Helmholtz Resonator Calculator: Bass Traps, Exhausts, Bottles

The cavity is an adiabatic acoustic compliance and the neck is an inertance. For a circular neck, $A=\pi r^2$ and $L_\mathrm{eff}=L+k_\Sigma r$. The tuning frequency is

Rienstra and Hirschberg, An Introduction to Acoustics, §5.2.3, attached PDF pp. 131–133 (printed pp. 122–124), give the mass–spring equation and approximately 0.85r per flanged end or 0.61r per unflanged thin-wall end. The presets sum the two ends; they assume a much larger cavity cross section than the neck. Custom means a combined coefficient, not a coefficient per end.

Air sound speed is estimated by $c=343\sqrt{T_K/293.15}$ m/s. The square-root temperature dependence is in NASA CR-4671, attached p. 49; 343 m/s at room temperature is also used in the UNSW worked example. A known sound speed bypasses this approximation.

With acoustic resistance defined as pressure divided by volume velocity, $M_a=\rho L_\mathrm{eff}/A$, $C_a=V/(\rho c^2)$ and $Q=\omega_0 M_a/R_a=1/(2\zeta)$. Resistance includes all losses in the chosen acoustic loading. NASA CR-4671 §2.4, attached pp. 55–57, supplies the resistance/Q relation and a numerical example. Its mechanical resistance must be divided by $A^2$ to obtain the acoustic resistance used here.

For constant resistance and constant driving pressure, the normalized dissipated power is derived here from $Z=R_a+j(\omega M_a-1/(\omega C_a))$:

Solving $P_\mathrm{rel}=1/2$ gives $f_\pm=f_0(\sqrt{4+Q^{-2}}\pm Q^{-1})/2$ and exactly $\Delta f=f_+-f_-=f_0/Q$. The lower edge is evaluated in a numerically stable equivalent form. The band center is geometric, $\sqrt{f_-f_+}=f_0$, not the arithmetic average. The amplitude decay time is $Q/(\pi f_0)$ for the underdamped constant-resistance model. Required volume is the frequency equation rearranged with the neck fixed.

Screen the actual cavity at tuning and again at the upper half-power edge. With $D=\max(D_c,L_\mathrm{eff},d)$, the tool's one-tenth-wavelength ceiling is $f_\mathrm{screen}=0.1c/D$. The response plot ends at this ceiling; mathematical band edges remain reported with a warning if they exceed it. Target sizing is screened independently using $V_\mathrm{target}$ and $f_\mathrm{target}$: require $V_\mathrm{target}\leq D_c^3$, a neck air volume no greater than 10% of that required volume, and $D f_\mathrm{target}/c\leq0.1$. The entered largest cavity dimension is retained as the proposed design envelope; update it when resizing the cavity. The status beside the required-volume result summarizes these tests. The volume-tuning chart leaves gaps where those geometry and compactness screens fail, and an unsuitable target is omitted from that chart.

• One rigid, airtight cavity and one straight circular neck; small acoustic amplitudes, adiabatic cavity compression and approximately uniform cavity pressure.
• No mean flow, neck taper, interacting ports, compliant walls or frequency-dependent changes in the end correction.
• Damping ratio or total acoustic resistance is a user-supplied estimate. A constant resistance near resonance is assumed for the response and half-power band; the geometry does not determine damping.
• Density is only needed in resistance mode and is entered for the actual gas state. In damping-ratio mode, the displayed inertance/compliance use the hidden reference density; frequency and bandwidth do not depend on it.
• Room-temperature air is anchored at 343 m/s and 20 °C; humidity, gas composition and temperature variation within the device are omitted.
• The animation represents steady excitation at the tuning frequency, with arbitrary exaggerated displacement and slow playback. Its cavity outline is an equivalent cube cross section, not a construction drawing.

Use for first estimates of acoustically compact resonators. The one-tenth-wavelength warning and 10% neck-volume warning are conservative screening choices of this tool, not accuracy guarantees or limits quoted from a standard. Long thin cavities can develop modes even when the equivalent volume looks small; enter their actual largest dimension. This dimension also defines the cavity envelope used for target sizing. A target volume that cannot fit that envelope, has too much neck air volume, or fails compactness is explicitly flagged beside the sizing result. Changing target frequency does not bypass these screens. End corrections are approximate and can differ with flange size, neck/cavity area ratio, nearby surfaces and edge shape.

The model requires positive effective length and Q > 0.5. Broad, strongly damped responses are especially sensitive to the assumed constant resistance. The full reported band is checked at its upper half-power edge. The response chart is restricted to frequencies within the one-tenth-wavelength screen; a truncated curve does not establish a complete usable band. The volume-tuning chart omits candidates outside the geometry and compactness screens. The chart describes normalized power under constant pressure drive. It is not an absolute absorption coefficient, an exhaust transmission-loss curve or a prediction of room bass reduction. Radiation into different surroundings changes coupling and total damping.

For hot exhaust, use a sound speed and density appropriate to gas temperature and composition. Mean flow, strong pressure oscillations, turbulence and thermal gradients require a more detailed model and measurements. No durability, pressure-vessel, back-pressure or emissions assessment is included. Installation work belongs to a qualified person.