= Acoustic wavelength measurement
{title2=$\lambda=2d$}
An <acoustic wavelength measurement> makes an invisible spatial pattern readable with a ruler. The authored guide <Interesting and inexpensive experiments for high school physics by Joe Wolfe> places a loudspeaker near a pipe's open end, drives it with an oscillator, and slides a <microphone> attached to a metre rule through the pipe. An <oscilloscope> or computer displays the <microphone> <signal>. Its oscillation <amplitude> rises and falls with position; adjacent minima locate <pressure> <nodes of a standing wave> a distance $d=\lambda/2$ apart. A cap gives a <pressure> <antinode> at the closed end; an open end has a <pressure> node just outside it. These are <pressure> patterns, distinct from <displacement> patterns. The guide advises <wavelengths> longer than the pipe diameter, sufficient pipe length to locate both a node and an <antinode>, and <frequency spectrum> analysis to distinguish the intended tone from other sounds.<Interesting and inexpensive experiments for high school physics by Joe Wolfe>[pp. 10–11, Standing waves. Reflection and transmission. Speed of sound; apparatus, microphone scan, and pressure-node distinction]{ref}
The inference follows from <superposition> of two oppositely directed <acoustic plane waves>. In an ideal uniform, lossless pipe, taking equal <pressure> <amplitudes> $P$ and a rigid cap at $x=0$ gives
$$
p'(x,t)=P\cos(kx-\omega t)+P\cos(kx+\omega t)=2P\cos(kx)\cos(\omega t).
$$
The time-varying <pressure> perturbation $p'$ has local oscillation <amplitude> $2|P\cos(kx)|$. Its zeros are separated by $\pi/|k|=\lambda/2$. Unequal incident and reflected <amplitudes> leave nonzero minima, but their spacing remains $\lambda/2$ in this ideal model. A <microphone> detects that <pressure> variation through a changing electrical <signal>, rather than watching a <wave crest> move. Record the tone's <frequency> too, then infer <phase speed> $v_p=2fd$.
For proposed controls, keep the drive <frequency> and <amplitude> fixed during a scan, verify the <frequency> on the recorded <signal>, and repeat the scan in both directions. Fit several minimum positions $x_j=a+jd$ using <linear regression> to reduce dependence on one pair of positions. Check the ruler scale and use one consistent reference point on the <microphone>. Compare with the source switched off to locate background <measurement noise>; use <frequency spectrum> analysis to check for additional tones, and reduce drive if the trace ceases to resemble a <sinusoid>. Record air <temperature> and avoid strong air motion. These are suggested controls, not additional procedures or observations attributed to the guide.
The pipe must support approximately <plane waves> over the measured region: transverse structure at shorter <wavelengths>, the <microphone>'s finite size, and disturbance by the ruler can compromise the interpretation. Weak reflection gives shallow minima, while <damping> and other reflections can shift the extrema of the simple two-wave model. Measure interior spacings rather than assuming the first minimum is exactly at the pipe lip. A short scan may not include enough minima to estimate $d$ reliably, and drift during sequential scanning can imitate spatial variation.
For an explicitly hypothetical calculation, $f=1000\,\mathrm{Hz}$ and $d=0.1715\,\mathrm m$ give $\lambda=0.3430\,\mathrm m$, $K\simeq18.3183\,\mathrm{rad\,m^{-1}}$, and $v_p=343.0\,\mathrm{m\,s^{-1}}$. No such measurements are claimed here. If independent small <measurement errors> in $f$ and fitted $d$ have <standard deviations> $\sigma_f$ and $\sigma_d$, first-order <error propagation> gives $\sigma_{v_p}/v_p\simeq\sqrt{(\sigma_f/f)^2+(\sigma_d/d)^2}$. A systematic scale error or changing air conditions requires separate assessment.
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