Measuring Waves in Fluids and Solids

What can we measure about a wave?

A wave carries a disturbance through a medium, such as water or a stretched string. The material oscillates, but the disturbance can travel along it. Watching the movement carefully allows us to measure three different quantities.

Wavelength, $\lambda$, is the distance between corresponding points on neighbouring waves. For water waves, this can be the distance from one crest to the next. Frequency, $f$, is the number of complete waves passing a fixed point each second. Wave speed, $v$, describes how quickly the disturbance travels, rather than how quickly one particle moves up and down.

During each complete oscillation, a travelling wave advances by one wavelength. This connects the three measurements:

$$v = f\lambda$$

Use frequency in hertz (Hz) and wavelength in metres (m) to obtain speed in metres per second (m/s). A frequency of 5 Hz means five complete oscillations each second.

Choosing and reading measuring apparatus

Choose an instrument whose range covers the measurement and whose scale is fine enough to distinguish the changes you are investigating. A metre rule with millimetre divisions is suitable for measuring a string or several water wavelengths. A stopwatch is suitable for timing a sequence of waves, although your reaction time limits the accuracy of manually starting and stopping it.

Place a ruler parallel to the length being measured and view its markings directly from above. Use the zero mark, not necessarily the physical end of the ruler. If that end is damaged, measure between two clear scale readings and subtract the initial reading from the final reading.

The same principles apply to other laboratory measurements:

For water in a measuring cylinder, read the bottom of the curved liquid surface at eye level once the water has settled. Choose a cylinder with a suitable capacity and closely spaced divisions rather than estimating a small volume in a large beaker. For temperature, wait until the thermometer reading is steady before recording it.

Record readings in a table with units in the column headings. Keep the detail justified by the instrument's scale. Useful conversions include $1\text{ m}=100\text{ cm}=1000\text{ mm}$ and $1\text{ mL}=1\text{ cm}^3$.

Observing water waves in a ripple tank

A ripple tank is a shallow transparent tray of water. A vibrating dipper produces repeated disturbances at the surface. A straight dipper produces approximately straight, parallel wavefronts, which are convenient for measuring wavelength. A lamp makes the wave pattern visible on a screen beneath the tank.

Set the tank level and add a shallow layer of water, no more than about 1 cm deep for this method. Measure the water depth with a ruler: total volume alone does not show whether the depth is uniform. Turn on the light and the vibrating dipper, then adjust the equipment until regular wavefronts are clearly visible.

To measure wavelength, measure a distance spanning several complete crest-to-crest intervals. Divide that distance by the number of intervals:

$$\lambda = \frac{\text{distance spanning several wavelengths}}{\text{number of wavelengths}}$$

For example, measuring from the first crest to the sixth crest includes five wavelengths. Using several intervals gives a larger total distance to measure, reducing the relative effect of uncertainty at its ends.

If measuring a projected pattern, the screen scale must correspond to distances in the water: an enlarged projection would otherwise give an enlarged wavelength. A calibrated scale or a direct measurement of the water pattern avoids this problem.

To measure frequency, choose one fixed point in the path of the travelling waves and count the complete waves passing it during a measured time:

$$f = \frac{N}{t}$$

Here, $N$ is the number of complete waves and $t$ is the time in seconds. Counting over a longer time reduces the relative effect of starting and stopping the stopwatch. This works only if the waves pass slowly enough to count reliably.

Record frequency and wavelength, calculate $v=f\lambda$, and repeat the measurements. Repeated results help you judge whether the measurements are consistent; a mean can reduce the influence of random variation.

Making a moving pattern easier to measure

Rapidly moving wavefronts are difficult to locate accurately with a ruler. A stroboscope flashes at regular intervals. When its flash frequency is correctly matched to the wave frequency, each flash shows the wave at the same stage of its motion, so the pattern appears stationary. This makes the wavelength easier to measure, and the matched frequency can be read from the stroboscope.

If investigating how wavelength changes with frequency, change the dipper frequency while keeping the water depth and temperature the same. These are control variables because changing the water conditions could change the wave speed. Under unchanged conditions, increasing frequency produces a shorter wavelength rather than necessarily a faster wave.

Keep electrical equipment and connections away from water, keep hands dry and deal with spills promptly under teacher supervision. Check for photosensitive epilepsy before using flashing lights; use a non-flashing method if necessary. Switch off the apparatus before dismantling it.

Measuring waves in a solid: a stretched string

A stretched string provides a solid medium in which waves are easy to observe. A signal generator drives a vibration generator attached to the string. The signal generator supplies an adjustable frequency, while the string is held under steady tension.

Measure the vibrating length of string with a metre rule. Slowly increase the driving frequency until a clear standing wave forms: a stable pattern with points that remain still and loops that oscillate between them. The stable pattern is the sign that the apparatus is ready for a wavelength measurement. Read the frequency from the signal generator rather than trying to count very rapid string movements with a stopwatch.

Each complete loop between neighbouring stationary points spans half a wavelength. If a vibrating length $L$ contains $n$ complete loops:

$$\lambda = \frac{2L}{n}$$

A one-loop pattern therefore has a wavelength twice the vibrating length. A two-loop pattern has a wavelength equal to that length. Count the loops, calculate wavelength and use $v=f\lambda$ to find the speed of waves on the string.

Repeat at other frequencies that produce clear standing-wave patterns. Keep the same string, vibrating length and tension. The speed depends on the properties of the string and its tension: with these unchanged, higher-frequency patterns have shorter wavelengths. Your calculated speeds should therefore be similar, allowing for measurement uncertainty.

Secure the supports and attachments before starting, keep hands clear of moving equipment and do not overstretch the string. Switch off the vibration generator before changing the arrangement.

Why apparatus suitability matters

The ripple tank makes travelling wavefronts visible, so a ruler can measure their spacing and a stopwatch can time their passage when they are slow enough. A stroboscope helps when the pattern moves too quickly to measure directly.

On a rapidly vibrating string, a standing-wave pattern turns the wavelength measurement into a measurement of a stable length and a count of loops. The signal generator provides the frequency. In both investigations, suitable apparatus makes the relevant quantity observable and measurable; the wave-speed equation then connects the measurements.