Acoustic Horn Calculator

1 — Webster: 1D horn physics

Assume pressure is uniform across cross-section, varies only along axis $x$. Conservation of mass + momentum gives Webster:

$$\frac{1}{S}\frac{\partial}{\partial x}\left(S\frac{\partial p}{\partial x}\right) = \frac{1}{c^2}\frac{\partial^2 p}{\partial t^2}$$

$S(x)$ is area profile. Solve for harmonic $e^{j\omega t}$ → spatial ODE whose solutions propagate only if $\omega$ above cutoff set by area flare rate. Ignore transverse modes — that is the core approximation. Valid up to $ka\approx 3–5$ depending on flare angle, beyond you need BEM.

2 — Three profiles, three bargains

Exponential: $S(x)=S_0 e^{mx}$. Classic Keele / Klipsch. Area doubles every ln2/m. Loads driver with frequency-independent resistance above cutoff, then dies below:

$$f_c = \frac{m c}{4\pi}$$

Great LF loading, terrible directivity — beamwidth shrinks as $\propto 1/f$ because curved walls repeatedly reflect higher-order modes.

Conical: $S(x)=\pi(r_0 + x\tan\theta)^2$. Straight walls. Poor loading — no sharp $f_c$, just gradual transformer — but wall angle sets beamwidth directly: full angle $\approx 2\theta$. That is constant directivity by geometry.

Oblate Spheroidal (OSWG): $S(x)=S_0(1+(x/a)^2)$ where $a$ governs throat curvature. Derived by Geddes: define oblate spheroidal coordinates $\xi,\eta$ so wavefronts are spherical caps. Matches flat piston to spherical expansion at mouth with minimal discontinuity, killing HOMs that plague exponential and rapid-flare horns. Best CD + low HOM, slight price in LF loading vs exponential.

3 — Directivity, ka, and why size matters

Mouth directivity kicks in when mouth circumference comparable to wavelength. Formal parameter:

$$ka = \frac{2\pi a_{mouth}}{\lambda} = \frac{\omega a_{mouth}}{c}$$

$ka < 1$ → mouth omnidirectional, waveguide does little. $ka\approx 2–3$ → pattern control starts. $ka >5$ → strong beam. For constant directivity you want $ka_{throat}$ small enough that driver doesn't beam before waveguide takes over, and mouth $ka$ large enough to hold pattern to lowest desired frequency: $f_{min}\approx c/(2\cdot mouth\_diameter)$. That is why HF beaming shows as ___ in this canvas? The tool visual scales 1:1 so you can read mouth vs throat ratio immediately.

Beamwidth $\approx 2\arcsin\left(\frac{0.707\lambda}{2a}\right)$ for rigid disk, but actual horn beam $\approx 2\theta$ conical above $ka\approx1$.

4 — Playbook: choosing OSWG vs conical vs exponential

• Studio monitor CD: Use oblate spheroidal. Set throat to driver diameter (25.4 mm for 1"), length 150–250 mm, half-angle 40–50°. Mouth ends $\approx 180–220$ mm gives pattern control to ~1.2 kHz.
• PA long throw: Conical with 30° half-angle gives 60° nominal (−6 dB) beam that stays ~60° from 2 kHz to 16 kHz. Pay with LF loading — add separate mid horn below.
• Bass horn: Exponential wins. Choose $m$ from desired $f_c$ via $f_c=m c/(4\pi)$. Example: $f_c=80$ Hz → $m=4\pi f_c /c \approx 2.93$ $m^{-1}$. Then length sets mouth size: $S_{mouth}=S_0 e^{mL}$.
• HOM check: If flare changes slope abruptly (throat kink), you create higher-order modes that bounce and create time smear. OSWG curvature continuity fixes this — watch profile in canvas: smooth second derivative = good.