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Strings, Standing Waves and Harmonics

Joe Wolfe / J.Wolfe@unsw.edu.au.

 

Introduction: vibrations, strings, pipes, percussion....

How do we make musical sounds? To make a sound , we need something that vibrates. If we want to make musical notes you usually need the vibration to have an almost constant frequency: that means stable pitch. We also want a frequency that can be easily controlled by the player. In electronic instruments this is done with electric circuits or with clocks and memories. In non-electronic instruments, the stable, controlled vibration is produced by a standing wave. Here we discuss the way strings work. This also a good introduction for studying wind instruments, because vibrating strings are easier to visualise than the vibration of the air in wind instruments. Both are less complicated than the vibrations of the bars and skins of the percussion family.

Travelling waves in strings

The strings in the violin, piano and so on are stretched tightly and vibrate so fast that it is impossible to see what is going on. If you can find a long spring (a toy known as a 'slinky' works well) or several metres of flexible rubber hose you can try a few fun experiments which will make it easy to understand how strings work. (Soft rubber is good for this, garden hoses are not really flexible enough.) First hold or clamp one end and then, holding the other end still in one hand, stretch it a little (not too much, a little sag won't hurt). Now pull it aside with the other hand to make a kink, and then let it go. (This, in slow motion, is what happens when you pluck a string.) You will probably see that the kink travels down the "string", and then it comes back to you. It will suddenly tug your hand sideways but, if you are holding it firmly, it will reflect again.

First you will notice that the speed of the wave in the string increases if you stretch it more tightly. This is useful for tuning instruments - but we're getting ahead of ourselves. It also depends on the "weight" of the string - it travels more slowly in a thick, heavy string than in a light string of the same length under the same tension.

Next let's have a close look at the reflection at the fixed end. You'll notice that if you initially pull the string to the left, the kink that travels away from you is to the left, but that it comes back as a kink to the right - the reflection is inverted. This effect is important not only in string instruments, but in winds and percussion as well. When a wave encounters a boundary with something that won't move or change (or that doesn't change easily), the reflection is inverted.

Plucked strings

If you pluck one of the string on a guitar or bass, you are doing something similar, although here the string is fixed at both ends. You pull the string out at one point, then release it as shown. The motion that follows is interesting, but complicated. The initial motion is shown below. However, the high frequency components of the motion (the sharp bends in the string) quickly disappear -- which is why the sound of a guitar note becomes more mellow a second or more after you pluck it.

 

A sketch of the reflection of travelling kinks caused by plucking a string. At the instants represented by (f) and (m), the string is straight so it has lost the potential energy associated with pulling it sideways, but it has a maximum kinetic energy. Note that, at the reflections, the phase of the kink is changed by 180°: from up to down or vice versa. Notice also how the kinks 'pass through' each other when they meet in the middle.

Why is the reflection inverted? Well, if we assume that it is clamped or tied to a fixed object, the point of reflection didn't actually move. But look at the motion of the string by comparing the different times represented in the left hand sketches. Note that the string behind the kink is moving back towards the undisturbed position (down in the sketch). As the kink approaches the end, it becomes smaller and, when it reaches the immovable end, there is no kink at all - the string is straight for an instant. But the string still has its downwards momentum, and that carries it past the position of rest, and produces a kink on the other side, which then moves back in the other direction.

As mentioned above, this motion is only observed immediately after the pluck. As the high frequency components lose energy, the sharp kinks disappear and the shape gradually approaches that of the fundamental mode of vibraiton, which we discuss below.

A bowed string behaves rather differently

First, it has a continuous source of energy, and so can maintain the same motion indefinitely (or at least until one runs out of bow. Second, the string shape required to match the uniformly moving bow is different.

A sketch of the reflection of travelling kinks caused by bowing a string. See the animation and an explanation of the bow-string interaction in Bows and strings

Travelling waves and standing waves

An interesting effect occurs if you try to send a simple wave along the string by repeatedly waving one end up and down. If you have found a suitable spring or rubber hose, try it out. Otherwise, look at these diagrams.

The figure at right is the same diagram represented as a time sequence - time increases from top to bottom. You could think of it as representing a series of photographs of the waves, taken very quickly. The red wave is what we would actually see in a such photographs.

Suppose that the right hand limit is an immoveable wall. As discussed above, the wave is inverted on reflection so, in each "photograph", the blue plus green adds up to zero on the right hand boundary. The reflected (green) wave has the same frequency and amplitude but is travelling in the opposite direction.

At the fixed end they add to give no motion - zero displacement: after all it is this condition of immobility which causes the inverted reflection. But if you look at the red line in the animation or the diagram (the sum of the two waves) you'll see that there are other points where the string never moves! They occur half a wavelength apart. These motionless points are called nodes of the vibration, and they play an important role in nearly all of the instrument families. Halfway between the nodes are antinodes: points of maximum motion. But note that these peaks are not travelling along the string: the combination of two waves travelling in opposite directions produces a standing wave.

This is shown in the figure. Note the positions (nodes) where the two travelling waves always cancel out, and the others (antinodes) where they add to give an oscillation with maximum amplitude.

You could think of this diagram as a representation of the fifth harmonic on a string whose length is the width of the diagram. This brings us to the next topic.

Harmonics and modes

The string on a musical instrument is (almost) fixed at both ends, so any vibration of the string must have nodes at each end. Now that limits the possible vibrations. For instance the string with length L could have a standing wave with wavelength twice as long as the string (wavelength = 2L) as shown in the first sketch in the next series. This gives a node at either end and an antinode in the middle.

This is one of the modes of vibration of the string ("mode of vibration" just means style or way of vibrating). What other modes are allowed on a string fixed at both ends? Several standing waves are shown in the next sketch.

 

A sketch of the first four modes of vibration of an idealised* stretched string with a fixed length. The vertical axis has been exaggerated.

Let's work out the relationships among the frequencies of these modes. For a wave, the frequency is the ratio of the speed of the wave to the length of the wave: f = v/wavelength. Compared to the string length L, you can see that these waves have lengths 2L, L, 2L/3, L/2. We could write this as 2L/n, where n is the number of the harmonic.

The fundamental or first mode has frequency f₁ = v/wavelength = v/2L,
The second harmonic has frequency f₂ = v/l₂ = 2v/2L = 2f₁
The third harmonic has frequency f₃ = v/l₃ = 3v/2L = 3f₁,
The fourth harmonic has frequency f₄ = v/l₄ = 4v/2L = 4f₁, and, to generalise,

The n^th harmonic has frequency f_n = v/l_n = nv/2L = nf₁.

All waves in a string travel with the same speed, so these waves with different wavelengths have different frequencies as shown. The mode with the lowest frequency (f₁) is called the fundamental. Note that the nth mode has frequency n times that of the fundamental. All of the modes (and the sounds they produce) are called the harmonics of the string. The frequencies f, 2f, 3f, 4f etc are called the harmonic series. This series will be familiar to most musicians, particularly to buglers and players of natural horns. If for example the fundamental is the note C3 or viola C (a frequency of 131 Hz), then the harmonics would have the pitches shown in the next figure. These pitches have been approximated to the nearest quarter tone. The octaves are exactly octaves, but all other intervals are slightly different from the intervals in the equal tempered scale.

The figure shows the musical notation for the first twelve harmonics on a C string. When you play the sound file, listen carefully to the pitch. The seventh and eleventh harmonics fall almost halfway between notes on the equal tempered scale, and so have been notated with half sharps.

You can produce these pitches on a stretched string: it's easiest on the low strings of a guitar, cello or bass*. Touch the string lightly at a point 1/n of its length from the end (where n is 1, 2, 3 etc), then bow the string close to the end. Alternatively, touch the string very lightly at a point 1/n of its length from the end, pluck the string close to the end and release the first finger as soon as you have plucked. Touching the string produces a node where you touch, and so you excite (mainly) the mode which has a node there. You will find that you can play bugle tunes using harmonics two to six of a string.

(* If you have just done this experiment, you may have noticed some peculiarities. The twelfth fret, which is used to produce the octave, is less than half way along the length of the string, and so the position where you touch the string to produce the 2nd harmonic is not directly above the octave fret. I said "idealised" string above, meaning a string that is completely flexible and so can bend easily at either end. In practice, strings have a finite bending stiffness and so their effective length (the "L" that should be used in the above formulae) is a little less than their physical length. This is one of the reasons why larger strings usually have a winding over a thin core, and why the G string on a classical guitar has poor tuning on the higher frets. There is also an effect due to the extra stretching of a string when it is pushed down to the fingerboard, an effect which is considerable on steel strings.)

An exercise for guitarists. On a guitar tuned in the usual way, the B string and high E string are approximately tuned to the 3rd and 4th harmonics of the low E string. If you pluck the low E string anywhere except one third of the way along, the B string should start to vibrate, driven by the vibrations in the bridge from the harmonic of the first string. If you pluck the low E string anywhere except one quarter of the way along, the top E string should be driven similarly.

Harmonic tuning on guitars