---
title: Sound Waves
module: Oscillations and Waves
moduleNumber: 7
lessonNumber: 8
order: 708
summary: >
  Sound is a pressure wave so small that a loud tone displaces air molecules by less
  than the width of an atom, yet a microphone reads it easily — because pressure, not
  displacement, is what the ear and the instrument sense. The acoustic impedance
  $Z=\rho c$ ties pressure, density, and particle velocity together, fixes the
  intensity a wave carries, and sets the reference for the decibel, a logarithm that
  tames a $10^{12}$ range in power. We derive the sound speed from the gas's
  stiffness, convert between pressure and intensity levels, and treat the measurement
  itself — calibration, geometry, background, averaging — as part of the physics.
topics: [Waves]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 15 — Traveling Waves; §§15-1–15-3"
---

## Longitudinal fields and sound speed

Sound in a gas consists of small longitudinal motions of the gas about its equilibrium
state. A loudspeaker diaphragm, a tuning fork, or a vibrating surface first pushes
nearby gas. That local compression raises pressure slightly. Adjacent gas responds,
and the disturbance propagates through the medium. Individual molecules travel only
small distances about local equilibrium positions; the sound pattern can travel many
metres while the molecular excursion remains microscopic.

At a fixed instant, a longitudinal wave contains alternating regions of compression
and rarefaction. Compression means that more mass occupies a small volume than in
the equilibrium gas. Rarefaction means less mass occupies the corresponding volume.
Both descriptions refer to departures from the ambient state. Atmospheric pressure
and equilibrium density remain much larger than their oscillatory changes in ordinary
sound measurements.

$$
% caption: A longitudinal pulse in a gas tube. Parcels bunch into a compression and thin into a rarefaction; each parcel oscillates a little along the axis while the compression pattern travels toward the receiver.
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$$

The same wave can be represented by several physical variables. The longitudinal
displacement $\xi(x,t)$ gives the displacement of a small material element from its
equilibrium position. Particle velocity $u(x,t)$ gives the time derivative of that
displacement. Gauge pressure $p'(x,t)$ gives the pressure change above or below
ambient pressure. Density change $\rho'(x,t)$ gives the corresponding density
departure. The prime distinguishes a fluctuating quantity from its equilibrium value:
$p=p_{\mathrm{amb}}+p'$ and $\rho=\rho_0+\rho'$.

No single representation is intrinsically more complete. A microphone responds
mainly to pressure. A laser vibrometer can measure surface or particle displacement.
A model of energy flow often uses pressure and particle velocity together. State the
measured variable before converting an amplitude or a level. A quoted “sound
amplitude” without that information has no unambiguous unit or physical meaning.

A sinusoidal plane wave travelling in the positive $x$ direction has displacement
form

$$
\xi(x,t)=\xi_0\cos(kx-\omega t+\phi_0),
\qquad
k=\frac{2\pi}{\lambda},
\qquad
\omega=2\pi f.
$$

Here $\xi_0$ is the displacement amplitude, $k$ is the wavenumber, $\lambda$ is
wavelength, $\omega$ is angular frequency, and $f$ is frequency. A fixed material
parcel has a fixed equilibrium coordinate $x$ and oscillates back and forth. A fixed
phase point, such as a pressure maximum, moves at the wave speed $v=\omega/k=f\lambda$.
Keeping those two motions separate avoids a common error: the gas does not stream
from source to receiver at the sound speed.

Sound is ordinarily treated as a small-signal wave. The displacement, density
change, and gauge pressure are sufficiently small that linear relations apply.
Nonlinear effects become important at large pressure amplitudes, in shock waves, or
very close to strongly driven sources. Unless stated otherwise, the relations below
use linear acoustics.

> **Definition (Plane progressive sound wave).** A wave whose surfaces of equal
> phase are parallel planes and whose acoustic variables depend on one propagation
> coordinate. The approximation applies within a small region of a broad beam or far
> from a source when wavefront curvature is negligible across the measurement area.

Plane-wave language describes local behavior. A point source emits curved wavefronts,
yet a small microphone diaphragm samples only a small patch of one wavefront. The
patch can often be approximated as plane if its diameter is much smaller than the
local radius of curvature and the wavelength-dependent spatial variation across the
diaphragm is negligible.

### Displacement, particle velocity, pressure, and density

Different sound variables have fixed derivative relations. Differentiate the
displacement wave with respect to time:

$$
u(x,t)=\frac{\partial \xi}{\partial t}
=\omega\xi_0\sin(kx-\omega t+\phi_0).
$$

Particle-velocity amplitude is therefore $u_0=\omega\xi_0$. At a displacement
maximum, the particle velocity is zero. At the equilibrium crossing, the magnitude
of particle velocity is largest. This is the same local phase relation as simple
harmonic motion. It gives the motion of one gas parcel; wavefront propagation is a
separate quantity.

$$
% caption: At a fixed location, displacement and particle velocity run a quarter cycle apart. Where the displacement reaches a peak the particle is momentarily at rest; where the displacement crosses zero the particle speed is greatest.
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$$

Pressure follows from compression. Take two neighboring equilibrium locations,
separated by $\d x$. Their instantaneous separation is

$$
\d x+\left[\xi(x+\d x,t)-\xi(x,t)\right]
\simeq \d x\left(1+\frac{\partial\xi}{\partial x}\right).
$$

The fractional length change is $\partial\xi/\partial x$. In a narrow tube with
fixed cross-sectional area, it also gives the fractional volume change of the gas
element. Compression has $\partial\xi/\partial x<0$: the right boundary has moved
less far right than the left boundary, reducing the interval between them. Expansion
has $\partial\xi/\partial x>0$.

The bulk modulus $B$ relates a small pressure rise to fractional volume decrease:

$$
p'=-B\frac{\partial\xi}{\partial x}.
$$

For the sinusoidal displacement above,

$$
p'(x,t)=Bk\xi_0\sin(kx-\omega t+\phi_0).
$$

The pressure amplitude is $p_0=Bk\xi_0$. Pressure is in phase with particle
velocity for a progressive wave moving in the positive direction. Their signs are
the same with the phase convention used here: a parcel moving toward positive $x$
has positive particle velocity and is part of a compression that has positive gauge
pressure. Reversing propagation direction reverses the sign relation.

Mass conservation gives the density relation. A material element with a shorter
length contains the same mass in less volume, so its density increases. To first
order,

$$
\rho'(x,t)=-\rho_0\frac{\partial\xi}{\partial x}.
$$

Combining this with the pressure relation produces $p'=(B/\rho_0)\rho'$. In a
linear fluid, $B/\rho_0=v^2$, so pressure and density departures have the same sign.
The oscillatory excess density in a compression therefore coincides with positive
gauge pressure.

$$
% caption: Pressure and displacement along a right-moving plane wave at one instant. Pressure peaks in the compressions, while the displacement wave is shifted by a quarter wavelength, crossing zero exactly where the pressure is extreme.
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$$

The displacement amplitude can be tiny even when a sound is readily measured.

> **Worked example.** At $f=1.00\ \mathrm{kHz}$, a particle-velocity amplitude of
> $u_0=1.00\times10^{-4}\ \mathrm{m\,s^{-1}}$ corresponds to a displacement amplitude
>
> $$
> \xi_0=\frac{u_0}{\omega}=\frac{u_0}{2\pi f}
> =\frac{1.00\times10^{-4}\ \mathrm{m\,s^{-1}}}{2\pi(1.00\times10^{3}\ \mathrm{Hz})}
> =1.59\times10^{-8}\ \mathrm{m},
> $$
>
> smaller than an atomic diameter. A microphone resolves the associated pressure
> fluctuation without resolving that displacement directly.

Microscopic displacement should not be confused with negligible transported energy;
the wave transfers energy through the medium continuously.

### Sound speed and the material response of a gas

The speed of a small pressure disturbance in a fluid is

$$
v=\sqrt{\frac{B}{\rho_0}},
$$

where $B$ is the bulk modulus appropriate to the compression process and $\rho_0$
is equilibrium density. A large bulk modulus resists compression and raises wave
speed. Greater mass density raises inertia and lowers wave speed when the modulus is
held fixed. The expression has the expected dimensions because $B$ has units
$\mathrm{N\,m^{-2}}$ and $B/\rho_0$ has units $\mathrm{m^2\,s^{-2}}$.

The acoustic compression and expansion of a gas occur rapidly enough that little
heat enters or leaves a small parcel during one cycle. The appropriate bulk modulus
is therefore the adiabatic value, $B=\gamma p_{\mathrm{amb}}$. The isothermal value
$p_{\mathrm{amb}}$ applies to a constant-temperature compression. An ideal gas then
gives

$$
v=\sqrt{\frac{\gamma R T}{M}},
$$

where $T$ is absolute temperature, $M$ is molar mass, $R$ is the gas constant, and
$\gamma=C_p/C_V$. Dry air near room temperature has $\gamma\simeq1.40$,
$M\simeq 2.90\times10^{-2}\ \mathrm{kg\,mol^{-1}}$, and sound speed near
$343\ \mathrm{m\,s^{-1}}$ at $20\ ^\circ\mathrm{C}$.

$$
% caption: Ideal-gas sound speed grows as the square root of absolute temperature. Ambient pressure cancels from the ideal-gas result at fixed composition, so temperature is the controlling variable.
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$$

At fixed gas composition and temperature, ideal-gas sound speed is nearly independent
of ambient pressure. Both $B=\gamma p$ and $\rho=pM/(RT)$ increase proportionally
with pressure, leaving their ratio unchanged. Ambient pressure still affects density,
impedance, microphone loading, nonlinear behavior, attenuation, and source coupling.
Sound speed and acoustic impedance therefore have different pressure dependence.

Near room temperature over a modest interval, the approximation

$$
v_{\mathrm{air}}\simeq(331+0.60\,T_{\mathrm{C}})\ \mathrm{m\,s^{-1}}
$$

is a numerical check, with $T_{\mathrm{C}}$ in degrees Celsius. The
square-root expression remains the better model when temperature range, composition,
or uncertainty matter. Record whether the temperature is air temperature near the
propagation path or a remote weather-station value; a heated room can have a vertical
temperature gradient large enough to affect a precise time-of-flight measurement.

A time-of-flight record gives estimated speed $v=L/\Delta t$, where $L$ is sensor
separation and $\Delta t$ is the propagation delay. Trigger jitter, uncertain
sensor acoustic centers, reflections, and temperature gradients contribute to the
uncertainty. A repeated pulse sequence permits an average delay and a scatter
estimate. A sinusoidal source needs a phase method or a broadband modulation because
a delay measured from a single periodic trace is ambiguous by integer periods.

## Energy, intensity, and propagation loss

A sound wave transfers energy through a medium. The local energy has two
interchangeable forms. Particle motion carries kinetic energy. Compression stores
elastic energy because the gas has been displaced from its equilibrium volume. In a
harmonic progressive wave, the two contributions vary through the cycle and have
equal time averages. The mean total energy density can therefore be written in
several equivalent ways:

$$
\left\langle e\right\rangle
=\frac{1}{2}\rho_0u_0^2
=\frac{p_0^2}{2\rho_0v^2}
=\frac{p_{\mathrm{rms}}^2}{\rho_0v^2}.
$$

The angle brackets denote an average over complete cycles. Subscript $0$ denotes a
peak amplitude, whereas “rms” denotes root-mean-square amplitude. A sinusoid has
$p_{\mathrm{rms}}=p_0/\sqrt{2}$ and $u_{\mathrm{rms}}=u_0/\sqrt{2}$. Do not mix
peak pressure with rms particle velocity in the same formula. A factor of two error
in energy or intensity usually traces to that mismatch.

The energy density is local: it describes energy stored in one cubic metre at a
particular region of the sound field. It does not by itself state how rapidly energy
passes through a surface. That rate depends on wave speed and propagation direction.
A plane progressive wave has time-averaged intensity

$$
I=\left\langle p'(x,t)u(x,t)\right\rangle
=v\left\langle e\right\rangle
=\frac{1}{2}\rho_0v u_0^2
=\frac{p_0^2}{2\rho_0v}
=\frac{p_{\mathrm{rms}}^2}{\rho_0v}.
$$

Intensity has SI units $\mathrm{W\,m^{-2}}$. The product $p'u$ is an instantaneous
power flux: pressure times particle velocity gives watts per square metre. Its cycle
average is positive for a progressive wave because pressure and particle velocity are
in phase. In a standing-wave region, their phase relation differs, and the local
cycle-averaged flux can be small even where pressure amplitude is large. The
plane-progressive formulas need that condition stated whenever they are used to infer
intensity from pressure.

$$
% caption: In a short time the wave sweeps a slab through a patch of area $A$. The slab's depth is the distance the wave travels, so the energy delivered per unit area per unit time is the wave speed times the mean energy density.
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The relation $I=v\langle e\rangle$ follows from a control-surface argument. During a
short interval $\Delta t$, a plane wave moves through a distance $v\Delta t$. A
surface patch of area $A$ therefore receives energy from a slab of volume
$A v\Delta t$. If the mean energy density is $\langle e\rangle$, that slab contains
$\langle e\rangle A v\Delta t$. Dividing by $A\Delta t$ produces
$I=v\langle e\rangle$. The argument measures energy crossing a surface normal to the
direction of propagation. An oblique surface receives a smaller flux per area by the
projected-area factor.

Power is intensity integrated over area. Through a surface $S$ enclosing a source,

$$
P=\iint_S \vec I\cdot\d\vec A.
$$

When intensity is uniform and perpendicular to a flat receiver of area $A$, this
reduces to $P=IA$. A microphone pressure reading samples the local acoustic field.
Total source power requires an area integration over a suitable enclosing surface.

A sinusoidal tone has intensity proportional to the square of each of these
amplitudes:

$$
I\propto p_0^2,\qquad I\propto p_{\mathrm{rms}}^2,\qquad I\propto u_0^2,
\qquad I\propto \xi_0^2 f^2.
$$

Doubling pressure amplitude multiplies intensity by four. Doubling displacement
amplitude at fixed frequency also multiplies intensity by four. Doubling frequency
at fixed displacement amplitude multiplies particle velocity amplitude by two and
intensity by four. A claim that a sound “doubled in amplitude” is incomplete unless
the variable is named. A doubling of rms pressure, particle velocity, diaphragm
displacement, electrical drive voltage, and emitted power are different experimental
statements.

> **Worked example.** Take air with $\rho_0=1.20\ \mathrm{kg\,m^{-3}}$ and
> $v=343\ \mathrm{m\,s^{-1}}$. A measured rms pressure of $0.200\ \mathrm{Pa}$ in a
> locally progressive field carries intensity
>
> $$
> I=\frac{p_{\mathrm{rms}}^2}{\rho_0 v}
> =\frac{(0.200\ \mathrm{Pa})^2}
> {(1.20\ \mathrm{kg\,m^{-3}})(343\ \mathrm{m\,s^{-1}})}
> =9.72\times10^{-5}\ \mathrm{W\,m^{-2}}.
> $$
>
> The same reading gives a particle-velocity rms value
>
> $$
> u_{\mathrm{rms}}=\frac{p_{\mathrm{rms}}}{\rho_0v}
> =\frac{0.200\ \mathrm{Pa}}{(1.20\ \mathrm{kg\,m^{-3}})(343\ \mathrm{m\,s^{-1}})}
> =4.86\times10^{-4}\ \mathrm{m\,s^{-1}}.
> $$
>
> Both conversions assume the plane-wave impedance $\rho_0v$. Close to a small source,
> near a wall, or inside a resonant enclosure, pressure and particle velocity acquire
> a phase difference and the simple ratio becomes inadequate.

Intensity is a vector quantity before averaging over a surface. Its direction is the
direction of energy transport. In a plane progressive sound wave, pressure and
particle-velocity phasors align, and the mean intensity vector points along the
wavevector. In a complex field, use the time average of pressure times the component
of particle velocity normal to the surface. A pressure-only microphone cannot
determine intensity direction by itself; a paired pressure–particle-velocity probe or
an array measurement resolves the missing directional information.

The average power of a source does not equal the electrical power printed on a
loudspeaker amplifier. Electrical input power divides among acoustic radiation,
heating in the voice coil, mechanical loss, enclosure vibration, and circuitry.
Acoustic output power must be inferred from acoustic measurements on a specified
surface or reported from a calibrated source characterization. Name which power is
being reported: electrical input, acoustic radiation, or power incident on a receiver.

### Geometric spreading, point sources, and propagation loss

A compact source in an open region can be approximated by a point source if the
receiver distance is large compared with the source dimensions. If it radiates
uniformly in every direction and loses negligible energy to absorption, the same
acoustic power crosses every sphere centered on the source:

$$
P=I(r)\,4\pi r^2,
\qquad
I(r)=\frac{P}{4\pi r^2}.
$$

The inverse-square relation comes from geometric spreading. A sphere with twice the
radius has four times the area, so the same power is distributed over four times the
surface. It does not describe molecular energy loss from the wave; it describes an
area increase. Absorption adds an additional decrease whose magnitude depends on
frequency, humidity, gas composition, and path length.

$$
% caption: An isotropic point source sends the same power through every surrounding sphere. Doubling the radius quadruples the area, so the intensity at $2r$ is one quarter of the intensity at $r$.
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$$

The level change between two distances follows immediately:

$$
\frac{I_2}{I_1}=\left(\frac{r_1}{r_2}\right)^2,
\qquad
10\log_{10}\left(\frac{I_2}{I_1}\right)
=-20\log_{10}\left(\frac{r_2}{r_1}\right).
$$

Doubling distance from an ideal point source reduces intensity by a factor of four
and reduces intensity level by approximately $6.02\ \mathrm{dB}$. Ten times the
distance reduces level by $20\ \mathrm{dB}$. These values apply to a single
free-field source. They cannot be transferred unchanged into a room where direct
sound combines with reflected sound.

A source mounted in a large rigid plane radiates primarily into a hemisphere. With
uniform hemispherical radiation,

$$
I(r)=\frac{P}{2\pi r^2}.
$$

At a given distance and total acoustic power, the hemispherical intensity is twice
the isotropic full-sphere value. A loudspeaker, a mouth, or a machine enclosure
usually has its own directional pattern. A report should state the direction and
distance of every microphone relative to the source axis.

$$
% caption: The same source power spreads over a full sphere in open space but over only a half sphere above a rigid plane. Halving the area doubles the far-field intensity at the same radius.
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$$

Absorption and obstacles alter the point-source model. A phenomenological attenuation
form is

$$
I(r)=\frac{P}{4\pi r^2}\,10^{-\alpha r/10},
$$

when $\alpha$ is an attenuation coefficient expressed in decibels per metre for a
specified frequency and propagation condition. This representation separates
geometric spreading from path attenuation. It should be fitted only over a region
where source directionality and reflection conditions are controlled. An arbitrary
level-versus-distance slope does not establish an absorption coefficient.

$$
% caption: A level-versus-log-distance scan separates two effects. Pure geometric spreading gives a straight decline; extra path attenuation bends the measured curve steadily below the spreading line.
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\node[acc] at (4.35,1.95) {spreading only};
\node[black] at (2.55,1.02) {with path loss};
\end{tikzpicture}
$$

Near a source, the measured field can contain reactive energy that alternately
stores and returns energy to the source. The pressure-to-velocity ratio then differs
from $\rho_0v$, and pressure alone does not give a reliable intensity estimate.
The transition distance depends on source size and wavelength. Avoid claiming
free-field intensity from a microphone placed against a loudspeaker grille, inside a
duct termination, or a few centimetres from a vibrating panel without an appropriate
near-field model.

An open-space measurement also needs a background record. Turn the source off while
leaving microphone gain, geometry, and time weighting unchanged. If the background
level is close to the source-on level, subtraction must be done with linear mean
square pressure or intensity, never by arithmetic subtraction of decibel values.
The later decibel section gives the needed conversion.

## Decibel levels and acoustic measurements

Sound intensity spans an enormous physical range. A logarithmic ratio compresses that
range while preserving multiplicative changes. The intensity level $\beta_I$ is
defined relative to a reference intensity $I_0$ by

$$
\beta_I=10\log_{10}\left(\frac{I}{I_0}\right)\ \mathrm{dB}.
$$

In air-acoustics examples, $I_0=10^{-12}\ \mathrm{W\,m^{-2}}$ is commonly used as
the reference intensity. A sound at $10^{-6}\ \mathrm{W\,m^{-2}}$ has level
$60\ \mathrm{dB}$ on that scale because its intensity ratio is $10^6$. The decibel
is a ratio label. It does not carry the dimensions of watts per square metre, pascals,
or acoustic power.

The factor of 10 in the definition follows the use of an energy-like quantity. For
two intensity values,

$$
\Delta\beta_I
=10\log_{10}\left(\frac{I_2}{I_1}\right).
$$

An intensity ratio of 2 gives $+3.01\ \mathrm{dB}$, a ratio of 10 gives
$+10\ \mathrm{dB}$, and a ratio of 100 gives $+20\ \mathrm{dB}$. A $-10\ \mathrm{dB}$
change means one tenth of the original intensity. The sign describes a ratio; it
does not mean that physical intensity became negative.

$$
% caption: Each 10 dB step multiplies intensity by ten. The right column lists the intensity ratio for each level, so decibels combine by multiplying ratios, never by adding the printed numbers.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\useasboundingbox (-.7,-.5) rectangle (6.8,3.5);
\draw[->,black] (1.15,.5)--(1.15,3.0);
\foreach \yy/\db/\ratio in {.78/0/1,1.28/10/10,1.78/20/100,2.28/30/1000,2.78/40/10000}{
  \draw[acc,very thick] (1.15,\yy)--(5.6,\yy);
  \node[left,black] at (1.05,\yy) {\db};
  \node[right] at (5.7,\yy) {\ratio};
}
\node[above,rotate=90,black] at (.5,1.75) {level in dB};
\node[above] at (5.95,3.0) {ratio};
\end{tikzpicture}
$$

Microphones measure pressure, so sound-pressure level is often more direct:

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

The conventional reference pressure in air is
$p_{\mathrm{ref}}=20\ \mathrm{\mu Pa}=2.0\times10^{-5}\ \mathrm{Pa}$. The factor is
20 instead of 10 because intensity is proportional to pressure squared in a
progressive plane wave. When both references are matched by
$I_0=p_{\mathrm{ref}}^2/(\rho_0v)$, sound-pressure level and intensity level have
nearly equal numerical values under plane-wave free-field conditions.

Pressure level does not universally equal intensity level. The relation can fail near
boundaries, in small rooms, in ducts, and close to sources because acoustic impedance
is not necessarily the real quantity $\rho_0v$. A pressure meter then still reports a
valid pressure level, but it does not alone establish net intensity or acoustic
power. Record the quantity measured instead of relabeling every microphone reading as
intensity.

The phrase “twice as loud” belongs to perception and cannot be read directly from
one decibel difference. Loudness depends on frequency, duration, spectral content,
listener, and listening environment. Here decibels describe a physical
pressure or intensity ratio. A report can separately state an A-weighted sound level
or a subjective listening result, provided the physical quantity and weighting are
identified.

### Combining independent sources

Independent sources add in linear intensity or mean-square pressure. Their decibel
levels must be converted before addition:

$$
I_{\mathrm{tot}}=\sum_{j=1}^N I_j,
\qquad
L_{\mathrm{tot}}
=10\log_{10}\left(\sum_{j=1}^N10^{L_j/10}\right).
$$

For $N$ equal, independent sources at the same receiver location,

$$
L_{\mathrm{tot}}=L_{\mathrm{one}}+10\log_{10}N.
$$

Two equal sources give a $3.01\ \mathrm{dB}$ increase. Ten equal sources give a
$10\ \mathrm{dB}$ increase. Two sources with levels differing by more than about
$10\ \mathrm{dB}$ produce a total only slightly above the stronger source because
the lower linear intensity is a small fraction of the total.

$$
% caption: Equal independent sources add as intensity, so the total level rises by $10\log_{10}N$. Two sources give about $+3$ dB and ten give $+10$ dB above one; the printed dB values are never summed directly.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\useasboundingbox (-.55,-.5) rectangle (6.8,2.4);
\draw[->,black] (.7,.6)--(6.15,.6);
\draw[->,black] (.7,.6)--(.7,2.25);
\node[below] at (3.4,.5) {source count};
\node[above,rotate=90] at (.58,1.5) {total level};
\draw[acc,very thick] plot[domain=1:10,samples=140] ({.8+.5*\x},{1.05+.3388*ln(\x)});
\foreach \n in {1,2,10}
  \draw[fill=acc!12,draw=acc,line width=.4pt] ({.8+.5*\n},{1.05+.3388*ln(\n)}) circle (.06);
\node[below] at (1.3,.6) {1};
\node[below] at (1.8,.6) {2};
\node[below] at (5.8,.6) {10};
\node[anchor=west] at (1.95,1.34) {$+3$ dB};
\node[anchor=west] at (5.0,2.0) {$+10$ dB};
\end{tikzpicture}
$$

Coherent sources require a pressure-phase calculation. For two harmonic tones at the
same frequency, the instantaneous pressures add:

$$
p_{\mathrm{tot}}(t)=p_1(t)+p_2(t).
$$

Equal pressures with the same phase produce twice the pressure amplitude and four
times the intensity, a $6.02\ \mathrm{dB}$ increase. Equal pressures in opposite
phase cancel in the ideal model. These cases depend on stable relative phase at the
receiver. Independent noise sources, separate machines with drifting phase, and
long-time energy averages use intensity addition.

$$
% caption: Two equal harmonic pressures at one point add according to their relative phase. In phase they give twice the amplitude and four times the intensity ($+6$ dB); exactly out of phase they cancel. Independent-source level addition does not apply to a phase-locked pair.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\useasboundingbox (-.45,-1.85) rectangle (6.8,2.05);
\draw[black] (.7,.65)--(3.0,.65);
\draw[acc,very thick] plot[domain=.8:2.9,samples=130] (\x,{.65+.40*sin(171.43*(\x-.8))});
\draw[black,very thick,dashed] plot[domain=.8:2.9,samples=130] (\x,{.65+.40*sin(171.43*(\x-.8))});
\draw[black] (.7,-.95)--(3.0,-.95);
\draw[acc,very thick] plot[domain=.8:2.9,samples=130] (\x,{-.95+.80*sin(171.43*(\x-.8))});
\node[above] at (1.85,1.35) {in phase};
\node[anchor=west] at (2.95,.65) {pair};
\node[anchor=west] at (2.95,-.95) {sum};
\draw[black] (3.7,.65)--(6.0,.65);
\draw[acc,very thick] plot[domain=3.8:5.9,samples=130] (\x,{.65+.40*sin(171.43*(\x-3.8))});
\draw[black,very thick,dashed] plot[domain=3.8:5.9,samples=130] (\x,{.65-.40*sin(171.43*(\x-3.8))});
\draw[black] (3.7,-.95)--(6.0,-.95);
\draw[black,very thick] (3.8,-.95)--(5.9,-.95);
\node[above] at (4.85,1.35) {opposite phase};
\node[anchor=west] at (5.95,.65) {pair};
\node[anchor=west] at (5.95,-.95) {sum};
\end{tikzpicture}
$$

Background correction is another addition problem. Let $L_{\mathrm{on}}$ be the
level with source and background present, and let $L_{\mathrm{bg}}$ be the level
with the target source absent. If the sources are independent over the measurement
average,

$$
L_{\mathrm{src}}
=10\log_{10}\left(10^{L_{\mathrm{on}}/10}
-10^{L_{\mathrm{bg}}/10}\right).
$$

The subtraction occurs inside the linear sum. If source-on level is only slightly
above background, the difference of two nearly equal quantities has large relative
uncertainty. Report the source-on level and background level along with the corrected
result. A correction derived from a short quiet interval cannot represent an
intermittent background such as traffic, ventilation cycling, or speech.

Free-field point sources permit direct geometric-spreading and level arithmetic.

> **Worked example.** A $75.0\ \mathrm{dB}$ reading at $r_1=2.00\ \mathrm{m}$
> predicts, under the ideal point-source model, a reading at $r_2=8.00\ \mathrm{m}$ of
>
> $$
> L_2=L_1-20\log_{10}\!\left(\frac{r_2}{r_1}\right)
> =75.0\ \mathrm{dB}-20\log_{10}(4)=63.0\ \mathrm{dB},
> $$
>
> a drop of $12.0\ \mathrm{dB}$ for the fourfold distance increase.

The prediction is a model result, not a calibration certificate. It needs a source
axis, an open propagation path, comparable environmental conditions, and a far-field
distance range. A measurement that differs from the prediction may indicate
directionality, a reflecting surface, air attenuation, source-output drift, or
microphone placement error. Retain the residuals alongside the ideal curve.

### Microphone measurements and calibration

A microphone converts an acoustic pressure variation at its diaphragm into an
electrical signal. Its sensitivity $S$ is commonly specified in volts per pascal at a
reference frequency and incidence condition. If a calibrated microphone has
sensitivity $S=50.0\ \mathrm{mV\,Pa^{-1}}$ and the electronics have voltage gain
$G$, a sinusoidal pressure with rms value $p_{\mathrm{rms}}$ produces an rms output
approximately

$$
V_{\mathrm{out,rms}}=G\,S\,p_{\mathrm{rms}}.
$$

The approximation omits several instrument properties: microphone frequency response,
phase response, directional response, preamplifier gain, output impedance, noise, and
maximum input pressure. A specification sheet does not replace calibration under the
measurement configuration. The signal path must be known from diaphragm to stored
sample.

Broadband-measurement sensitivity depends on frequency. Let $S(f)$ be the complex
frequency response including amplitude and phase. A sampled
voltage spectrum $V(f)$ corresponds to pressure spectrum $P(f)=V(f)/(G S(f))$ only
within the calibrated bandwidth. Dividing by a response value outside its valid
range can strongly amplify electrical noise. A measurement report should give the
frequency band retained in the analysis and state whether a response correction was
applied.

The sensor itself can be characterized for several acoustic conditions. A pressure
response applies when the diaphragm is flush with a boundary or inside a suitable
coupler. A free-field response applies for a specified incidence direction in an
approximately reflection-free wave. A random-incidence response averages over many
arrival directions. These calibration conditions have different corrections at high
frequency because diaphragm geometry and acoustic scattering matter. Copying a
free-field calibration into a boundary measurement changes the pressure estimate.

### Establishing the pressure scale

A field calibration uses a source that produces a known rms pressure at the
microphone. If a calibrator has pressure level $L_{\mathrm{cal}}$ at a stated
frequency, the known pressure is

$$
p_{\mathrm{cal,rms}}
=p_{\mathrm{ref}}10^{L_{\mathrm{cal}}/20}.
$$

With observed calibrator voltage $V_{\mathrm{cal,rms}}$, the system scale factor is

$$
K=\frac{p_{\mathrm{cal,rms}}}{V_{\mathrm{cal,rms}}}
\quad\left[\mathrm{Pa\,V^{-1}}\right],
\qquad
p_{\mathrm{rms}}=K V_{\mathrm{rms}}.
$$

This factor includes microphone sensitivity and the active gain setting. It is valid
only while the same gain, input range, coupling, and meter mode remain in use. A
pre-run and post-run calibration comparison can identify drift, loose connectors,
battery loss, or an accidental gain change. If the two values disagree beyond the
instrument tolerance, retain both records and determine whether the run requires
repetition or a stated calibration uncertainty.

> **Worked example.** A $94.0\ \mathrm{dB}$ calibration tone corresponds to a known
> rms pressure
>
> $$
> p_{\mathrm{cal,rms}}
> =(20.0\times10^{-6}\ \mathrm{Pa})\,10^{94.0/20}
> =1.00\ \mathrm{Pa}.
> $$
>
> If the recorded rms voltage is $0.316\ \mathrm{V}$, the system scale factor is
> $K=p_{\mathrm{cal,rms}}/V_{\mathrm{cal,rms}}=3.16\ \mathrm{Pa\,V^{-1}}$. A later
> source record of $0.100\ \mathrm{V_{rms}}$ then corresponds to
> $p_{\mathrm{rms}}=KV_{\mathrm{rms}}=0.316\ \mathrm{Pa}$, giving
>
> $$
> L_p=20\log_{10}\!\left(\frac{0.316\ \mathrm{Pa}}
> {20.0\times10^{-6}\ \mathrm{Pa}}\right)=84.0\ \mathrm{dB}.
> $$

The numerical neatness arises from the selected values. In a real measurement, keep
more digits internally and round the reported level only after evaluating the
uncertainty. The calibration tone frequency is also part of the record because a
single-frequency calibration cannot establish an entire frequency response.

An rms meter uses a finite averaging interval. The rms pressure of a time record
$p(t)$ over interval $T$ is

$$
p_{\mathrm{rms}}(T)=
\sqrt{\frac{1}{T}\int_{t_0}^{t_0+T}p^2(t)\,\d t}.
$$

The averaging interval must contain enough cycles to stabilize a tonal rms estimate.
At $20\ \mathrm{Hz}$, a $100\ \mathrm{ms}$ window contains only two cycles; phase
placement can noticeably change an unwindowed estimate. At $1\ \mathrm{kHz}$, the
same interval contains one hundred cycles. Broadband and impulsive sounds need a
window length chosen for the physical question, then stated with the result.

Clipping cannot be repaired by applying a calibration factor after the fact. The
flattened waveform has lost amplitude and spectral information. Set the input range
with a short test recording or use an instrument with documented headroom. Electrical
noise also needs attention. A microphone self-noise specification and preamplifier
noise combine with acoustic background. Measure the quiet-system level with the
microphone installed and the source absent; determine whether the desired signal
exceeds that floor by a sufficient margin for the claimed uncertainty.

### Geometry, orientation, and spatial sampling

Acoustic results are functions of position. A microphone location requires three
coordinates, source distance, source-axis angle, height above the floor, and the
positions of nearby large boundaries. “One metre from the speaker” is incomplete if
the microphone could have been on-axis, behind the source, near a desk, or beside a
wall. A tape measure gives distance; it does not determine the acoustic geometry.

Spatial sampling interval depends on wavelength and the purpose of the scan. A
single-source far-field level may use one carefully documented point. Mapping a
reflected sound field requires multiple locations because pressure extrema can occur
over distances comparable with a fraction of a wavelength. Higher frequencies have
shorter wavelengths and require finer spatial spacing. The scan path should be marked
physically or located with coordinates; a hand-held microphone moved informally
through space cannot support a reproducible spatial map.

Microphone orientation matters when sensitivity varies with incidence angle. Align a
free-field microphone according to its calibration convention. Keep the operator and
support hardware out of the direct path when possible, because body reflection and
scattering can change local pressure. A tripod reduces motion and makes post-run
repositioning possible. Record whether a windscreen, extension cable, boundary mount,
or protective grille was installed because each can alter high-frequency response.

Repeated measurements separate source variation from meter scatter. At each planned
location, record several independent time windows, then report mean level, temporal
range, and the averaging rule. If the source output changes over time, randomize or
cycle measurement positions so that a position trend is not confused with source
drift. A measurement program with one reading per position cannot distinguish those
two causes.

## Exposure, data records, and quality checks

Intensity describes instantaneous average power flow per area. Acoustic exposure
adds that flow over time:

$$
H=\int_{t_0}^{t_0+T}I(t)\,\d t,
\qquad
\left[H\right]=\mathrm{J\,m^{-2}}.
$$

Constant progressive-wave intensity gives $H=IT$. Doubling duration doubles exposure
even though the instantaneous level is unchanged. A high-level short event
and a lower-level long event can therefore have comparable integrated energy. Duration
belongs in any exposure statement.

Equivalent continuous sound-pressure level compresses a varying pressure record into
one level with the same mean-square pressure over a specified interval:

$$
L_{\mathrm{eq},T}
=10\log_{10}\left[
\frac{1}{T}\int_{t_0}^{t_0+T}
\frac{p^2(t)}{p_{\mathrm{ref}}^2}\,\d t
\right].
$$

The interval $T$ is part of the quantity. A one-minute equivalent level and an
eight-hour equivalent level are different summaries even if their numerical values
happen to match. A time-weighted maximum, a percentile level, a peak pressure, and
an equivalent level answer different questions. Do not replace one with another
without saying so.

Frequency weighting is a separate choice. An unweighted record, often called
Z-weighted, retains the instrument response over its stated band. An A-weighted
record applies a frequency-dependent weighting before level integration. The suffix
must be preserved in notation: $L_{\mathrm{Aeq},T}$ and $L_{\mathrm{Zeq},T}$ do not
mean the same physical processing. A result reported only as “dB” leaves the
weighting, time behavior, and reference quantity unspecified.

Measurements of exposure should retain raw or sufficiently detailed time data when
possible. A sole daily equivalent level cannot distinguish continuous machinery
noise from a small number of impulses or a sequence of tonal
events. A compact report includes at least a time history or time-binned level table,
the equivalent level, the maximum detector mode, total interval, and any
frequency weighting. Those records allow later comparison without reconstructing the
experiment from memory.

### Required record fields

- **Source state:** source identity, drive condition, operating cycle, and any output
  monitor.
- **Environment:** air temperature, relative humidity when relevant, propagation
  space, nearby boundaries, and background-source state.
- **Geometry:** source coordinates and axis, microphone coordinates and orientation,
  surveyed distance, mounting method, and scan order.
- **Instrument chain:** microphone model or sensitivity file, calibration method,
  calibrator value, preamplifier gain, input range, sample rate, and analysis band.
- **Level definition:** pressure or intensity quantity, reference value, frequency
  weighting, time weighting, rms or peak convention, and averaging interval.
- **Uncertainty:** repeatability, calibration tolerance, background contribution,
  position tolerance, and any model assumptions used for power or range conversion.

An uncertainty statement should follow the calculation path. For the calibrated
voltage example, let the calibration factor have relative standard uncertainty
$u_K/K$, and let the rms voltage have relative standard uncertainty $u_V/V$. If
they are independent, the pressure relative uncertainty is approximately

$$
\left(\frac{u_p}{p}\right)^2
=\left(\frac{u_K}{K}\right)^2+\left(\frac{u_V}{V}\right)^2.
$$

For small relative changes, the corresponding pressure-level uncertainty is

$$
u_{L_p}\simeq\frac{20}{\ln 10}\frac{u_p}{p}
\simeq 8.686\frac{u_p}{p}\ \mathrm{dB}.
$$

A 5% relative pressure uncertainty gives approximately $0.43\ \mathrm{dB}$. That
conversion is local; larger uncertainties should be propagated through the logarithm
directly. A point-source power estimate also carries uncertainty from distance,
directionality, range selection, and environmental variation. Meter
precision alone does not quantify model uncertainty.

The measurement objective determines the required detail. A classroom demonstration
may need calibrated pressure at one documented point. A source-power estimate needs
an enclosing-surface design or a justified radiation model. A room survey needs a
spatial grid and a reflection-aware interpretation. An exposure record needs duration,
weighting, time integration, and source-state history. The same microphone can serve
each task, but the geometry, processing, and uncertainty model differ.

### Pressure survey and quality checks

A loudspeaker operates with a stable electrical drive in an open indoor
space. The measurement goal is a pressure-level map along one documented source axis,
followed by an exposure summary for a ten-minute operating interval. The goal does
not include a claim of total acoustic power, because a single axis scan does not
measure radiation over an enclosing surface. Stating that boundary at the outset
prevents a pressure survey from being reported as a power characterization.

Mount a calibrated microphone on a tripod with its reference point at the same height
as the loudspeaker acoustic center. Survey positions at $1.00\ \mathrm{m}$,
$2.00\ \mathrm{m}$, and $4.00\ \mathrm{m}$ along the forward axis. Measure the
temperature near the path, retain the source-drive monitor, and acquire a
source-off background interval at every position. Revisit the $1.00\ \mathrm{m}$
position after each new range. That return measurement tracks output drift while
leaving the range sequence explicit.

Suppose the calibrated one-minute source-on levels are $76.4\ \mathrm{dB}$,
$70.1\ \mathrm{dB}$, and $64.0\ \mathrm{dB}$ at the three ranges. The corresponding
background levels are $67.0\ \mathrm{dB}$, $67.6\ \mathrm{dB}$, and
$62.5\ \mathrm{dB}$. Background matters increasingly at larger range because the
target source becomes less dominant. Convert each pair to linear ratios before
subtraction:

$$
L_{\mathrm{src}}
=10\log_{10}\left(10^{L_{\mathrm{on}}/10}
-10^{L_{\mathrm{bg}}/10}\right).
$$

At $1.00\ \mathrm{m}$, the correction from $76.4\ \mathrm{dB}$ background-inclusive
level is small because background is $9.4\ \mathrm{dB}$ lower. At $4.00\ \mathrm{m}$,
the $1.5\ \mathrm{dB}$ difference means background occupies a large part of the
source-on mean-square pressure. The corrected source estimate has much larger
uncertainty at that range even if the meter display has the same resolution.

$$
% caption: Source-on and background levels are recorded separately at each range before correction. Their linear difference gives the source contribution; the shrinking gap at the far range warns that the corrected value there is much less certain.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
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\node[above,rotate=90] at (.78,2.15) {level};
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\node[below] at (5.2,.5) {far};
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The ideal free-field point-source trend from $1.00\ \mathrm{m}$ to
$2.00\ \mathrm{m}$ is $-6.02\ \mathrm{dB}$; from $1.00\ \mathrm{m}$ to
$4.00\ \mathrm{m}$ it is $-12.04\ \mathrm{dB}$. Compare those predictions with
background-corrected levels, not with uncorrected values. If the corrected results
depart from the ideal trend, list likely experimental causes in a separate
interpretation field: wall reflection, source directivity, source heating, geometry
error, background variability, or insufficient far-field range. Do not insert an
absorption coefficient from the three-point slope difference alone; the fitted slope
must be compared with its uncertainty and the stated propagation model.

The calibration check can be processed on the same scale. Assume the pre-run
calibrator check reads $94.0\ \mathrm{dB}$ after applying the system factor and the
post-run check reads $93.8\ \mathrm{dB}$. The $0.2\ \mathrm{dB}$ difference is a
traceable drift indicator. Take half the difference, or a documented alternative
based on laboratory practice, as one contribution to
calibration uncertainty. It should not be silently adjusted away by forcing both
checks to read the nominal value.

For the ten-minute exposure summary, acquire a continuous pressure time record with
the selected weighting documented. Calculate $L_{\mathrm{eq},600\ \mathrm{s}}$ from
the full interval. Segment the record into one-minute values as a diagnostic, then
compare their linear mean-square average with the ten-minute result. A large
difference between segments signals a changing source or background. The single
ten-minute equivalent level remains a valid energy-equivalent summary, while the
one-minute sequence preserves temporal variation needed for interpretation.

If the measurement uses A weighting, apply it before forming the mean-square
integral and report $L_{\mathrm{Aeq},600\ \mathrm{s}}$. If the goal is a physical
pressure comparison with a plane-wave intensity conversion, retain an unweighted
band-limited pressure record as well. A-weighted values are valuable summaries of
weighted content, but they are not substitutes for raw pressure when an impedance or
power calculation is required.

Before publication or comparison, perform four final checks. Verify that the source
state did not change across the relevant time interval. Verify that pre-run and
post-run calibration values agree within the stated uncertainty. Verify that no
stored waveform is clipped and that the source signal remains adequately above
background and instrument noise. Verify that each stated conversion—pressure to
intensity, range extrapolation, or acoustic power—has the geometry and propagation
conditions needed by its model. These checks connect a displayed decibel value to a
measurement that another reader can assess, reproduce, or challenge quantitatively.

## Measurement model boundaries

The common acoustic relations apply under different physical conditions.
Keeping those conditions attached to the formula is part of accurate reporting. The
relation $v=\sqrt{B/\rho_0}$ describes a small disturbance in a material with a
specified bulk response. The ideal-gas form uses an adiabatic compression and a
uniform gas composition. The plane-wave relation
$I=p_{\mathrm{rms}}^2/(\rho_0v)$ requires a locally progressive wave and a
pressure–particle-velocity relation close to the real impedance $\rho_0v$. The
point-source relation needs a far-field range, known radiation geometry, and a path
where reflections and absorption are either negligible or explicitly modeled.

The limitations do not invalidate a measurement. They identify the quantity that was
actually measured and the additional data required for a conversion. A pressure
reading beside a loudspeaker can compare repeatable drive settings.
It becomes an intensity estimate only after pressure–velocity behavior is established.
A range scan in a room still documents the received pressure distribution. It becomes
an inverse-square test only after direct and reflected contributions are separated or
controlled. This distinction preserves the experimental result while preventing
unsupported claims.

Temperature enters several stages of the analysis. It changes sound speed through
the gas equation, affects wavelength at a
fixed source frequency, and can alter air density in the impedance used for a
pressure-to-intensity conversion. For small temperature changes in an ideal gas,

$$
\frac{\Delta v}{v}\simeq\frac{1}{2}\frac{\Delta T}{T}.
$$

A careful report records the temperature and the temperature location. It need not
apply a correction when the intended result is measured sound-pressure level at
that condition. It does need a correction or uncertainty allowance when comparing
time-of-flight speed, wavelength, or intensity estimates across materially different
conditions.

The same discipline applies to decibels. A pressure level is a logarithmic pressure
ratio. An equivalent level is a time-integrated mean-square-pressure statistic. An
intensity level is a logarithmic power-flux ratio. An A-weighted level includes a
frequency weighting. Each remains meaningful when its reference, interval, and
processing are named. Ambiguity enters when a bare number is moved between those
definitions without the conditions that make the conversion valid.