ventilation from 62 to 49.96 dB in the ideal single-frequency model.
Real off-axis response varies with frequency, reflections arrive from many angles, and moving the microphone changes geometry. Do not claim a guaranteed 12 dB broadband improvement.
Sensitivity and preamp gain
At 82 dB SPL, pressure is
p = 20×10⁻6 × 10^(82/20) = 0.2518 Pa.
For sensitivity 15 mV/Pa, output is 3.777 mV. In dBV that is −48.46 dBV. Adding 35 dB gain gives −13.46 dBV, about 0.212 V RMS.
This calculation checks electrical scale but not clipping on peaks. Crest factor and interface maximum level remain necessary.
Proximity Effect and Pressure Gradient
Directional microphones often show increased low-frequency response at close distance, called proximity effect. Pressure-gradient operation responds partly to pressure difference across spaced acoustic paths. At low frequency and near a source, phase and amplitude gradients differ from the far-field calibration.
The amount depends on pattern, design and source. It can add warmth or create boominess. An omnidirectional pressure microphone ideally lacks the same gradient-based proximity effect, though real mounts and rooms still affect tone.
Distance choice therefore changes more than level. It changes room balance, low-frequency response, movement sensitivity and plosive risk.
Time Delay, Phase and Multiple Microphones
A path difference Delta d creates time delay
Delta t = Delta d/c.
For Delta d = 0.20 m and c = 343 m/s, delay is 0.000583 s, or 0.583 ms.
At frequency f, phase difference is
phi = 360° f Delta t.
At 1 kHz, phi ≈ 210°. At frequencies where path difference equals half a wavelength, signals can partially cancel when combined with similar levels.
Comb filtering
Adding a signal to a delayed version produces regularly spaced peaks and nulls. Null frequencies for equal-amplitude simple addition occur approximately where
f = (2k+1)/(2Delta t).
For Delta t = 0.583 ms, first null is about 857.6 Hz, followed by odd multiples.
Real reflections and microphone responses soften the pattern, but the mathematics explains hollow tone from nearly coincident delayed signals.
The three-to-one guideline
A practical guideline sometimes places microphones at least three times farther from each other than each is from its source, aiming to reduce leakage level. It is not a physical law and cannot guarantee phase compatibility. Source directivity, room, polar patterns and mix levels matter.
Worked Example 2: A Stereo Pair
Two omnidirectional microphones are spaced 0.40 m apart. A source is far enough away that a plane-wave approximation is reasonable and arrives 30° off the perpendicular bisector.
Path difference is approximately
Delta d = b sin theta = 0.40×0.5 = 0.20 m.
Delay is 0.583 ms. At 500 Hz, phase difference is
360×500×0.000583 = 104.9°.
At 2 kHz, phase difference wraps to about 419.8°, equivalent to 59.8° modulo 360, but phase-versus-frequency continues to create combing if channels are summed.
Interaural-style cue
The delay can help encode direction in stereo playback. Larger spacing increases delay cues but can reduce mono compatibility. There is no universal best spacing; technique serves the scene and output format.
Level differences with directional microphones
If the pair instead uses crossed cardioids at one point, spacing delay is near zero while polar responses create interchannel level differences. Coincident techniques trade time differences for level differences.
The mathematics helps compare designs: one manipulates Delta t, another R(theta), and near-coincident arrangements use both.
Polar-Plot Measurement and Interpolation
A manufacturer measures response at discrete angles and frequencies under stated conditions. Connecting points on a polar plot is interpolation. A deep null may shift between sampled frequencies or disappear in a reflective room.
Convert radial dB to amplitude ratio with
R = 10^(L/20).
A plotted −15 dB response has amplitude ratio 0.1778. Treating the radial distance linearly in dB can distort a hand calculation because polar plotting conventions vary.
Angular uncertainty
If microphone orientation is uncertain by ±5°, response near a steep null can vary far more than near the broad front lobe. Sensitivity to angle is derivative dR/dtheta.
For cardioid R = 0.5(1+cos theta),
dR/dtheta = −0.5 sin theta
with theta in radians. At 90°, a small angular error has maximum first-order amplitude effect; at 0° and 180°, the first derivative is zero, though behaviour near the rear null remains important in dB because R approaches zero.
Microphone Arrays and Beam Steering
Two or more microphones can use spatially different arrival times to favour one direction. This is the core idea behind a delay-and-sum array.
For two microphones separated by distance d, a plane wave arriving at angle theta relative to broadside has an approximate path difference
Delta x = d sin(theta).
The corresponding time difference is
Delta t = d sin(theta) / c,
where c is sound speed.
With d = 0.080 m, theta = 30 degrees and c = 343 m/s,
Delta t = 0.080 × 0.5 / 343 ≈ 0.0001166 s = 116.6 microseconds.
At a 48 kHz sample rate, one sample lasts 20.83 microseconds, so the delay is about 5.60 samples. Integer-sample shifting alone cannot match it exactly. A practical digital system uses a fractional-delay filter or accepts steering error.
Phase depends on frequency
At frequency f, the phase difference is
Delta phi = 2 pi f Delta t.
At 1 kHz, the example delay corresponds to about
2 pi × 1000 × 0.0001166 = 0.733 rad ≈ 42.0 degrees.
At 8 kHz, the phase difference is eight times larger. This frequency dependence means one fixed phase shift is not the same as a true time delay across a wide band.
Spatial aliasing
If microphone spacing is too large relative to wavelength, different arrival directions can produce ambiguous phase relationships. A useful conservative condition for avoiding grating lobes over a wide steering range is spacing no greater than about half the shortest wavelength of interest:
d ≤ lambda_min/2 = c/(2f_max).
For f_max = 8 kHz, this gives
d ≤ 343/(16,000) ≈ 0.0214 m, or about 21 mm.
An 80 mm spacing therefore cannot behave as an unambiguous ideal two-element sampler up to 8 kHz. It may still be useful, but the pattern becomes frequency dependent and can develop extra lobes.
Aperture and resolution
A larger array aperture can create a narrower main beam at a given wavelength, improving angular discrimination. Yet increasing element spacing can worsen spatial aliasing. Designers can add more elements so total aperture grows while local spacing remains small.
This is a classic engineering trade-off: aperture, number of channels, computation, frequency range, physical size and cost interact. Mathematics makes the trade-off explicit.
Room Reflections, Reverberation and Intelligibility
A microphone captures direct sound plus a sequence of reflections. A simple linear model writes the recorded signal as
y(t) = x(t) * h(t) + n(t),
where x(t) is the source, h(t) is the room impulse response, the asterisk denotes convolution and n(t) is additional noise.
The impulse response is a time map of the room's effect. Early reflections may strengthen or colour the direct sound. Later reflections form a decaying reverberant field. Moving the microphone changes path lengths and therefore changes both timing and phase.
Sabine's reverberation estimate
For a reasonably diffuse room, the Sabine approximation is
T60 = 0.161 V/A
in SI units, where V is room volume in cubic metres and A is equivalent absorption area in square metres. T60 is the estimated time for sound level to decay by 60 dB.
If V = 180 m³ and A = 60 m², then
T60 = 0.161 × 180/60 = 0.483 s.
If added treatment raises absorption area to 90 m²,
T60 = 0.161 × 180/90 = 0.322 s.
The model suggests a shorter decay. It does not predict every seat or frequency perfectly. Absorption is frequency dependent, rooms are not always diffuse, and irregular geometry matters.
Direct-to-reverberant ratio
Moving a microphone closer to a talker usually raises direct sound according to the distance model while the late reverberant field changes less. That improves the direct-to-reverberant ratio. This is one mathematical reason a close microphone can improve speech clarity even before any electronic processing.
Directionality may help if unwanted energy arrives from angles where the microphone has lower response. But room reflections arrive from many directions, and the polar pattern changes with frequency. Placement, room treatment and source behaviour remain part of the system.
Measuring an impulse response
Professionals may use calibrated sweeps, suitable playback, analysis software and controlled procedures. A student can explore the concept more modestly by convolving a clean recording with published or synthetic impulse responses. Comparing waveforms, energy decay curves and spectrograms reveals how one mathematical operation produces audible room character without unsafe sound levels.
A Complete Gain-and-Noise Budget
A gain budget follows the wanted signal and noise through each stage. Suppose a microphone produces −55 dBV for a quiet speech level at its position. A preamplifier adds 50 dB of gain, so the nominal output becomes
−55 dBV + 50 dB = −5 dBV.
In voltage,
V = 10^(−5/20) V ≈ 0.562 V RMS.
If the converter clips at +10 dBV, nominal headroom is 15 dB before peaks and uncertainty are considered.
Add noise in power, not amplitude
Suppose the microphone self-noise referred to the input is −118 dBV, and the preamplifier's input-referred noise over the chosen bandwidth is −122 dBV. Convert each to power-like ratios, add and convert back:
L_total = 10 log10(10^(−118/10) + 10^(−122/10)) ≈ −116.54 dBV.
The combined noise is higher than either contributor alone. The input signal-to-noise ratio is approximately
−55 − (−116.54) = 61.54 dB.
Adding 50 dB of ideal gain moves both wanted signal and input-referred noise upward together; it does not change that ratio. Later-stage fixed noise can make too little gain undesirable, while clipping makes too much gain undesirable. Gain staging manages both risks.
Peak factor matters
Speech and music have peaks above their average or RMS level. If nominal programme level is −18 dBFS and peaks reach 14 dB above it, peaks land at −4 dBFS, leaving 4 dB before digital full scale. A different speaker or performance may exceed that allowance.
This is why a meter reading is not merely “loud enough.” The operator needs a reference, an averaging method, a peak definition and a margin appropriate to the material.
Calibration and Traceability
Comparing recordings by waveform number alone is risky because microphone sensitivity, gain, converter scale and software settings differ. Calibration establishes a known relationship between physical input and recorded level.
A sound calibrator may generate a stated sound pressure at a stated frequency for compatible measurement microphones. The recorded value can then define a conversion between dBFS and sound-pressure level for that exact chain and gain setting. Calibration certificates, environmental limits and equipment compatibility matter in professional measurements.
For ordinary creative recording, absolute SPL calibration may not be required. Relative tests still need control. Keep source position, orientation, gain, distance and room conditions fixed; change one factor at a time; and document the settings.
Traceability means the measurement can be related through a documented chain to recognised references, with uncertainties stated. It does not mean every classroom demonstration must become a laboratory calibration. The transferable habit is to ask what the number is referenced to and whether another person could reproduce it.
Measurement Weighting and Exposure
Sound-level meters may use A, C or Z frequency weighting and fast, slow or impulse time weighting. A-weighted dBA is not interchangeable with unweighted peak pressure. Exposure limits combine level and duration according to a jurisdiction’s rule.
OSHA and NIOSH publish different occupational criteria in the United States. Other jurisdictions differ. This article does not choose a personal safe-listening limit. Use the current applicable authority and competent assessment.
Audio production meters such as LUFS, peak dBFS and SPL answer different questions. A digital recording level alone cannot reveal acoustic hearing exposure without calibration.
What the Models Leave Out
- Near-field source geometry and movement.
- Frequency-dependent polar response.
- Room reflections and modal behaviour.
- Wind, handling noise and stand vibration.
- Nonlinear distortion and maximum SPL.
- Microphone loading and preamp impedance.
- Analogue filter phase and converter latency.
- Source directivity and performer technique.
- Correlated noise and coherent interference.
- Perceptual weighting and hearing variability.
The equations create checkable expectations, not guaranteed recordings.
Common Misconceptions
“Gain makes the microphone hear farther”
Gain raises the captured signal and captured noise. Placement changes acoustic ratios.
“A cardioid hears only the front”
It attenuates directions according to frequency-dependent response; it is not an acoustic wall.
“Twice the distance is half as many decibels”
Under free-field pressure behaviour, doubling distance is about −6 dB, not half the dB number.
“Two 70 dB sounds make 140 dB”
Independent equal sources combine to about 73 dB.
“A null on one polar plot rejects every sound”
Null position and depth change with frequency and reflections.
“Higher bit depth removes room noise”
It changes quantisation range; it does not remove acoustic or analogue noise.
“Phase reversal means the rear is silent”
Figure-eight rear response has opposite polarity but substantial magnitude.
How Students Can Build Transferable Skill
Convert ratios both ways
Calculate dB from amplitude ratios, then recover ratios with 10^(L/20). Check round trips.
Plot ideal patterns
Graph omni, cardioid and figure-eight in polar coordinates. Mark nulls and −6 dB points.
Model distance
Use one measured reference level and predict direct level at several distances. Compare with safe low-level measurements in a quiet classroom and explain reflections.
Build a gain budget
Start with SPL, convert to pressure, apply sensitivity, add gain and compare with interface headroom. Keep every reference unit.
Simulate delay
Add a sine wave to a delayed copy in a spreadsheet or approved audio tool. Sweep frequency and find comb-filter nulls.
Audit a specification
Read sensitivity, self-noise, maximum SPL and polar plots from one manufacturer document. State test frequencies and references.
Keep pathways open
Microphone mathematics connects acoustics, music production, broadcasting, electronics, signal processing, theatre, communications and hearing science. It supports exploration but does not grant professional or clinical authority.
Guidance for Parents and Teachers
- Keep playback at comfortable, supervised levels.
- Use manufacturer plots instead of invented product claims.
- Separate acoustic SPL, analogue dBu/dBV and digital dBFS.
- Ask what reference every decibel uses.
- Compare closer placement with added electronic gain.
- Reward phase and time reasoning, not only level arithmetic.
- Secure stands and cables; never suspend equipment without qualified rigging.
- Use current hearing-safety guidance for any exposure activity.
A useful prompt is: “What changed before the signal reached the microphone?” This separates acoustic improvements—distance, angle, room—from electronic changes after capture.
A compact recording investigation
Let students record a constant, moderate test source from 0.25 m, 0.50 m and 1.00 m while keeping microphone orientation and gain fixed. Before recording, they should predict relative direct-sound changes of approximately 0, −6 and −12 dB from the nearest position.
Afterward, compare measured RMS values over the same time window. Differences from prediction invite explanation: room reflections, background noise, source directivity, automatic gain control, positioning error or a source that was not actually constant.
Require the report to state whether software applied normalisation, compression or noise reduction. These processes can erase the level relationship being tested. File format, sample rate and bit depth should be recorded, but a higher sample rate does not fix an uncontrolled acoustic setup.
The conclusion should distinguish model, evidence and interpretation. “The free-field model predicts a 6 dB loss per doubling; our room showed smaller changes beyond 0.5 m; reflected energy is a plausible reason” is more honest than declaring the inverse-distance law false.
Frequently Asked Questions
What is a microphone polar pattern?
A frequency-dependent representation of relative response versus direction under stated test conditions.
What does cardioid mean?
An ideal first-order pattern with maximum front response and a rear null, approximated by 0.5(1+cos theta).
How much level is lost when distance doubles?
About 6 dB for direct pressure from a point-like source in free field. Rooms and source geometry change the result.
Why are decibels logarithmic?
They express very large ratios compactly and turn multiplication of gains into addition.
Does more gain improve SNR?
It can overcome later electronic noise, but it does not improve the acoustic desired-to-noise ratio already at the capsule.
What is microphone sensitivity?
Output voltage for a stated sound pressure, often reported at 1 kHz in mV/Pa or dBV/Pa.
What causes comb filtering?
Combining similar signals with a delay creates frequency-dependent reinforcement and cancellation.
What is proximity effect?
Low-frequency rise at close distance in many pressure-gradient directional microphones.
Is dBFS the same as dB SPL?
No. dBFS references digital full scale; dB SPL references acoustic pressure.
Can this article set hearing exposure limits?
No. Use current applicable occupational or public-health guidance and qualified assessment.
Useful Next Reading
- Shure: Microphone Techniques for Recording
- Shure: Understanding Microphone Specifications
- NIOSH: Occupational Noise Exposure
- OSHA: Occupational Noise Exposure
- Why Mathematics? | Digital Audio Sampling, Nyquist Rate and Aliasing
- Why Mathematics? | Radio Receiver Tuning, Resonance and Bandwidth
- Why Mathematics? | Subtitles, Timecodes and Reading Speeds
Microphone mathematics turns recording choices into relationships that can be tested. Decibels track ratios, distance changes direct sound, polar functions describe direction, delay predicts phase, and a gain budget carries the signal into electronics. The most useful lesson is that capture begins in air. Good mathematics helps a student improve the acoustic situation before asking electronics to solve a problem that is already mixed into the signal.