§F
What is sound?
§F · Cheat sheet

Cheat sheet

Every key formula, in canonical order, at a glance. Each formula links to the lesson that derives it.

1 — The signal

Ideal-gas law

p=nkBT=ρRT/Mp = n k_B T = \rho R T / M

Kinetic pressure

p=nm⟨vx2⟩=13nm⟨v2⟩p = n m \langle v_x^2 \rangle = \tfrac13 n m \langle v^2 \rangle

Pressure as energy density

p=23up = \tfrac23 u

Equipartition

⟨12mv2⟩=32kBT\langle \tfrac12 m v^2 \rangle = \tfrac32 k_B T

Maxwell–Boltzmann speed law

f(v)=mkBTvexp⁡(−mv22kBT)f(v) = \frac{m}{k_B T} v \exp(-\frac{m v^2}{2 k_B T})

Mean-square displacement

⟨∣Δr(t)∣2⟩=2dDt\langle \lvert \Delta \mathbf{r}(t) \rvert^2 \rangle = 2 d D t

Stokes–Einstein diffusion

D=kBT/(6πηa)D = k_B T / (6\pi\eta a)

Sound as perturbation

p(r,t)=p0+p′(r,t)p(\mathbf{r},t) = p_0 + p'(\mathbf{r},t)

Acoustic wave equation

∂t2p′=c2∇2p′\partial_t^2 p' = c^2 \nabla^2 p'

2 — Oscillators

SHM equation of motion

x¨+ω02x=0\ddot x + \omega_0^2 x = 0

SHM general solution

x(t)=Xcos⁡(ω0t+φ)x(t) = X \cos(\omega_0 t + \varphi)

Natural frequency

ω0=k/m\omega_0 = \sqrt{k/m}

SHM total energy

E=12mx˙2+12kx2=12kX2E = \tfrac12 m \dot x^2 + \tfrac12 k x^2 = \tfrac12 k X^2

Phasor / real signal

x(t)=Re⁡[X~eiω0t], X~=Xeiφx(t) = \operatorname{Re}[\tilde X e^{i\omega_0 t}],\ \tilde X = X e^{i\varphi}

Time-average of product

⟨Re⁡[A~eiωt] Re⁡[B~eiωt]⟩=12Re⁡[A~B~∗]\langle \operatorname{Re}[\tilde A e^{i\omega t}]\, \operatorname{Re}[\tilde B e^{i\omega t}] \rangle = \tfrac12 \operatorname{Re}[\tilde A \tilde B^*]

Damped oscillator equation

x¨+2γx˙+ω02x=0\ddot x + 2\gamma \dot x + \omega_0^2 x = 0

Underdamped solution

x(t)=X0e−γtcos⁡(ωdt+φ)x(t) = X_0 e^{-\gamma t} \cos(\omega_d t + \varphi)

Damped frequency

ωd=ω02−γ2\omega_d = \sqrt{\omega_0^2 - \gamma^2}

Driven amplitude (phasor)

X~(ω)=F0/m(ω02−ω2)+2iγω\tilde X(\omega) = \dfrac{F_0/m}{(\omega_0^2 - \omega^2) + 2 i \gamma \omega}

Peak frequency

ωpeak2=ω02−2γ2\omega_\text{peak}^2 = \omega_0^2 - 2\gamma^2

Quality factor

Q≡ω0/(2γ)Q \equiv \omega_0 / (2\gamma)

Resonance bandwidth

Δω≈ω0/Q\Delta\omega \approx \omega_0 / Q

Lorentzian power shape

Pˉ(ω)∝1(ω−ω0)2+(ω0/2Q)2\bar P(\omega) \propto \dfrac{1}{(\omega - \omega_0)^2 + (\omega_0/2Q)^2}

Fourier synthesis of response

x(t)=∫X~(ω) eiωt dωx(t) = \int \tilde X(\omega)\, e^{i\omega t}\, d\omega

3 — From oscillator to wave

Discrete-chain equation of motion

m y¨n=κ (yn+1−2yn+yn−1)m\,\ddot y_n = \kappa\,(y_{n+1} - 2y_n + y_{n-1})

Chain mode frequencies

ωk=2κ/m sin⁡ ⁣(kπ2(N+1))\omega_k = 2\sqrt{\kappa/m}\,\sin\!\left(\frac{k\pi}{2(N+1)}\right)

1-D wave equation

∂t2y=c2 ∂x2y\partial_t^2 y = c^2\,\partial_x^2 y

Wave speed on a string

c=T/μc = \sqrt{T/\mu}

d’Alembert general solution

y(x,t)=F(x−ct)+G(x+ct)y(x,t) = F(x - ct) + G(x + ct)

d’Alembert initial-value formula

y=12[f(x−ct)+f(x+ct)]+12c∫x−ctx+ctg(ξ) dξy = \tfrac12[f(x-ct)+f(x+ct)] + \frac{1}{2c}\int_{x-ct}^{x+ct} g(\xi)\,d\xi

Wave kinematic relation

c=λ/T=λf, ω=ckc = \lambda/T = \lambda f,\ \omega = ck

Clamped / free-end reflection

G(η)=−F(−η) (clamped), G(η)=F(−η) (free)G(\eta) = -F(-\eta)\ \text{(clamped)},\ G(\eta) = F(-\eta)\ \text{(free)}

Clamped-string mode shape

yn(x,t)=Ansin⁡ ⁣(nπxL)cos⁡(ωnt+φn)y_n(x,t) = A_n \sin\!\left(\frac{n\pi x}{L}\right)\cos(\omega_n t + \varphi_n)

Mode frequencies

ωn=nπcL, fn=nc2L\omega_n = \frac{n\pi c}{L},\ f_n = \frac{nc}{2L}

4 — The acoustic wave equation

Continuity equation

∂tρ+∇⋅(ρv)=0\partial_t \rho + \nabla \cdot (\rho \mathbf{v}) = 0

Linearised continuity

∂tρ′+ρ0∇⋅v′=0\partial_t \rho' + \rho_0 \nabla \cdot \mathbf{v}' = 0

Euler’s equation

ρ DvDt=−∇p\rho\, \frac{D\mathbf{v}}{Dt} = -\nabla p

Linearised Euler

ρ0 ∂tv′=−∇p′\rho_0\, \partial_t \mathbf{v}' = -\nabla p'

Adiabatic equation of state

p′=(γp0/ρ0) ρ′≡c2ρ′p' = (\gamma p_0/\rho_0)\, \rho' \equiv c^2 \rho'

Speed of sound (ideal gas)

c2=γp0/ρ0=γRT0/Mc^2 = \gamma p_0/\rho_0 = \gamma R T_0/M

General-fluid sound speed

c2≡(∂p/∂ρ)sc^2 \equiv (\partial p/\partial \rho)_s

Acoustic wave equation

∂t2p′=c2∇2p′\partial_t^2 p' = c^2 \nabla^2 p'

Discrete-chain dispersion

ω(q)=2κ/m ∣sin⁡(qa/2)∣\omega(q) = 2\sqrt{\kappa/m}\, \lvert \sin(qa/2) \rvert

Kinetic-theory sound speed

c2=γkBT/mc^2 = \gamma k_B T/m

Sound speed vs thermal speed

c/vrms=γ/3c/v_\text{rms} = \sqrt{\gamma/3}

Acoustic Lagrangian density

L=12ρ0(∂tϕ)2−12ρ0c2(∇ϕ)2\mathcal{L} = \tfrac12 \rho_0 (\partial_t \phi)^2 - \tfrac12 \rho_0 c^2 (\nabla \phi)^2

Wave equation (velocity potential)

∂t2ϕ=c2∇2ϕ\partial_t^2 \phi = c^2 \nabla^2 \phi

Temperature rule of thumb

c≈331+0.6 TCc \approx 331 + 0.6\, T_\text{C}

Acoustic energy density

E=12ρ0∣v′∣2+12ρ0c2p′2\mathcal{E} = \tfrac12 \rho_0 \lvert \mathbf{v}' \rvert^2 + \tfrac{1}{2\rho_0 c^2} p'^2

Impedance and reflection

Z=ρ0c, R=(Z2−Z1)/(Z2+Z1)Z = \rho_0 c,\ R = (Z_2 - Z_1)/(Z_2 + Z_1)

5 — Energy, momentum, impedance

Plane harmonic wave

p′(r,t)=P0cos⁡(ωt−k⋅r)p'(\mathbf{r}, t) = P_0 \cos(\omega t - \mathbf{k} \cdot \mathbf{r})

Dispersion relation

ω=ck\omega = c k

Pressure–velocity relation

v~′=k^ρ0c p~′\tilde{\mathbf{v}}' = \dfrac{\hat{\mathbf{k}}}{\rho_0 c}\, \tilde p'

Acoustic energy density

E=12ρ0∣v′∣2+p′22ρ0c2\mathcal{E} = \tfrac12 \rho_0 \lvert\mathbf{v}'\rvert^2 + \dfrac{p'^2}{2\rho_0 c^2}

Mean energy density (plane wave)

⟨E⟩=P022ρ0c2\langle \mathcal{E} \rangle = \dfrac{P_0^2}{2\rho_0 c^2}

Acoustic intensity

I=p′ v′\mathbf{I} = p'\,\mathbf{v}'

Mean intensity (plane wave)

⟨I⟩=P022ρ0c k^\langle \mathbf{I} \rangle = \dfrac{P_0^2}{2\rho_0 c}\,\hat{\mathbf{k}}

Intensity–energy relation

⟨I⟩/⟨E⟩=c\langle I \rangle / \langle \mathcal{E} \rangle = c

Specific acoustic impedance

Z≡ρ0cZ \equiv \rho_0 c

Reflection coefficient

R=Z2−Z1Z2+Z1R = \dfrac{Z_2 - Z_1}{Z_2 + Z_1}

Sound intensity level

LI=10 log⁡10(I/Iref) dBL_I = 10\,\log_{10}(I/I_\text{ref})\ \text{dB}

Sound pressure level

Lp=20 log⁡10(P/Pref) dBL_p = 20\,\log_{10}(P/P_\text{ref})\ \text{dB}

Mean momentum density

⟨ρ′v′⟩=P022ρ0c3=⟨I⟩c2\langle \rho' v' \rangle = \dfrac{P_0^2}{2\rho_0 c^3} = \dfrac{\langle I \rangle}{c^2}

Radiation pressure (reflecting)

Prad=2⟨I⟩/cP_\text{rad} = 2\langle I \rangle / c

6 — Sources and radiation

Radial wave equation (spherical)

∂t2(rp)=c2 ∂r2(rp)\partial_t^2(rp) = c^2\,\partial_r^2(rp)

Outgoing monopole field

p(r,t)=f(r−ct)rp(r,t) = \dfrac{f(r-ct)}{r}

Spherical wave field

p(r,t)=P0arcos⁡(ωt−k(r−a)+φ)p(r,t) = \dfrac{P_0 a}{r}\cos(\omega t - k(r-a) + \varphi)

Inverse-square intensity

⟨I⟩=P02a22ρ0c r2∝1r2\langle I\rangle = \dfrac{P_0^2 a^2}{2\rho_0 c\, r^2} \propto \dfrac{1}{r^2}

Total radiated power

Prad=2πP02a2ρ0cP_\text{rad} = \dfrac{2\pi P_0^2 a^2}{\rho_0 c}

Cylindrical far-field wave

p(ρ,t)≈P0ρcos⁡(ωt−kρ+π/4)p(\rho,t) \approx \dfrac{P_0}{\sqrt{\rho}}\cos(\omega t - k\rho + \pi/4)

Cylindrical spreading intensity

⟨I⟩∝1/ρ\langle I\rangle \propto 1/\rho

Dipole far-field (cos⁡θ\cos\theta)

p≈ω2ρ0Q0d4πc rcos⁡θ cos⁡(ωt−kr)p \approx \dfrac{\omega^2\rho_0 Q_0 d}{4\pi c\, r}\cos\theta\,\cos(\omega t - kr)

Dipole intensity

⟨I⟩∝ω4Q02d2c2r2cos⁡2θ\langle I\rangle \propto \dfrac{\omega^4 Q_0^2 d^2}{c^2 r^2}\cos^2\theta

Piston directivity

D(θ)=2J1(kasin⁡θ)kasin⁡θD(\theta) = \dfrac{2 J_1(ka\sin\theta)}{ka\sin\theta}

Free-space Green’s function

G=δ(t−t′−∣r−r′∣/c)4π ∣r−r′∣G = \dfrac{\delta(t - t' - \lvert\mathbf{r}-\mathbf{r}'\rvert/c)}{4\pi\,\lvert\mathbf{r}-\mathbf{r}'\rvert}

Retarded-time field

p′(r,t)=14π∫s(r′, t−∣r−r′∣/c)∣r−r′∣ d3r′p'(\mathbf{r},t) = \dfrac{1}{4\pi}\displaystyle\int \dfrac{s(\mathbf{r}',\,t-\lvert\mathbf{r}-\mathbf{r}'\rvert/c)}{\lvert\mathbf{r}-\mathbf{r}'\rvert}\,d^3r'

7 — Boundaries, diffraction, and modes

Reflection coefficient (normal)

R=Z2−Z1Z2+Z1R = \dfrac{Z_2 - Z_1}{Z_2 + Z_1}

Transmission coefficient (normal)

T=2Z2Z2+Z1T = \dfrac{2 Z_2}{Z_2 + Z_1}

Power reflection / transmission

RP=∣R∣2, TP=4Z1Z2(Z1+Z2)2R_P = \lvert R \rvert^2,\ T_P = \dfrac{4 Z_1 Z_2}{(Z_1 + Z_2)^2}

Snell’s law for sound

sin⁡θic1=sin⁡θtc2\dfrac{\sin\theta_i}{c_1} = \dfrac{\sin\theta_t}{c_2}

Critical angle

θc=arcsin⁡(c1/c2)\theta_c = \arcsin(c_1 / c_2)

Reflection coefficient (oblique)

R=Z2cos⁡θi−Z1cos⁡θtZ2cos⁡θi+Z1cos⁡θtR = \dfrac{Z_2 \cos\theta_i - Z_1 \cos\theta_t}{Z_2 \cos\theta_i + Z_1 \cos\theta_t}

Quarter-wave matching impedance

Z2=Z1Z3Z_2 = \sqrt{Z_1 Z_3}

Circular aperture (Airy) intensity

I(θ)∝[2J1(kasin⁡θ)kasin⁡θ]2I(\theta) \propto \left[\dfrac{2 J_1(k a \sin\theta)}{k a \sin\theta}\right]^2

Rayleigh resolution criterion

sin⁡θ=1.22 λ/D\sin\theta = 1.22\, \lambda / D

Tube modes (open-open / closed-closed)

fn=nc2Lf_n = \dfrac{n c}{2 L}

Tube modes (closed-open)

fn=(2n−1)c4Lf_n = \dfrac{(2n-1) c}{4 L}

Rectangular cavity eigenfrequencies

f=c2nx2Lx2+ny2Ly2+nz2Lz2f = \dfrac{c}{2} \sqrt{\dfrac{n_x^2}{L_x^2} + \dfrac{n_y^2}{L_y^2} + \dfrac{n_z^2}{L_z^2}} n(f)≈4πVc3f2+πS2c2f+Le8cn(f) \approx \dfrac{4\pi V}{c^3} f^2 + \dfrac{\pi S}{2 c^2} f + \dfrac{L_e}{8 c}

Schroeder frequency

fS≈2000T60/Vf_S \approx 2000 \sqrt{T_{60}/V}

Sabine reverberation time

T60=0.161 VAT_{60} = \dfrac{0.161\, V}{A}

Diffuse-field energy density

E=4PcAE = \dfrac{4 P}{c A}

8 — The frequency picture

Short-time Fourier transform

X(t,ω)=∫−∞∞x(τ) w(τ−t) e−iωτ dτX(t,\omega) = \int_{-\infty}^\infty x(\tau)\, w(\tau - t)\, e^{-i\omega\tau}\, d\tau

Time–frequency uncertainty

Δt⋅Δω≥12\Delta t \cdot \Delta\omega \geq \tfrac12

First-order lowpass transfer function

H(ω)=11+iω/ωcH(\omega) = \dfrac{1}{1 + i\omega/\omega_c}

Cascaded filters multiply

Htotal(ω)=H2(ω) H1(ω)H_\text{total}(\omega) = H_2(\omega)\, H_1(\omega)

Resonant transfer function

H(ω)=1/m(ω02−ω2)+2iγωH(\omega) = \dfrac{1/m}{(\omega_0^2 - \omega^2) + 2i\gamma\omega}

Q as bandwidth

Q≡ω02γ=f0ΔfQ \equiv \dfrac{\omega_0}{2\gamma} = \dfrac{f_0}{\Delta f}

9 — Doppler and moving media

Doppler, source approaching

fo=fs1−vs/cf_o = \dfrac{f_s}{1 - v_s/c}

Doppler, moving observer

fo=fs (1+vo/c)f_o = f_s\,(1 + v_o/c)

Doppler, both moving

fo=fs c+voc−vsf_o = f_s\,\dfrac{c + v_o}{c - v_s}

Convected wave equation

1c2(∂t+U⋅∇)2p′=∇2p′\dfrac{1}{c^2}\left(\partial_t + \mathbf{U}\cdot\nabla\right)^{2} p' = \nabla^2 p'

Dispersion relation in flow

ω=U⋅k±c ∣k∣\omega = \mathbf{U}\cdot\mathbf{k} \pm c\,\lvert\mathbf{k}\rvert

Mach number

M=vs/cM = v_s/c

Mach cone half-angle

sin⁡α=cvs=1M\sin\alpha = \dfrac{c}{v_s} = \dfrac{1}{M}

Doppler, general (with medium)

fo=fs c+U+voc+U−vsf_o = f_s\,\dfrac{c + U + v_o}{c + U - v_s}

Vortex shedding (Aeolian) frequency

fshed=St⋅v/df_\text{shed} = \text{St}\cdot v/d

Lighthill analogy

1c2∂t2p′−∇2p′=∂i∂jTij\dfrac{1}{c^2}\partial_t^2 p' - \nabla^2 p' = \partial_i \partial_j T_{ij}

Lighthill U8U^8 power law

P∼ρU8D2/c5P \sim \rho U^8 D^2 / c^5

Refraction at a flow boundary

sin⁡θ1c1+U1cos⁡θ1=sin⁡θ2c2+U2cos⁡θ2\dfrac{\sin\theta_1}{c_1 + U_1\cos\theta_1} = \dfrac{\sin\theta_2}{c_2 + U_2\cos\theta_2}

10 — Attenuation and the nonlinear edge

Damped wave equation

∂ttp′=c2∇2p′+δ ∂t∇2p′\partial_{tt} p' = c^2 \nabla^2 p' + \delta\, \partial_t \nabla^2 p'

Exponential decay

p′(x,t)∼e−αxcos⁡(ωt−krx)p'(x,t) \sim e^{-\alpha x}\cos(\omega t - k_r x)

Absorption coefficient

α≈δω2/2c3\alpha \approx \delta \omega^2 / 2c^3

Classical absorption

αclassical=2π2f2ρ0c3[43η+(γ−1)κcp]∝f2\alpha_\text{classical} = \dfrac{2\pi^2 f^2}{\rho_0 c^3}\left[\tfrac{4}{3}\eta + (\gamma-1)\tfrac{\kappa}{c_p}\right] \propto f^2

Relaxation absorption

αrelax(ω)∝ω2τ1+(ωτ)2\alpha_\text{relax}(\omega) \propto \dfrac{\omega^2 \tau}{1 + (\omega\tau)^2}

Relaxation peak frequency

fr=1/(2πτ)f_r = 1/(2\pi\tau)

Total atmospheric absorption

αtotal(f)=αclassical+αN2+αO2\alpha_\text{total}(f) = \alpha_\text{classical} + \alpha_{\text{N}_2} + \alpha_{\text{O}_2}

Local sound speed

clocal≈c0+γ+12 v′c_\text{local} \approx c_0 + \tfrac{\gamma+1}{2}\, v'

Burgers equation

∂tv+(c0+βv) ∂xv=ν ∂xxv\partial_t v + (c_0 + \beta v)\,\partial_x v = \nu\,\partial_{xx} v

Shock formation distance

xshock∼c02βωv0=c0βMωx_\text{shock} \sim \dfrac{c_0^2}{\beta \omega v_0} = \dfrac{c_0}{\beta M \omega}

Rankine–Hugoniot (mass)

ρ1(u1−Us)=ρ2(u2−Us)\rho_1(u_1 - U_s) = \rho_2(u_2 - U_s)

Strong-shock density limit

ρ2/ρ1≤(γ+1)/(γ−1)=6\rho_2/\rho_1 \le (\gamma+1)/(\gamma-1) = 6