5 Primary Vibrators
5.1 Standing waves
You learned earlier that when a musician plays a note on an instrument, they supply it with energy that causes the primary vibrator to oscillate at certain specific frequencies. In Section 5 we are going to look at what determines these specific frequencies for some of the primary vibrators found in different instruments.
In Unit TA212_1 Sound for music technology: an introduction, we talk about travelling waves: that is, waves that propagate outwards away from their source, becoming weaker and weaker the further they travel. The example presented there is of a sound wave travelling away from a tuning fork.
I am now going to introduce you to standing waves. Standing waves are so called because they appear to stand still – that is, they don't appear to travel anywhere. You may be surprised to learn, therefore, that a standing wave can be made up of two travelling waves. Just like travelling waves, standing waves can be either longitudinal or transverse. However, for the purposes of this particular discussion, it is not important whether the travelling waves and the resultant standing wave are longitudinal or transverse.
Describe what is meant by:
a longitudinal wave
a transverse wave
A longitudinal wave is a wave where the displacement of the medium is in the direction of travel of the wave.
A transverse wave is a wave where the displacement of the medium is at right angles to the direction of travel of the wave.
Let us consider what happens when two travelling waves move in opposite directions. The resultant disturbance is the superposition of the forward and backward travelling waves. In other words, it is found by adding together the two waves.
Figure 6 shows two sinusoidal waves that have the same wavelength and amplitude but are travelling in opposite directions. The wave denoted by the solid line is travelling in the forwards direction (to the right); the wave denoted by the dashed line is travelling in the backwards direction (to the left). In each diagram, each wave has moved one eighth of a wavelength from the position shown in the previous diagram.
At the instant shown in Figure 6(a), the peaks and troughs of the forward and backward travelling waves coincide and the two waves are said to be exactly in phase. By adding together the two travelling waves, the resultant disturbance is found to be a sine wave that has the same wavelength but twice the amplitude of each of the travelling waves.
At the later time shown in Figure 6(b), the forward travelling wave has moved by an eighth of a wavelength to the right, and the backward travelling wave has moved a similar distance to the left. The resultant disturbance is like that of Figure 6(a) but with a smaller amplitude.
Advancing by the same amount of time again, the forward travelling wave has moved another eighth of a wavelength to the right, and the backward travelling wave another eighth of a wavelength to the left (see Figure 6(c)). Now the two waves are exactly out of phase. The peaks of the forward travelling wave coincide with the troughs of the backward travelling wave and vice versa. The resultant disturbance at this instant is zero everywhere.
Figure 6(d) shows the situation after the waves have advanced another eighth of a wavelength in the direction that they are travelling. The resultant disturbance is like that of Figure 6(b) but in the opposite sense, so that the positive displacements of Figure 6(b) are now negative, and vice versa.
By Figure 6(e), each travelling wave has moved half a wavelength (either to the left or right) from its starting position. They are now back in phase and the resultant disturbance is now as it was in Figure 6(a) but in the opposite sense.
If we kept advancing by the same time steps, we would see the resultant disturbances successively appear like those of (d), (c), (b) and (a) of Figure 6. We would then be back where we started originally and the whole cycle would start over again.
Run the Flash animation below. It should help you to see just how two travelling waves interact to form a standing wave.
Use the controls below (Play/Pause, Step, Reset) to run the flash animation.
The disturbance that results from adding together the forward and backward travelling waves is called a standing wave. Standing waves are characterised by there being locations where the displacement is always zero. These positions are called nodes (a good way to remember this is to think ‘no-deviation’ in displacement). The distance between adjacent nodes is half a wavelength. Halfway between adjacent nodes are locations where the displacement oscillates between a large positive value and a large negative value. These positions are called antinodes. The distance between a node and a neighbouring antinode is a quarter of a wavelength.
Figure 7 shows a standing wave. The solid curve shows the maximum displacement in one direction (i.e. the resultant disturbance at an instant when the two travelling waves forming the standing wave are fully in phase, as in Figure 6(a)). The dashed curve shows the displacement one half-cycle later (i.e. the resultant disturbance when the two travelling waves are next fully in phase as in Figure 6(e)). How many nodes and how many antinodes are shown in the figure?
Nodes are places where the displacement is always zero. There are four of these shown in the figure.
Antinodes are places where the displacement oscillates between a large positive value and a large negative value. They occur midway between the nodes, and there are three such places shown in the figure.
I mentioned earlier that when a note is played on a musical instrument the primary vibrator, and indeed the instrument as a whole, vibrates strongly at certain specific frequencies – the primary vibrator's natural frequencies. What is actually happening is that standing waves are being set up in the primary vibrator at these frequencies.
You should recall that the primary vibrator of a stringed instrument is the string and the primary vibrator of a wind instrument is the air column. Over the next two sections we'll see how standing waves are set up on a string and in an air column, and it will become apparent why this occurs only at certain simply related frequencies. This will lead us to discover why stringed instruments and wind instruments produce pitched notes.
After this, in Section 5.13, we shall look at standing-wave formation in the primary vibrators of some other instruments. For example, we shall look at a circular membrane (the primary vibrator in timpani and other types of drum), a circular plate (the primary vibrator of a cymbal) and a rectangular bar (the primary vibrator of the xylophone). We shall see again that standing waves are set up only at certain frequencies. However, you will learn that for these primary vibrators, the natural frequencies are not simply related. This will lead us to understand why for many percussion instruments the notes produced have a less well-defined sense of pitch.