Resonance Air Column Apparatus

What this figure shows. A vertical cylindrical glass tube about one metre long is drawn mounted on a stand with a ruled scale running alongside it from 0 at the bottom to 100 at the top, open at its upper end A where a tuning fork is held and vibrating just above the opening, and connected at its lower end B via a flexible rubber tube down to a water reservoir R whose height can be raised or lowered to control the water level inside the glass tube. Three successive resonance water-column lengths are marked on the scale and labelled with braces as $\lambda/4$ (the shortest, first resonance), $3\lambda/4$ (the second resonance) and $5\lambda/4$ (the third resonance), with a node (N) shown at the water surface and antinodes (A) shown near the open top for each case. The figure is the complete physical apparatus referenced throughout this subsection's derivation: it shows exactly how the same fixed tuning fork produces resonance at several different, evenly-spaced water levels as the reservoir is raised, each spaced $\lambda/2$ apart, which is the direct experimental …

Worked out. A frequency generator fixed at 343 Hz is held above a 1.0 m high tube, and water is pumped in slowly to fill it; taking the speed of sound as 343 m/s, the task is to find the minimum height of water needed to first achieve resonance. The wavelength is $\lambda=v/f=343/343=1.0\ \text{m}$, so the successive resonance air-column lengths (measured from the open top down to the water surface) are $L_1=\lambda/4=0.25\ \text{m}$, $L_2=3\lambda/4=0.75\ \text{m}$, and $L_3=5\lambda/4=1.25\ \text{m}$ -- but since the tube is only 1.0 m tall, the third (and any higher) resonance cannot physically occur inside it. So the LAST resonance obtainable is the second one, at air-column length 0.75 m, which corresponds to a water height (measured …

Worked out. A student's resonance-column experiment shows the air column resonating in its fundamental mode with a tuning fork at a length of 0.2 m, and resonating with the SAME fork's first overtone when the length is increased to 0.7 m; the task is to compute the end correction. Using the end-correction formula derived by combining the fundamental and first-overtone resonance conditions, $e=(L_2-3L_1)/2$, with $L_1=0.2\ \text{m}$ and $L_2=0.7\ \text{m}$ substituted directly, gives $e=(0.7-3\times0.2)/2=(0.7-0.6)/2=0.1/2=0.05\ \text{m}$. The result illustrates that the end correction, though small compared with the resonance lengths themselves, is not negligible and must be explicitly accounted for whenever an experiment relies on a single resonance length rathe …

Worked out. A resonance air column apparatus with an adjustable piston shows two successive resonances at column lengths 20 cm and 85 cm when driven by a tuning fork of frequency 256 Hz, and the task is to compute the speed of sound in air at room temperature. Since two successive resonances are always separated by exactly half a wavelength, the wavelength is found from $\lambda=2\Delta L=2(L_2-L_1)=2\times(85-20)\ \text{cm}=2\times65\ \text{cm}=130\ \text{cm}=1.30\ \text{m}$. The speed of sound then follows directly from $v=f\lambda=256\times1.30\approx332.8\ \text{m/s}$, a value that agrees closely with the accepted speed of sound in air near room temperature, and the calculation needs no separate knowledge of the (unknown) end correction at all, since taking the differe …