Imagine a string stretched between two fixed points, for example, a guitar string. Pulling the middle of the string and then releasing it produces a stationary wave. There is a node at each of the fixed ends and an antinode in the middle. Releasing the string produces two progressive waves travelling in opposite directions. These are reflected at the fixed ends. The reflected waves combine to produce the stationary wave.
Figure 14.3 shows how a stationary wave can be set up using a long spring. A stationary wave is formed whenever two progressive waves of the same amplitude and wavelength, travelling in opposite directions, superpose. Figure 14.5 uses a displacement–distance graph ( $s–x$ ) to illustrate the formation of a stationary wave along a long spring (or a stretched length of string):
- At time $t = 0$ , the progressive waves travelling to the left and right are in phase. The waves combine constructively, giving an amplitude twice that of each wave.
- After a time equal to one-quarter of a period $\left( {t = \frac{T}{4}} \right)$ , each wave has travelled a distance of one quarter of a wavelength to the left or right. Consequently, the two waves are in antiphase (phase difference $= {180^ \circ }$ ). The waves combine destructively, giving zero displacement.
- After a time equal to one-half of a period $\left( {t = \frac{T}{2}} \right)$ , the two waves are back in phase again. They once again combine constructively.
- After a time equal to three-quarters of a period $\left( {t = \frac{{3T}}{4}} \right)$ , the waves are in antiphase again. They combine destructively, with the resultant wave showing zero displacement.
- After a time equal to one whole period $(t = T)$ , the waves combine constructively. The profile of the spring is as it was at $t = 0$ .
This cycle repeats itself, with the long spring showing nodes and antinodes along its length. The separation between adjacent nodes or antinodes tells us about the progressive waves that produce the stationary wave.
A closer inspection of the graphs in Figure 14.5 shows that the separation between adjacent nodes or antinodes is related to the wavelength λ of the progressive wave. The important conclusions are:
- separation between two adjacent nodes (or between two adjacent antinodes) = $= \frac{\lambda }{4}$
- separation between adjacent node and antinode $= \frac{\lambda }{2}$
The wavelength $\lambda$ of any progressive wave can be determined from the separation between neighbouring nodes or antinodes of the resulting stationary wave pattern.
(This separation is $\frac{\lambda }{2}$ .) This can then be used to determine either the speed v of the progressive wave or its frequency f by using the wave equation:

$v = f\lambda$

KEY EQUATION

$v = f\lambda$

The wave equation, where v is the speed of the wave, f is the frequency and $\lambda$ is the wavelength.

**Figure 14.5:** The blue-coloured wave is moving to the left and the red-coloured wave to the right. The
principle of superposition of waves is used to determine the resultant displacement. The profile of the
long spring is shown in green

It is worth noting that a stationary wave does not travel and therefore has no speed. It does not transfer energy between two points like a progressive wave. Table 14.1 shows some of the key features of a progressive wave and its stationary wave.

**Table 14.1:** A summary of progressive and stationary waves.
	Progressive wave	Stationary wave
wavelength	$\lambda$	$\lambda$
	f	f
speed	v	zero

1) A stationary (standing) wave is set up on a vibrating spring. Adjacent nodes are separated by $25 cm$ .
Calculate:
a: the wavelength of the progressive wave
b: the distance from a node to an adjacent antinode.

Observing stationary waves

Here we look at experimental arrangements for observing stationary waves, for mechanical waves on strings, microwaves and sound waves in air columns.

Stretched strings: Melde’s experiment

A string is attached at one end to a vibration generator, driven by a signal generator (Figure 14.6). The other end hangs over a pulley and weights maintain the tension in the string. When the signal generator is switched on, the string vibrates with small amplitude. Larger amplitude stationary waves can be produced by adjusting the frequency.

**Figure 14.6:** Melde’s experiment for investigating stationary waves on a string

The pulley end of the string cannot vibrate; this is a node. Similarly, the end attached to the vibrator can only move a small amount, and this is also a node. As the frequency is increased, it is possible to observe one loop (one antinode), two loops, three loops and more. Figure 14.7 shows a vibrating string where the frequency of the vibrator has been set to produce two loops.
A flashing stroboscope is useful to reveal the motion of the string at these frequencies, which look blurred to the eye. The frequency of vibration is set so that there are two loops along the string; the frequency of the stroboscope is set so that it almost matches that of the vibrations. Now we can see the string moving ‘in slow motion’, and it is easy to see the opposite movements of the two adjacent loops.

**Figure 14.7:** When a stationary wave is established, one half of the string moves upwards as the other half moves downwards. In this photograph, the string is moving too fast to observe the effect

This experiment is known as Melde’s experiment, and it can be extended to investigate the effect of changing the length of the string, the tension in the string and the thickness of the string.

Microwaves

Start by directing the microwave transmitter at a metal plate, which reflects the microwaves back towards the source (Figure 14.8). Move the probe receiver around in the space between the transmitter and the reflector and you will observe positions of high and low intensity. This is because a stationary wave is set up between the transmitter and the sheet; the positions of high and low intensity are the antinodes and nodes, respectively.

reflecting sheet / prode / microwave transmitter / meter — **Figure 14.8:** A stationary wave is created when microwaves are reflected from the metal sheet

If the probe is moved along the direct line from the transmitter to the plate, the wavelength of the microwaves can be determined from the distance between the nodes. Knowing that microwaves travel at the speed of light c ( $3 \times 0 \times {10^8}\,m\,{s^{ - 1}}$ ), we can then determine their frequency f using the wave equation:

$c = f\lambda$

An air column closed at one end

A glass tube (open at both ends) is clamped so that one end dips into a cylinder of water. By adjusting its height in the clamp, you can change the length of the column of air in the tube (Figure 14.9). When you hold a vibrating tuning fork above the open end, the air column may be forced to vibrate and the note of the tuning fork sounds much louder. This is an example of a phenomenon called resonance. The experiment described here is known as the resonance tube.

**Figure 14.9:** A stationary wave is created in the air in the tube when the length of the air column is adjusted to the correct length

For resonance to occur, the length of the air column must be just right. The air at the bottom of the tube is unable to vibrate, so this point must be a node. The air at the open end of the tube can vibrate most freely, so this is an antinode. Hence, the length of the air column must be one-quarter of a wavelength (Figure 14.10a). (Alternatively, the length of the air column could be set to equal threequarters of a wavelength – see Figure 14.10b.)
Take care! The representation of stationary sound waves can be misleading. Remember that a sound wave is a longitudinal wave, but the diagram we draw is more like a transverse wave. Figure 14.11a shows how we normally represent a stationary sound wave, while Figure 14.11b shows the direction of vibration of the particles along the wave.

Open-ended air columns

The air in a tube that is open at both ends will vibrate in a similar way to that in a closed column. Take an open-ended tube and blow gently across the top. You should hear a note whose pitch depends on the length of the tube. Now cover the bottom of the tube with the palm of your hand and repeat the process. The pitch of the note now produced will be about an octave lower than the previous note, which means that the frequency is approximately half of the original frequency.

**Figure 14.10:** Stationary wave patterns for air in a tube with one end closed.
b	a

It is rather surprising that a stationary wave can be set up in an open column of air in this way. What is going on? Figure 14.12 compares the situation for open and closed tubes. An open-ended tube has two open ends, so there must be an antinode at each end. There is a node at the midpoint.
For a tube of length l you can see that in the closed tube the stationary wave formed is one-quarter of a wavelength, so the wavelength is 4l, whereas in the open tube it is half a wavelength, giving a wavelength of 2l. Closing one end of the tube thus doubles the wavelength of the note and so the frequency halves.

**Figure 14.11:** a The standard representation of a stationary sound wave may suggest that it is a transverse wave. b A sound wave is really a longitudinal wave, so that the particles vibrate as shown.
a	b

**Figure 14.12:** Stationary wave patterns for sound waves in a a closed tube, and b an open tube

2) Look at the stationary (standing) wave on the string in Figure 14.7. The length of the vibrating section of the string is $60 cm$ .
a: Determine the wavelength of the progressive wave and the separation of the two neighbouring antinodes.
The frequency of vibration is increased until a stationary wave with three antinodes appears on the string.
b: Sketch a stationary wave pattern to illustrate the appearance of the string.
c: Calculate the wavelength of the progressive wave on this string.

3) a: Sketch a stationary wave pattern for the microwave experiment in Practical Activity 14.1. Clearly show whether there is a node or an antinode at the reflecting sheet.
b: The separation of two adjacent points of high intensity is found to be $14 mm$ . Calculate the wavelength and frequency of the microwaves.

4) Explain how two sets of identical but oppositely travelling waves are established in the microwave and air column experiments described in Practical Activity 14.1.

Stationary waves and musical instruments (extension)

The production of different notes by musical instruments often depends on the creation of stationary waves (Figure 14.13). For a stringed instrument, such as a guitar, the two ends of a string are fixed, so nodes must be established at these points. When the string is plucked half-way along its length, it vibrates with an antinode at its midpoint. This is known as the fundamental mode of vibration of the string. The fundamental frequency is the minimum frequency of a stationary wave for a given system or arrangement.

**Figure 14.13:** When a guitar string is plucked, the vibrations of the strings continue for some time afterwards. Here, you can clearly see a node close to the end of each string

Similarly, the air column inside a wind instrument is caused to vibrate by blowing, and the note that is heard depends on a stationary wave being established. By changing the length of the air column, as in a trombone, the note can be changed. Alternatively, holes can be uncovered so that the air can vibrate more freely, giving a different pattern of nodes and antinodes.
In practice, the sounds that are produced are made up of several different stationary waves having different patterns of nodes and antinodes. For example, a guitar string may vibrate with two antinodes along its length. This gives a note having twice the frequency of the fundamental, and is described as a harmonic of the fundamental. The musician’s skill is in stimulating the string or air column to produce a desired mixture of frequencies.
The frequency of a harmonic is always a multiple of the fundamental frequency. The diagrams show some of the modes of vibration of a fixed length of string (Figure 14.14) and an air column in a tube of a given length that is closed at one end (Figure 14.15).