Wave-phenomena data studio

A digital record is a list of samples taken fₛ times per second for a duration T, and those two numbers bound everything you can infer from it. Sampling at fₛ can represent frequencies only up to the Nyquist limit fₛ/2; a component above it is not lost but folded to a false frequency |f − n fₛ|. At fₛ = 8.00 kHz a 5.20 kHz overtone appears at 2.80 kHz, indistinguishable from a real 2.80 kHz tone, which is why an anti-alias filter must act before sampling, never after. The duration sets the other limit: a spectrum computed from a window of length T has bins 1/T apart, so a 0.250 s window gives 4.00 Hz bins and cannot separate two tones 3 Hz apart. A peak read from its tallest bin is known to about Δf/(2√3) = 1.15 Hz, the a/√3 rule applied to a half-width of half a bin. The two limits are independent: sampling faster does nothing for resolution, and recording longer does nothing for bandwidth.

Two in-phase sources d apart produce minima wherever the path difference δ is a half-integer number of wavelengths, δ = (m + ½)λ. A microphone slid along a line at distance L records the minima positions yₘ, and the honest inversion converts each into δₘ = √(L² + (yₘ + d/2)²) − √(L² + (yₘ − d/2)²) before anything is fitted. Plot δₘ against m + ½: the slope is λ, and the intercept tests the model, because a phase offset φ₀ between the sources shifts every minimum by −φ₀λ/2π. A non-zero intercept is the evidence that the sources were not in phase. The first-year shortcut yₘ = (m + ½)λL/d is the small-angle limit, and a survey reaching y/L = 0.8 is nowhere near it: for d = 0.800 m and L = 2.000 m the m = 2 minimum of a 0.200 m wave sits at y = 1.62 m, where the shortcut returns λ = 0.260 m, 30% high. Uncertainty flows through the derivative dδ/dy = sin θ₁ − sin θ₂, which shrinks with order, so the outermost minima carry the most weight in the fit.

Two microphones recording one 1250 Hz tone give two channels, and the cross-correlation peak between them locates the lag Δt of one behind the other. The phase difference the section's formulas want is Δφ = 2π f Δt, and it inherits u(Δφ) = 2π f u(Δt). A lag of 0.160 ± 0.010 ms gives Δφ = 1.257 ± 0.079 rad, one fifth of a cycle. But a pure tone's correlation repeats every period, 0.800 ms here, so the lag is measured only modulo the period and the phase only modulo 2π. Geometry removes the ambiguity: with the microphones 0.120 m apart the true lag cannot exceed d/v = 0.350 ms, less than half a period, so the 0.160 ms peak is the only candidate. The general condition is d < λ/2, and λ = 0.274 m passes. The delay then gives the direction of arrival: sin α = vΔt/d = 0.457, so the source lies 27.2° off the perpendicular, with u(α) = (v/d) u(Δt)/cos α = 0.032 rad = 1.8°.

Two equal tones 2.67 Hz apart add to 2 cos(πΔf t) cos(2πf̄ t): a carrier at the mean frequency whose amplitude passes through zero 2.67 times a second. Nothing in the signal says whether it is beats or two lines; the analysis window does. A 0.20 s window holds 0.53 of a beat, and its spectrum shows one line at f̄ whose height changes from window to window. A 1.00 s window holds 2.67 beats, and the same signal now shows two lines 2.67 bins apart. Both pictures are correct; the second has resolved what the first only modulated. The count in time is the more precise route: N beats spanning T_count give |f₁ − f₂| = N/T_count, with an uncertainty set by two edge timings, not by any bin. What the count cannot give is the sign: 440.0 Hz beating at 2.67 Hz means a partner at 442.67 or 437.33 Hz. Lengthen the window until the lines split, or detune one source and watch whether the beat slows or quickens: either measurement decides.

A source moving past a fixed microphone leaves a spectrogram track that starts on the approach plateau fₐ = f₀v/(v − vₛ), glides through f₀ at closest approach and settles on fᵣ = f₀v/(v + vₛ). Dividing the plateaus cancels f₀: vₛ = v(fₐ − fᵣ)/(fₐ + fᵣ), and f₀ = 2fₐ fᵣ/(fₐ + fᵣ) comes free. With equal uncertainties u_f on both plateaus, propagation gives u(vₛ) ≈ v u_f/(√2 f₀), so 1.2 Hz on a 491 Hz horn costs 0.6 m s⁻¹. The glide is where the window must be chosen with care. Writing f′ = f₀v/(v − vₛ cos θ) with θ the angle between the velocity and the line to the microphone, and using dθ/dt = vₛ/b at closest approach, the received frequency there changes at df′/dt = −f₀vₛ²/(vb). For 14.2 m s⁻¹ at b = 6.0 m that is 48 Hz s⁻¹, so a 0.25 s window sees the line move 12 Hz, three bins, and smears it. The blur stays under a bin only when T_w² |df′/dt| < 1, here T_w < 0.14 s. The plateaus, 3 s out, glide at 0.1 Hz s⁻¹ and tolerate the long window.

A recorded 500 Hz line whose strength rises and falls 2.9 times a second has three candidate causes, each a different model. Beats: a second source at 500 ± 2.9 Hz, with pulsing at the same rate at every microphone position. Interference: two coherent sources set up a stationary pattern in space, and a microphone on a cart at 1.00 m s⁻¹ crossing nodes λ/2 = 0.343 m apart reports 2.92 Hz pulses that stop the instant the cart stops. Doppler: relative motion shifts the line's frequency, not its strength; a source rocking toward and away from the microphone shows a track wobbling in frequency under a constant envelope. The discriminating measurements are cheap: move the microphone and see whether the amplitude changes (interference); lengthen the window until the line splits (beats); look for a frequency glide that reverses at closest approach (Doppler). Only once the model is chosen does the recovered parameter carry meaning, and it is reported with the uncertainty the record allows, 0.29/T on a bin-read peak or the timing resolution on a count, propagated through the inversion.