= Kundt tube spacing fit
{c}
{title2=$x_j=a+jd,\quad c=2fd$}
Write the measured positions of consecutive maxima of <sound pressure> amplitude as $x_j=a+jd+\varepsilon_j$. The arbitrary origin $a$ is fitted along with the half-<wavelength> $d$ by <linear regression>; forcing the line through zero would turn a ruler-origin offset into an erroneous slope. For independent equal-<variance> position errors of standard deviation $\sigma_x$,
$$
\widehat d=\frac{\sum_j(j-\bar j)(x_j-\bar x)}{\sum_j(j-\bar j)^2},\qquad
\sigma_d=\frac{\sigma_x}{\sqrt{\sum_j(j-\bar j)^2}}.
$$
As an original example, choose $f=1000\,\mathrm{Hz}$, $a=24\,\mathrm{mm}$ and $d=171.5\,\mathrm{mm}$. Add stipulated position errors $(1,-1,0,0,-1,1)\,\mathrm{mm}$ at $j=0,\ldots,5$. Their sum and index-weighted sum vanish, so the fitted values remain $\widehat a=24\,\mathrm{mm}$ and $\widehat d=171.5\,\mathrm{mm}$, giving \b[$c=343.0\,\mathrm{m/s}$]. These are synthetic inputs, not measurements of a particular air sample.
With stipulated $\sigma_x=1\,\mathrm{mm}$ and independent drive-frequency uncertainty $\sigma_f=1\,\mathrm{Hz}$, first-order <uncertainty propagation> gives
$$
\sum_{j=0}^{5}(j-\bar j)^2=17.5,\qquad
\sigma_d\simeq0.2390\,\mathrm{mm},\qquad
\sigma_c^2=(2f\sigma_d)^2+(2d\sigma_f)^2,
\qquad \boxed{\sigma_c\simeq0.5884\,\mathrm{m/s}}.
$$
This statistical calculation omits common scale-calibration error, thermal drift and mode misidentification. Counting more positions reduces independent localization error; it cannot correct a wrong family of extrema or an incorrectly calibrated ruler.
\Image[/kundt-tube-pressure-motion-and-spacing-fit.png]
{title=Pressure and fluid motion give shifted patterns but the same half-wavelength spacing}
{description=Original lossless mode and synthetic regression example. Absolute spatial mode shapes show the positions of maxima; signed time traces at $kx=\pi/4$ show their relative phase. The six positions and residuals are stipulated model data. No historical dust pattern or measured trace is reproduced.}
{source=kundt-tube-pressure-motion-and-spacing-fit.py}
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