Measuring waves

Quantity	Apparatus Selected	Typical Resolution	Technique to Minimise Error
Length	Metre ruler / Micrometer screw gauge / Vernier calipers	Ruler: 1 mm <br> Micrometer: 0.01 mm	Use a micrometer for very thin objects like wires. Use a fiducial marker (e.g., a pin) to mark clear start/end points.
Mass	Digital balance	0.1 g or 0.01 g	Tare (zero) the balance before placing the object to remove systematic errors. Always dry wet objects before weighing.
Time	Stopwatch / Light gates	Stopwatch: 0.01 s	Time 10 or 20 oscillations and calculate a mean to reduce the impact of human reaction time. Use light gates for high-speed motion.
Volume	Measuring cylinder / Displacement can	1 ml or 2 ml	Read the scale at eye level. For irregular solids, use a displacement can and measure the displaced water.
Temperature	Thermometer (Analogue or Digital)	1°C, 0.5°C, or 0.1°C	Stir liquids thoroughly to ensure a uniform temperature before taking the reading.
Area	1 cm² grid paper	1 cm² per square	Trace irregular objects on the grid. Count all squares that are >50% covered.

Minimising Measurement Errors

A clock that is always exactly five minutes fast will make you consistently early, demonstrating how a small flaw in an instrument shifts every single result. This type of flaw is called a zero error, which is a systematic error where a measuring instrument displays a non-zero value when the true quantity is zero. To correct this, you must apply a mathematical correction to your final readings.

A parallax error is another common systematic error caused by viewing a scale from an angle rather than straight on. To avoid this, always read scales with your eye line directly perpendicular (90°) to the measurement marking. When measuring liquids in a cylinder, always read from the bottom of the curved surface, known as the meniscus.

For timing experiments, typical human reaction time is between 0.2 s and 0.5 s. This delay supersedes the 0.01 s resolution of a stopwatch, meaning high-resolution stopwatches can give a false sense of precision for very short events.

Working with Zero Corrections

A student uses vernier calipers to measure the thickness of a metal cylinder. Before placing the cylinder in the jaws, the calipers read $0.03\text{ mm}$. The final reading with the cylinder is $14.56\text{ mm}$. Calculate the true thickness.

Step 1: Identify the zero error and the measured reading.

• Zero Error = $+0.03\text{ mm}$
• Measured Reading = $14.56\text{ mm}$

Step 2: Subtract the zero error from the measured reading.

• $14.56\text{ mm} - 0.03\text{ mm}$

Step 3: Calculate the final answer.

• True thickness = $14.53\text{ mm}$

Measuring Waves in Fluids (Ripple Tank)

If you throw a pebble into a still pond, the expanding rings of water behave exactly like the invisible light and sound waves travelling around us. We can study these properties using a ripple tank, which is a shallow glass tank filled with a shallow layer of water (less than 1 cm deep).

A motor-driven oscillating paddle (dipper) creates straight waves across the water surface. A lamp placed above the tank casts shadows of the wavefronts onto a white screen below, where light patches represent wave troughs and dark patches represent wave crests.

To measure the wavelength ($\lambda$), use a ruler to measure the total distance across 10 complete wavefronts on the screen, then divide that distance by 10. To measure the frequency ($f$), count how many complete waves pass a fixed point in 10 seconds and divide by 10.

Alternatively, frequency can be read directly from the signal generator powering the paddle. To make measuring wavelength easier, a stroboscope can be used to flash light at the exact same frequency as the waves, making the pattern appear "frozen" on the screen.

Measuring Waves in Solids (Vibrating String)

When a guitarist plucks a string, it vibrates so fast it becomes a blur, but at specific frequencies, clear, stable shapes appear along the wire. These stable shapes are called stationary waves, and they form when an incoming wave reflects off a fixed boundary and interferes with itself.

To observe this, attach a string to a vibration generator (driven by a signal generator) at one end, pass it over a wooden bridge and pulley at the other, and hang masses to provide constant tension. Turn on the signal generator and adjust the frequency until clear "loops" appear on the string.

The points on the string that do not move are called nodes, and the points of maximum vibration are called antinodes. Crucially, the length of a single loop represents only half a wavelength ($1/2 \lambda$). To find the full wavelength, you must measure the length of one loop using a metre ruler and multiply it by 2.

Calculating Wave Speed

Thunder and lightning happen simultaneously, but we calculate the speed of the storm by timing the delay because sound travels much slower than light. For any wave, whether in a ripple tank or on a string, the wave speed can be calculated if you know its frequency and wavelength.

$$v = f \times \lambda$$

Where:

• $v$ = wave speed in metres per second (m/s)
• $f$ = frequency in Hertz (Hz)
• $\lambda$ = wavelength in metres (m)

Worked Example

A student sets up a stationary wave on an elastic cord. The signal generator displays a frequency of 45 Hz. The student measures the total length of 3 vibrating loops to be 1.2 m. Calculate the speed of the wave.

Step 1: Find the length of one single loop.

• Length of 1 loop = $1.2\text{ m} / 3 = 0.4\text{ m}$

Step 2: Calculate the full wavelength ($\lambda$).

• One loop is half a wavelength, so $\lambda = 0.4\text{ m} \times 2 = 0.8\text{ m}$

Step 3: Substitute the values into the wave equation.

• $v = f \times \lambda$
• $v = 45\text{ Hz} \times 0.8\text{ m}$

Step 4: Calculate the final answer with units.

• $v = 36\text{ m/s}$