Near-Field Measurement Limits

A near-field measurement, the microphone a few millimetres from the cone or the port, is only valid up to a frequency, \(f_\text{max}\), set by the size of the radiator. This page explains where that ceiling comes from, and why the near-field / far-field splice has to hand over to the far-field measurement above it. It is not needed to use the wizard.

The near-field technique in one line

The near-field method (Keele, 1974) measures the sound pressure a few millimetres in front of the diaphragm and treats it as a scaled copy of the far-field on-axis response. That works because, for an acoustically small source, the near-field pressure and the far-field pressure differ only by a frequency-independent constant.

For a rigid circular piston of radius \(a\) in an infinite baffle, the exact on-axis pressure at distance \(d\) is

$$ |p(d)| = 2 \rho c U_0 \left|\sin\left[\frac{k}{2}\left(\sqrt{d^2+a^2}-d\right)\right]\right| ,\qquad k=\frac{2\pi f}{c} $$

with \(U_0\) the piston surface velocity. Two limits:

• Near (\(d \to 0\)): \(|p_\text{near}| \approx 2 \rho c U_0 \sin(ka/2)\).
• Far (\(d = r \gg a\)): \(|p_\text{far}| \approx \rho c U_0 \dfrac{k a^2}{2r}\) — the ordinary \(p_\text{far} = \dfrac{j\omega\rho Q}{2\pi r}\) of a baffled source of volume velocity \(Q = U_0\pi a^2\).

Their ratio is

$$ \frac{|p_\text{near}|}{|p_\text{far}|} \approx \frac{2 \rho c U_0 \sin(ka/2)}{\rho c U_0 k a^2/(2r)} \xrightarrow{ka\ll 1} \frac{2r}{a} . $$

The \(\sin(ka/2)\to ka/2\) small-angle step is the whole game: it makes the rising \(\propto f\) trend cancel between numerator and denominator, leaving a constant \(2r/a\). Measure near, multiply by that constant, and you have the far-field magnitude response — as long as \(ka \lesssim 1\). LinFIR uses

$$ f_\text{max} = \frac{c}{2\pi a} = \frac{c}{2\sqrt{\pi S_d}} \qquad\Longleftrightarrow\qquad ka = 1 . $$

Above it, \(\sin(ka/2)\) leaves its linear region, the near-field pressure grows its own interference ripple, and the far field starts developing directivity — so the ratio is no longer a constant and the method breaks.

The reactive (evanescent) field

Why is a small source "special"? Decompose the diaphragm's surface velocity into spatial Fourier components with transverse wavenumber \(k_\parallel\). Each component launches a plane wave at angle \(\sin\theta = k_\parallel/k\):

• \(k_\parallel < k\): a real angle exists — the component radiates.
• \(k_\parallel > k\): no real angle — the field is evanescent, decaying away from the surface as \(e^{-\gamma z}\) with \(\gamma = \sqrt{k_\parallel^{2}-k^{2}}\).

A feature of transverse size \(\ell\) carries \(k_\parallel \sim \pi/\ell\), so its evanescent field is gone within \(z \sim \ell\). The edge of the piston — where the velocity jumps from \(U_0\) to \(0\) — and any cone break-up inject large \(k_\parallel\); all of that is evanescent unless the feature is bigger than a wavelength.

When \(ka \ll 1\) the whole radiator is smaller than \(\lambda\): essentially none of its spatial content satisfies \(k_\parallel < k\), so the field right at the surface is dominated by the reactive/evanescent part — air being pushed sideways between regions of the cone and springing back, storing kinetic energy without radiating it. This shows up in the piston's radiation impedance \(Z_\text{rad} = \rho c[R_1(2ka) + j X_1(2ka)]\):

$$ R_1(2ka) \approx \frac{(ka)^2}{2}, \qquad X_1(2ka) \approx \frac{8 ka}{3\pi}, \qquad \frac{X_1}{R_1} \approx \frac{16}{3\pi ka}. $$

For \(ka \ll 1\) the reactance swamps the resistance: the source is a poor radiator that mostly sloshes air around locally. The two become comparable — the radiation-impedance "knee" — at \(ka\) of order 1, the same place the near-field validity ends.

Crucially, in that reactive regime the on-axis near-field pressure is still the simple, position-tolerant quantity above (\(\propto U_0 \sin(ka/2)\)); the messy evanescent structure lives off-axis and in the tangential field. That is exactly why the trick works — and why it stops working once real radiation, with its own propagating interference, takes over.

Connection to directivity

Directivity is the angular shape of the radiating (\(k_\parallel < k\)) part of the field, seen in the far zone. Track it against \(ka\):

• \(ka \lesssim 1\): only the lowest-\(k_\parallel\) sliver of the source's spectrum radiates, and it looks like a monopole — the speaker is omnidirectional.
• \(ka \sim 1\text{-}3\): more of the spectrum crosses below \(k\); the pattern starts to narrow.
• \(ka \gg 1\): the pattern locks into the piston beaming law \(D(\theta) = \dfrac{2 J_1(ka\sin\theta)}{ka\sin\theta}\).

So the near-field measurement is trustworthy precisely in the band where the loudspeaker has essentially no directivity. The onset of directivity and the breakdown of the near-field proxy are the same event, both governed by \(ka \approx 1\): directivity is the structure of the radiated field, and that structure only becomes well defined once the evanescent field — which decays exponentially with distance, \(e^{-\gamma z}\), over a few source dimensions — has faded relative to the propagating field.

Why the splice is built the way it is

• Below \(f_\text{max}\): the speaker is omnidirectional and the near-field pressure is a clean, scaled copy of the far-field response. The splice takes the near-field magnitude and phase here — there is no directivity to lose.
• Above \(f_\text{max}\): the near-field reading is corrupted by its own interference and the speaker is starting to beam. The splice must use the real far-field measurement, which carries the diffraction and directivity the near field never had.
• The transition band lives in the overlap where both are still acceptable. The wizard caps it at \(f_\text{max}\) (and at the far-field floor) for that reason; the unlock toggle lets you override it when no valid overlap exists, at a known cost to accuracy.