Acoustics Calculators

From kinetic theory: $$c = \sqrt{\gamma R T_k / M}$$ where $\gamma=1.4$, $R$ is gas constant, $M$ molar mass, $T_k = T+273.15$ K. For dry air at 1 atm this collapses to ISO 9613-1:

$$c(T) = 331.3 \sqrt{1 + \frac{T}{273.15}}$$

At 20 C that's 343.2 m/s. Picture O2 and N2 darting faster when warm — a pressure pulse gets handed off quicker. Humidity makes $M$ lighter so $c$ rises less than 0.5% even at 100% RH / 40 C, negligible for audio alignment. Pressure itself cancels: denser but also stiffer. This $c$ feeds wavelength, delay, and Helmholtz cards. If you leave temperature at 20 C while your venue is 30 C, your delay is about 1.7% short — 17 mm per metre of path, audible as comb in line-array rigs.

The wave repeats every $\lambda$, so it must satisfy $$c = f \lambda, \quad \lambda = \frac{c}{f}, \quad T_{per} = \frac{1}{f}$$

That's it. No hidden constant. A 440 Hz A is about 0.78 m — about a guitar body. 20 Hz is 17 m, bigger than most rooms, which is why bass pressurizes instead of propagates. 20 kHz is 17 mm, smaller than a tweeter dome, which is why it beams. If speaker spacing is greater than $\lambda/2$, you get spatial aliasing; if a port is less than $\lambda/10$, lumped models still work. Keep that mental ruler handy.

Time of flight is trivial but central:

$$d = c t, \quad t = \frac{d}{c}, \quad N_{samp} = t f_s$$

1 ms is 34.3 cm at 20 C. Haas says less than 30 ms arrivals fuse; greater than 50 ms they echo. For alignment, match subs and mains within 0.5 ms at crossover — about 17 cm. The calculator also gives samples at 48 kHz because your DSP delay line lives in samples, not meters. Temperature matters again: a summer outdoor stage needs 2% less electrical delay than winter.

Sound level meters read pressure squared averaged over time. Intensities add for uncorrelated sources (two drummers, two HVAC units). So convert each dB back to power, sum, convert back:

$$L_{\Sigma} = 10 \log_{10} \left( \sum_{i=1}^{N} 10^{L_i/10} \right)$$

Equal sources: $$L_{tot} = L_{single} + 10\log_{10} N$$. Two 80 dB incoherent = 83.0 dB. Ten = 90 dB. If perfectly coherent and in-phase (same signal, same wire), pressures add linearly:

$$L_{coh} = 20 \log_{10} \left( \sum_i 10^{L_i/20} \right) = L_{single} + 20\log_{10} N$$

That gives +6 dB per doubling. Reality lives between — correlated low-frequency room modes give almost +6, diffuse mid frequencies +3. Don't blindly add decibels.

Decibel is always 10 log of a power ratio. Power scales as amplitude squared. So for field quantities (pressure $p$, voltage $V$, velocity):

$$L_p = 20 \log_{10} \left| \frac{p}{p_0} \right|, \qquad L_W = 10 \log_{10} \frac{W}{W_0}$$

Inversions: $$\frac{p}{p_0} = 10^{L/20}, \quad \frac{W}{W_0} = 10^{L/10}$$ Doubling power = 10 log10 2 = +3.01 dB. Doubling pressure = 20 log10 2 = +6.02 dB. Same physical change, different bookkeeping because power goes as $p^2$. Voltage gain in audio follows 20 log; amplifier power follows 10 log. Toggle matters — 6 dB vs 3 dB mistakes blow drivers.

Sabine (1898) imagined energy sloshing uniformly, losing a tiny fraction A/S per bounce. Mean free path 4V/S, reflections per second cS/4V, exponential decay $E(t)=E_{0} e^{-cAt/4V}$. Set $10\log_{10}(E/E_{0})=-60$ dB:

$$T_{60}^{Sab} = 0.161 \frac{V}{A}, \quad A = \sum S_i \alpha_i = \bar{\alpha} S$$

0.161 = 55.26/c ~ $24\ln10 /c$ — it's just unit conversion for 60 dB and $c\approx343$. Works for $\bar{\alpha} < 0.2$ live rooms.

Eyring (1930) fixes high absorption by not approximating $\ln(1-\alpha)\approx -\alpha$. Treat energy as geometric series: after $n$ reflections $E_n = E_0 (1-\bar{\alpha})^n$. Solve for 60 dB drop:

$$T_{60}^{Eyr} = 0.161 \frac{V}{-S \ln(1-\bar{\alpha})}$$

When $\bar{\alpha}\ll1$, $- \ln(1-\bar{\alpha})\approx\bar{\alpha}$ and Eyring tends to Sabine. When $\bar{\alpha}=0.6$, Eyring can be 2x shorter — and more realistic. Both ignore air absorption (about 0.01 dB/m at 2 kHz, grows as $f^2$), frequency-dependent $\alpha$, and non-diffuse geometry. That's why real rooms measured with swept sine differ by plus or minus 20%.

Two derived tools: room constant and critical distance. $$R = \frac{S \bar{\alpha}}{1-\bar{\alpha}}, \quad d_c = \sqrt{\frac{Q R}{16\pi}}$$ $d_c$ is where direct $\propto1/r^2$ equals reverberant field — beyond it you hear room, not speaker. $Q$ is source directivity. Schroeder's crossover:

$$f_s = 2000 \sqrt{\frac{T_{60}}{V}}$$

Below $f_s$ you hear individual eigenmodes as booms; above, a smooth decay. Constant 2000 assumes about 10 modes overlap within a half-power bandwidth — rules of thumb vary 1500 to 4000, we pick 2000. Use it to know when to fix geometry vs add absorption.

Blow across a bottle: air plug in neck is mass $m_{air}=\rho S L_{eff}$, cavity volume is spring with compliance $V/\rho c^{2}$. Resonance of that mass-spring:

$$f_H = \frac{c}{2\pi} \sqrt{\frac{S}{V L_{eff}}}$$

End correction because moving air extends past physical ends. For one flanged end (box wall) + one free end: $$L_{eff} = L + 0.85 r$$ with $r=\sqrt{S/\pi}$. Both ends flanged gives L+1.7r; both free gives L+1.2r. The 0.85r is Rayleigh's radiation reactance — extra air dragged along.

Inversion for box tuning: $$L_{eff} = \frac{S c^2}{4\pi^2 f_b^2 V}, \quad L = L_{eff} -0.85r$$ That's the port length card. Assumptions: $L\ll\lambda$, $V^{1/3}\ll\lambda$ (lumped), circular unflared port, rigid lossless walls, no fill. Real ports with flare act longer at high SPL (end correction changes with velocity), damping material lowers effective $V$ by 10-20%, and driver Vas adds compliance. Start here, then measure and trim by 15%.

If source moves toward observer, each crest is emitted from a closer point — wavelengths compress. Observer moving toward meets crests faster. For $v_s$ source toward observer, $v_o$ observer toward source (positive toward each other):

$$f' = f \frac{c + v_o}{c - v_s}$$

If $v_s\to c$, denominator goes to 0 and $f'\to\infty$ — boom, shock forms. Negative $v_s$ means receding, drops pitch. At audio speeds (10 m/s car is 3% of $c$) shift is about 0.3 semitones per 10 m/s — subtle but Doppler for rotating Leslie horn at 5 m/s rim speed is why it choruses. Keep $c$ from temperature to avoid phantom cents.

A limp wall's inertia resists being driven by sound. Random-incidence transmission loss for single leaf:

$$TL \approx 20\log_{10}(m f) - 47.2$$ dB

$m$ surface mass kg/m2, $f$ Hz. 6 dB per octave, 6 dB per doubling mass — mass times frequency law. Real panels dip at coincidence $f_{c} = c^{2}/2\pi \sqrt{m/B}$ where bending wave matches trace wavelength; above that stiffness helps again. Double-leaf adds mass-spring-mass resonance. Use Mass Law as upper bound for limp, heavy, damped walls.

NR (Noise Rating) ISO 1996: linear curves $L_n = a_n + b_n NR$ where $n$ indexes octave band 31.5 Hz to 8 kHz, $a_n,b_n$ tabulated constants. Your measured octave SPL must sit below the curve to claim that NR. NR 25 is concert hall, 35 bedroom, 45 open office. It's a single-number mask for annoyance and speech interference, not a physical law.