6 Radiation
All the primary vibrators we discussed in the previous section can to some extent communicate vibrations to the surrounding air and hence radiate sound. However, some radiate sound better than others. Air columns, for example, radiate sound quite well. Even though only around 1% of the energy possessed by a vibrating air column is radiated away, this is enough to produce a clearly audible note.
Similarly, circular membranes and circular plates are also good sound radiators. They have a large surface area and so move a large amount of air when they vibrate.
A vibrating string, on the other hand, produces very little sound by itself, as it is too thin to disturb very much air. It is unable to create the large pressure differences required at the ear to produce a loud sound. So, in all instruments that have a string as the primary vibrator there must be some means of improving the efficiency of the sound radiation. This is usually done by coupling the string to one or more secondary vibrators. In the case of the piano, the string is coupled to a soundboard. In the case of the violin, it is coupled to the violin body and the air contained within the body. So, how does this help?
To help answer this question, let us first look at what happens when a tuning fork is struck. Striking a tuning fork causes its prongs to vibrate, but very little sound is produced because the prongs move back and forth through the air without particularly disturbing it. As a result, the tuning fork cannot be heard more than a few centimetres away. However, if the base of the tuning fork is pressed against a flat surface such as a table top, the sound may be heard throughout the room. The tuning fork causes the table top to vibrate and the large vibrating area of the table radiates sound much better than the fork itself (Figure 25). Clearly we can't get something for nothing and indeed we don't. The quieter sound produced by the tuning fork on its own dies away very slowly, whereas the louder sound produced when the tuning fork is in contact with the table top is damped much more rapidly.
Coupling a string to a secondary vibrator such as a piano soundboard or violin body achieves the same effect as pressing a tuning fork against a table top. The vibrations of the string force the secondary vibrator to vibrate. These vibrations are then radiated effectively to the surrounding air.
Another primary vibrator that sometimes needs the help of a secondary vibrator to improve the sound radiation is the rectangular bar. In instruments such as the xylophone and vibraphone, each bar is coupled to an air column enclosed within a cylindrical tube that is suspended below the bar. When a bar is struck, its vibrations force the air column to vibrate too. The vibrations of the air column are then radiated effectively to the surrounding air.
This all seems fairly straightforward. In some instruments, the primary vibrator enlists the help of a secondary vibrator to ensure that the sound produced by the instrument is sufficiently loud. The primary vibrator oscillates strongly at its resonance frequencies and, in turn, forces the secondary vibrator to vibrate at these frequencies. The secondary vibrator then radiates the vibrations to the surrounding air.
Hang on a minute! Won't the secondary vibrator have its own set of resonance frequencies at which it will vibrate strongly? The answer is yes. But surely we need the secondary vibrator to vibrate strongly at the frequencies at which it is being driven, i.e. the resonance frequencies of the primary vibrator!
In the case of a xylophone, this is no problem. The secondary vibrators are the air columns. The lengths of the air columns on a xylophone are chosen so that their resonance frequencies coincide with those of the xylophone bars.
But what about a piano? The soundboard is the secondary vibrator. If there were just one string attached to it, it might be possible to design the soundboard so that its resonance frequencies coincided with those of the string. However, in reality there are many strings of different lengths, tensions and masses attached to the soundboard. As each string has a different set of natural frequencies, it doesn't seem very likely that we can find a soundboard whose resonance frequencies coincide with those of all the strings. On a violin there are only four strings attached to the secondary vibrators (the violin body and contained air). However, their lengths and therefore their natural frequencies are being constantly altered by the player. So, again, it seems pretty unlikely that the violin body and air cavity could be designed to have resonance frequencies that coincide with all the possible string natural frequencies.
So, what do we want from a piano soundboard and from a violin body? Well, really what we want is for these secondary vibrators to vibrate reasonably strongly at all frequencies. In other words, we want them to have a relatively uniform response curve covering the whole frequency range used by the instrument. This is achieved by designing the soundboard or violin body in such a way that there are many overlapping broad peaks in the frequency-response curve (see Figure 26).
To sum up, in cases where the primary vibrator requires assistance to make sure that the sound produced by the instrument is sufficiently loud, secondary vibrators are employed to improve the radiation of the sound.
This improvement needs to be at the resonance frequencies of the primary vibrator. This is sometimes achieved by ensuring that the resonance frequencies of the primary and secondary vibrators coincide (e.g. the xylophone). Or, at other times, it is achieved by using a secondary vibrator that has as uniform a frequency response as possible (e.g. the piano and violin).