Dimensional analysis

Units, scaling, order-of-magnitude estimation.

The cheapest sanity check in physics, and the most underused technique in applied math. Almost every formula derived in the Sound book can be guessed up to a dimensionless prefactor by counting units. This chapter shows how, with practical examples.

The seven base dimensions

The SI system has seven base dimensions. For acoustics we mostly need four: length L, time T, mass M, temperature Θ. The others (electric current I, amount N, luminous intensity J) rarely appear.

Every physical quantity has a dimension: a product of powers of base dimensions.

Quantity	Symbol	Dimension	SI unit
Length	$\ell$	$\mathrm{L}$	m
Time	$t$	$\mathrm{T}$	s
Mass	$m$	$\mathrm{M}$	kg
Temperature	$T$	$\Theta$	K
Velocity	$v$	$\mathrm{L\,T^{-1}}$	m/s
Acceleration	$a$	$\mathrm{L\,T^{-2}}$	m/s²
Force	$F$	$\mathrm{M\,L\,T^{-2}}$	N
Energy	$E$	$\mathrm{M\,L^2\,T^{-2}}$	J
Power	$P$	$\mathrm{M\,L^2\,T^{-3}}$	W
Pressure	$p$	$\mathrm{M\,L^{-1}\,T^{-2}}$	Pa
Density (volume)	$\rho$	$\mathrm{M\,L^{-3}}$	kg/m³
Density (linear)	$\mu$	$\mathrm{M\,L^{-1}}$	kg/m
Specific energy	—	$\mathrm{L^2\,T^{-2}}$	J/kg
Wavenumber	$k$	$\mathrm{L^{-1}}$	1/m
Angular frequency	$\omega$	$\mathrm{T^{-1}}$	1/s
Intensity	$I$	$\mathrm{M\,T^{-3}}$	W/m²
Acoustic impedance	$Z$	$\mathrm{M\,L^{-2}\,T^{-1}}$	Pa·s/m
Boltzmann constant	$k_B$	$\mathrm{M\,L^2\,T^{-2}\,\Theta^{-1}}$	J/K

A formula is dimensionally consistent only if both sides have the same dimensions. This is necessary for correctness but not sufficient — there could still be wrong numerical factors. But it’s a check that costs nothing and catches half the algebraic errors people make.

Buckingham π theorem

Wave speed ∼ K^½ · ρ^−½

Combined dimensions: [M^0.00 L^1.00 T^-1.00] — target is [L T⁻¹].

The interactive above lets you pick the physical quantities you think a wave speed depends on, and solves the dimensional system for the exponents. Try bulk modulus + mass density to derive $c \sim \sqrt{K/\rho}$ for sound; try tension + linear mass density to get $c \sim \sqrt{T/\mu}$ for a string. Most pairs of incompatible quantities fail outright — no combination of them has units of velocity. The success conditions are themselves informative.

If a physical relation involves $n$ quantities built from $k$ independent base dimensions, the relation can be re-expressed in terms of $n - k$ dimensionless combinations (the π groups).

Two consequences we use:

A formula with no free dimensionless combinations is fully determined up to a constant by the dimensions of the inputs.
The dimensionless combinations themselves often have physical names: Reynolds number ( $Re = \rho v L / \mu$ ), Mach number ( $M = v/c$ ), Strouhal number ( $St = f L / v$ ).

Worked example: the speed of sound by dimensional reasoning

Suppose $c$ depends on bulk modulus $K$ ( $\mathrm{M\,L^{-1}\,T^{-2}}$ ) and density $\rho$ ( $\mathrm{M\,L^{-3}}$ ) only. Seek $c \sim K^a\, \rho^b$ . Match dimensions:

\mathrm{L\,T^{-1}} \;=\; \big(\mathrm{M\,L^{-1}\,T^{-2}}\big)^a \cdot \big(\mathrm{M\,L^{-3}}\big)^b \;=\; \mathrm{M}^{a+b}\, \mathrm{L}^{-a-3b}\, \mathrm{T}^{-2a}.

Three equations:

M: $a + b = 0$
L: $-a - 3b = 1$
T: $-2a = -1$

Solving: $a = 1/2$ , $b = -1/2$ . So

c \;\sim\; \sqrt{K / \rho}.

The prefactor turns out to be exactly 1 (you need fluid mechanics or kinetic theory to nail it down — see Sound 4.9), but the functional form fell out for free.

▶ The same logic for any wave speed

For transverse waves on a string: $c$ depends on tension $T$ ( $\mathrm{M\,L\,T^{-2}}$ ) and linear mass density $\mu$ ( $\mathrm{M\,L^{-1}}$ ). Same procedure: $c \sim T^a \mu^b$ , dimensional match gives $a = 1/2$ , $b = -1/2$ :

c \;\sim\; \sqrt{T / \mu}.

For flexural waves on a plate: $c$ depends on plate stiffness $D$ ( $\mathrm{M\,L^2\,T^{-2}}$ ) and density per area $\sigma$ ( $\mathrm{M\,L^{-2}}$ ) — and on frequency $\omega$ ( $\mathrm{T^{-1}}$ ). The dispersion relation has one free dimensionless combination, so we can predict the form only up to that combination:

c \;\sim\; \omega^{1/2}\, (D/\sigma)^{1/4}.

This is the dispersion of bending waves — different from non-dispersive longitudinal acoustic waves. The dimensional argument already tells us they are not in the same family.

Useful order-of-magnitude numbers

For sanity-checking derivations. Memorise these:

Quantity	Value
Speed of sound in air at 20°C	343 m/s
Speed of sound in water at 20°C	1480 m/s
Speed of sound in steel	~5100 m/s
Atmospheric pressure	$1.013 \times 10^5$ Pa
Air density	1.2 kg/m³
Water density	1000 kg/m³
Air acoustic impedance	412 Pa·s/m
Water acoustic impedance	$1.5 \times 10^6$ Pa·s/m
Thermal speed of air molecule (20°C)	~500 m/s
Boltzmann’s constant $k_B$	$1.38 \times 10^{-23}$ J/K
Conversational pressure amplitude	$10^{-2}$ Pa
Conversational acoustic intensity	$10^{-7}$ W/m²
Threshold of hearing (1 kHz) intensity	$10^{-12}$ W/m²
Threshold of pain	1 W/m²
Wavelength at 1 kHz in air	34 cm
Wavelength at 1 kHz in water	1.5 m
Wavelength at 1 MHz in tissue	1.5 mm

These let you spot-check derivations in seconds. If a derived speed of sound comes out as 34 m/s, you’ve dropped a factor of 10 somewhere.

Important dimensionless numbers

The most-used dimensionless combinations in acoustics:

Mach number: $M = v / c$ . Compressibility starts mattering when $M$ approaches 1.
Strouhal number: $St = f L / v$ . Sets the characteristic frequency of flow-induced sound (Aeolian tones).
Reynolds number: $Re = \rho v L / \mu$ . Sets the regime of fluid flow (laminar → turbulent at $Re \sim 10^4$ ).
Helmholtz number: $ka = 2\pi a / \lambda$ . How big is the source compared to the wavelength?
Knudsen number: $Kn = \lambda_{mfp} / L$ . How discrete is the medium on the scale of interest?

Each has its own role; each is dimensionless; each tells you which regime of physics applies.

Scaling laws and rules of thumb

Many acoustic results are scaling laws. Once the dimensional form is in hand, the qualitative consequences follow:

Wavelength varies inversely with frequency.
Intensity from a monopole falls as $1/r^2$ (inverse square law).
Sound power radiated by a turbulent jet scales as $\rho U^8 D^2 / c^5$ (Lighthill’s $U^8$ law).
The Schroeder frequency goes as $\sqrt{T_{60}/V}$ .
Doppler shift fractional change equals Mach number for non-relativistic motion.

Each of these is “obvious” once you’ve worked out the dimensions. None of them is obvious without.

⏳ The history — Rayleigh, Buckingham, and the dimensionless number

Lord Rayleigh’s 1877 Theory of Sound used dimensional reasoning throughout — to guess scaling laws, to check derivations, to argue that certain phenomena could only depend on dimensionless combinations of parameters. He didn’t formalise the technique; he just used it everywhere. By the early 1900s “Rayleigh’s method” was an informal craft.

In 1883 Osborne Reynolds, studying flow through pipes, identified what we now call the Reynolds number $Re = \rho U L / \mu$ — a dimensionless group whose value distinguished laminar from turbulent flow regardless of the absolute scale of the pipe. This was the first time a named dimensionless number was understood as the physically-meaningful parameter of a problem.

In 1914 Edgar Buckingham (US Bureau of Standards) formalised what Rayleigh had been doing: if a physical relationship involves $n$ variables with $k$ independent dimensions, it can be rewritten as a relation among $n - k$ dimensionless groups. The Buckingham π theorem turned dimensional analysis from craft into a recipe. Almost every dimensionless number in physics — Mach, Reynolds, Prandtl, Strouhal, Helmholtz — emerged from this framework.

What we use it for

Sanity-checking every derivation in the Sound book.
Deriving the speed of sound up to a constant — multiple times, once per route in Sound chapter 4.
Estimating whether diffraction matters at a given obstacle (compare the wavelength to obstacle size).
Comparing relative magnitudes in the nonlinear corrections of Sound chapter 10.
Understanding the Lighthill $U^8$ scaling for jet noise (Sound 9.5).