Xmax

Short answer: treat the driver as a baffled piston and convert its maximum displacement into a maximum volume-velocity; use the monopole far-field formula to turn that into RMS sound pressure, then convert to dB. Below are the exact formulas, a quick example, and the usual caveats.

Formula (on-axis, far field, 1 m reference)

Let

• $S_d$ = piston area (m²)
• $X_{\max }$ = one-sided peak excursion (m)
• $f$ = frequency (Hz), $\omega =2\pi f(rad/s)$
• $r$ = distance (m) (use $r=1$ m for the usual 1 m SPL reference)
• ${\rho }_0$ = air density ≈ 1.21 kg/m³
• $p_{ref}=20\times 10^{-6}Pa$ (SPL at 0 dB)

Volume-velocity (RMS) for a sinusoid:

$$Q_{\text{rms}}=S_d\cdot \frac{\omega X_{\max }}{\sqrt{2}}$$

Far-field RMS pressure (monopole approximation for a small baffled piston):

$p_{\text{rms}}=\frac{{\rho }_0\omega Q_{\text{rms}}}{4\pi r}=\frac{{\rho }_0{S}_d{\omega }^2{X}_{\max }}{4\pi r\sqrt{2}}$

Maximum SPL limited by excursion (RMS → dB):

$${\mathrm{SPL}}_{\text{excursion}}(f)=20{\log }_{10}\left(\frac{p_{\text{rms}}}{p_{ref}}\right)=20{\log }_{10}\left(\frac{{\rho }_0{S}_d{\omega }^2{X}_{\max }}{4\pi r\sqrt{2}{p}_{ref}}\right)=20lo{g}_{10}(\frac{{\rho }_0\pi S_d{f}^2{X}_{\max }}{r\sqrt{2}{p}_{ref}})=20lo{g}_{10}(134397.208879\cdot S_d{f}^2{X}_{max})$$

If you want peak dB (not RMS) add +3 dB to the RMS value.

Inverse - what Xmax you need for a target SPL at frequency f

Solve for $X_{\max }$:

$$X_{\max }=\frac{4\pi r\sqrt{2}{p}_{ref}}{{\rho }_0{S}_d{\omega }^2}\times 10^{\mathrm{SPL}/20}$$

Quick numeric example

Take $S_d=0.03{\mathrm{m}}^2$ (≈ 8″ driver), $X_{\max }=6\mathrm{mm}=0.006\mathrm{m}$, $f=50\mathrm{Hz}$, $r=1\mathrm{m}$:
Using the formulas above gives $p_{\text{rms}}\approx 1.21\mathrm{Pa}$ and
$\mathrm{SPL}\approx 95.6\mathrm{dB}$ (RMS, 1 m). Peak SPL ≈ 98.6 dB.

Important caveats (don’t ignore these)

• This is a theoretical excursion limit only. It ignores electrical power limits, thermal power handling, voice-coil nonlinearity, suspension nonlinearity, and driver resonance behavior. In practice you’ll hit other limits earlier.

• The mono far-field/piston approximation is valid when the driver behaves like a piston (low frequency compared to breakup, and the driver is mounted on a large baffle or in an enclosure). At higher frequencies directivity and piston breakup change the result.

• Room/woofer placement, enclosure type (sealed, ported), and acoustic loading change the SPL for a given displacement; a ported box can give much higher SPL near tuning frequency for the same cone excursion.

• The formula assumes on-axis and free-field; off-axis or room reflections change measured SPL.

• Use RMS SPL for comparisons to standards; humans perceive peak bursts differently.