Physics 9702/51 — October/November 2021

Background Concept

A stationary (standing) wave forms when waves reflect back and forth in a tube and superpose. In an open tube, the air at each open end can move freely, so each open end is a displacement antinode (and approximately a pressure node). For a given mode, the tube length $L$ fixes the wavelength pattern, and the resonant frequency is linked to the wave speed by

$$v = f\lambda.$$

The question proposes an empirical relationship between frequency $f$ and tube diameter $d$:

$$\frac{v}{f} = 2L + kd.$$

This has the form “wavelength-like quantity” on the left (since $v/f = \lambda$) equals a term involving the tube length plus a correction proportional to diameter. The aim of the experiment is to see if the data follow this straight-line form and to determine the constants $k$ and $v$ from measurements.

Understanding the Question

You have multiple cylindrical tubes available (different diameters). The tube is open at both ends and you must excite the stationary wave pattern shown (antinodes at both ends and a node at the centre: the fundamental mode for an open-open tube).

You must:

• design a practical set-up to find the resonant frequency $f$ for each tube,
• measure $d$ (and ensure/measure $L$),
• control other variables so any change in $f$ can be attributed to $d$,
• analyse results in a way that lets you find numerical values for $k$ and $v$.

Approach

1. Choose variables: make $d$ the independent variable (pick different tubes), measure the dependent variable $f$ at resonance for the same mode, and keep $L$ (and temperature) constant.
2. Collect data: for each tube, measure $d$ accurately with vernier calipers and determine the resonant frequency by sweeping the driving frequency and identifying a maximum response.
3. Linearise: because $v$ is unknown, avoid plotting $v/f$. Rearrange to a straight-line form that uses measurable quantities only:

$$\frac{1}{f} = \frac{2L}{v} + \frac{k}{v} d.$$

Then a plot of $1/f$ against $d$ should be a straight line. The intercept gives $2L/v$ and the gradient gives $k/v$, so you can find $v$ then $k$.

Step-by-Step Reasoning

1. Apparatus choice

• A signal generator provides a variable frequency electrical signal.
• A loudspeaker converts this to a sound wave incident on the tube.
• A microphone + oscilloscope / sound level meter measures sound amplitude, so resonances appear as clear maxima.
• Vernier calipers measure internal diameter $d$; a metre rule measures $L$.

1. Ensuring the correct resonance mode
   The pattern shown corresponds to the fundamental for an open-open tube (one node near the centre). In practice, the fundamental is usually the lowest-frequency strong resonance. You must use the same mode for all tubes, otherwise the relationship between $f$ and geometry changes.

2. Measurements for each tube

• Measure $d$ using vernier calipers. Because tubes may not be perfectly circular, take two readings at $90^\circ$ and average.
• Measure $L$ (or pre-select/cut tubes so $L$ is the same for all). If $L$ varies, the change in $f$ would be due to both $L$ and $d$, so you would not be testing the suggested relationship cleanly.
• Sweep frequency slowly; record the frequency $f$ at maximum microphone amplitude (resonance). Repeat to reduce random error and average.

1. Control variables (why they matter)

• Length $L$ constant: the formula contains $2L$, so changing $L$ would shift all results.
• Temperature constant: the speed of sound $v$ depends on temperature; if temperature drifts, the whole graph shifts, corrupting both gradient and intercept.
• Same end conditions and positions: keep both ends unobstructed and keep speaker/microphone positions fixed so the coupling to the tube is comparable.
• Same mode: ensures you are comparing like with like.

1. Linearisation and graph
   Start with

$$\frac{v}{f} = 2L + kd.$$

Divide by $v$:

$$\frac{1}{f} = \frac{2L}{v} + \frac{k}{v}d.$$

This matches the straight-line form $y = c + mx$ if you let:

• $y = 1/f$,
• $x = d$,
• intercept $c = 2L/v$,
• gradient $m = k/v$.

So:

• Plot $1/f$ (y-axis) against $d$ (x-axis).
• If the points lie close to a straight line, the relationship is supported.

1. Finding $v$ and $k$
   From the intercept $c$:

$$v = \frac{2L}{c}.$$

Then from the gradient $m$:

$$k = mv.$$

(Units check: $1/f$ has units of s and $d$ has units m, so gradient has units $\text{s m}^{-1}$. Multiplying by $v$ ($\text{m s}^{-1}$) gives dimensionless $k$, consistent with $2L + kd$ being a length.)

1. Safety

• Avoid high sound levels and prolonged exposure (hearing risk).
• Secure tubes so they do not roll/fall; manage cables to reduce trip hazards.

Key Takeaways

• A Paper 5 planning answer must clearly state variables, a workable method, controls, and a graph-based analysis.
• When a constant (here $v$) is unknown, rearrange the equation into a straight-line form using only measurable quantities.
• Gradient and intercept can be used to determine multiple constants when the linear form is chosen well.

Common Mistakes

• Allowing $L$ to vary between tubes (then changes in $f$ are not solely due to $d$).
• Not stating how resonance is identified (must mention scanning frequency and using a maximum microphone signal).
• Using different harmonics for different tubes (mixes modes and ruins the comparison).
• Choosing a graph that needs $v$ (e.g. plotting $v/f$ without knowing $v$).
• Omitting how $k$ and $v$ are obtained from the graph (must link gradient/intercept to constants).

Things to Be Careful About

• Measure internal diameter $d$ (not external).
• Keep ends truly open (do not block with the microphone; hold it just outside the tube).
• Use enough different diameters to make a convincing straight line (at least 6) and repeat $f$ readings.
• Keep temperature steady or record it; large temperature change changes $v$ and alters results.
• Use consistent units (convert $d$ and $L$ to m if calculating $v$ in $\text{m s}^{-1}$).