IN07 · Air beyond the open rim

Resonance Tube & Sound Speed

Why is the loudest resonance shifted from the tube’s geometric length? Air just outside an open rim also moves, so the wave reflects as if the tube extended a little farther. Resonance therefore depends on an effective length. Comparing measured resonances at several geometric lengths separates sound speed from this end correction.

Slide the end plate, move the microphone and compare open and closed tubes. Save sweeps at several lengths to infer sound speed and end correction.

IN07 · Sound reaches beyond the rim

Slide the end, move the pickup

Drag the silver end or green pickup; the probe keeps its fractional cavity position as length changes. The speaker stays outside the open mouth.

Left boundaryAir-column length

0.423417\,\mathrm{m}

Drive frequency

1000\,\mathrm{Hz}

Microphone along the cavity

0.08

Source pressure amplitude

0.025\,\mathrm{Pa}

Live calibrated pickup

0.81\,\mathrm{Pa}

\phi=-90^\circ

Three calibrated length records are ready.

Temperature, mouth & readout errors

Air temperature in degrees CelsiusMouth radius in millimetresPickup gain factorRuler zero offset in millimetresVoltage noise boundDetector seed

After changing pickup gain or ruler zero, calibrate again before the next acquisition. Earlier records retain their own calibration.

Model assumptions

One-dimensional linear dry-air acoustics with a weak, non-obstructing microphone. Thin unflanged ends use a constant 0.6133-radius correction only within a dimensionless mouth-size parameter of at most 0.3. A chosen conservative guard excludes lengths below four radii; it is not an accuracy specification. A virtual extension and passive resistance approximate end inertia and aggregate loss; this is not full radiation CFD. Temperature sets the PASCO approximate reference speed. Humidity, mean flow, viscothermal dispersion, nonlinear motion and probe loading are omitted. Particle displacement and tube width are enlarged. Complex pickup voltage, ruler/counter indications, calibration, temperature, seed and version are frozen with each scan. Sound speed and end correction need distinct measured lengths; one fixed length only identifies their ratio. Fits use raw peak positions and do not claim confidence intervals.

Simulated microphone observations

Find peaks, then compare lengths

Inspect saved acquisition

Resolved peaks in this scan

4

Compatible saved scans

3

Sound speed from records

344\,\mathrm{m/s}

Correction per open end

6.13\,\mathrm{mm}

Peaks searched in the saved voltage samples
Peak	Frequency / length	Pressure amplitude	Assigned mode

Calibration and inference limits

L=\frac{q_n}{f_n}c-N_o\delta

Different geometric lengths separate the slope, sound speed, from the intercept, end correction. Many orders at one fixed length only determine speed divided by effective length. Frequency scans identify the fundamental within a broad speed interval; length scans reject ambiguous or missing order sequences. The displayed fit is a deterministic estimate, not a confidence interval.

Acoustic model and primary sources

One-dimensional linear acoustics, weak non-obstructing probe and a thin unflanged mouth. The low-frequency correction is 0.6133 radii per open end, used only while the dimensionless mouth-size parameter is at most 0.3. A virtual extension and a passive dimensionless resistance of 0.025 represent end inertia and aggregate loss. They are a teaching termination model, not a complete radiation calculation. A chosen conservative guard requires a length of at least four radii to avoid very short, wide cavities; this is not an accuracy specification. The left effective open end is pressure-release. Dry-air sound speed uses the PASCO temperature approximation. Viscothermal dispersion, humidity, mean flow, nonlinear sound and probe loading are omitted. Particle displacement and tube width are enlarged; all particles use one displacement gain within each frame.

ka=2\pi fa/c\leq0.3,\qquad L/a\geq4

Start with the essentials

Focus question: Why is the loudest resonance shifted from the tube’s geometric length?
One-sentence intuition: The open end carries extra air inertia; changing tube length separates that extra length from sound speed.

Core mathematical model

One closed end selects odd quarter waves

f_n=\frac{(2n-1)c}{4(L+\delta)}

The rigid end fixes air displacement; the effective open end approximately fixes pressure. Each extra mode adds half a wavelength.

Two open ends add two corrections

f_n=\frac{nc}{2(L+2\delta)}

Both thin unflanged mouths contribute the same low-frequency extension in this model.

Pressure follows compression

p=-\rho c^2\frac{\partial\xi}{\partial x},\qquad u=\frac{\partial\xi}{\partial t}

Pressure responds to the spatial compression of particle displacement. Particle velocity is the time derivative, so pressure and velocity have a different time phase from pressure and displacement.

Fit slope and intercept from measured peaks

L=\frac{q_n}{f_n}c-N_o\delta

The known boundary chooses the quarter-wave or half-wave factor. Distinct recorded lengths identify a slope and an intercept.

Common difficulties

A pressure node is not still air

Typical misconceptionA quiet microphone marks a place where nothing moves.

Better mental modelThe microphone senses pressure. At a pressure node, air can move most strongly while its compression stays small.

Run the experiment

01
Move the actual end
Drag the silver end plate slowly, keeping drive frequency fixed.
What to observe: The pressure colours and particle motion brighten near resonances. Adjacent resonant lengths differ by half a wavelength.
02
Find a quiet pickup
Choose A quiet microphone, then drag the green probe away from that point.
What to observe: A pressure node gives little microphone voltage even though nearby air moves strongly.
03
Save a frequency and length sweep
Use Save frequency sweep, then Save length sweep.
What to observe: The saved curves contain complex pickup voltage and calibrated axes. Changing the live apparatus leaves those samples unchanged.
04
Separate the two unknowns
Choose Measure three lengths after changing temperature or boundary.
What to observe: The three setups supply different lengths at the same temperature and radius. Repeated orders at one fixed length cannot supply the missing geometric information.
05
Check the pickup and scales
Change Pickup gain factor or Ruler zero offset, then Calibrate pickup & scales before another acquisition.
What to observe: Later scans use the new saved reference observations. Earlier scans retain their own calibration.

Resonance Tube & Sound Speed

Find peaks, then compare lengths

The reflection reaches beyond the rim

Core mathematical model

One closed end selects odd quarter waves

Two open ends add two corrections

Pressure follows compression

Fit slope and intercept from measured peaks

Common difficulties

A pressure node is not still air

Run the experiment

Move the actual end

Find a quiet pickup

Save a frequency and length sweep

Separate the two unknowns

Check the pickup and scales

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