In this section, we will first look at an instrument which is a bit of cheat. The “musical saw” is ordinarily played with a violin bow, but its behaviour when tapped gives a very illuminating insight into the physics of a particular kind of shell vibration, which we will then apply to a genuine tuned percussion instrument, the Caribbean steel pan.
The first demonstration is well worth trying for yourself. Take an ordinary carpenter’s hand-saw. Sit down comfortably, then brace the handle of the saw with your knees, grip the narrow end of the saw in one hand, and bend it so that it takes up an S-shape: see Fig. 1 for the general idea. Now with something like a pencil in your other hand, tap the saw right in the middle of the S. If you have bent it by a suitable amount, you should hear a surprisingly long-ringing sound, with a note that you can change by adjusting the severity of the S-bend. But now keep hold of the saw and let it straighten out: if you tap again you will hear only a dull thud or rattling sound. In case you can’t try this yourself, Sound 1 gives a simple recorded example: I first tap the saw without bending it, then again with the S-bend in place.
What is going on? How do we create this clear ringing note seemingly out of nowhere by curving the saw-blade like this? Figure 2 shows a laboratory test, which allows the pattern of vibration to be visualised using a holographic method. The “saw” is being held in the required S-shape by some clamps, and then it is being made to vibrate at the resonant frequency we heard in the pencil-tapping experiment, using a vibration actuator which is out of sight behind the saw in the picture. Using a cunning trick involving a laser (which is where the green colour comes from), the dark curves show a contour map of the vibration amplitude. The next link describes how it is done.
The two broad white bands running along the length of the saw are nodal lines, where the vibration goes to zero. So we can work out what is happening. If you concentrate on a strip across the width of the saw in the vibrating region, the motion is just like the lowest mode of the free-free bending beam see in Fig. 1 of section 3.2: down in the middle and up on both edges, then reversing. Motion of this kind is happening over a chunk in the middle of the saw, but then it mysteriously fades away as you move towards either end.
To understand this fading, we need to remember the discussion of corrugated sheet in section 3.2. The effect of the curvature was to make the sheet much stiffer when you bend across the corrugations, compared to bending parallel to the corrugations. Now recall that our saw is in an S-shape. It has an inflection in the middle of the S, where the curvature is zero, but as you move away from that point in either direction you encounter stronger and stronger curvature. This region around the inflection is precisely where the vibration is occurring in Fig. 2.
We can imagine setting up vibration near the inflection, with the pattern of free-free bending across the width of the saw. Now imagine this pattern of vibration trying to propagate along the saw, away from the inflection. It encounters the growing curvature, and this increases the stiffness which produces the restoring force for the bending vibration. Eventually, it can reach a point where this curvature stiffening effect gets so strong that it is no longer possible to vibrate across the width at this frequency. Our travelling vibration pattern cannot get past this point: it experiences a kind of total internal reflection, and is turned round and travels the other way. It goes back to the inflection, then out on the other side. But it meets increasing curvature again in that direction, until it reaches another point of total internal reflection. So the vibration is confined to this region around the inflection, bounded by the two points where the curvature reaches a critical value and causes reflection. That is the pattern in Fig. 2. This kind of system is known generically as a waveguide. The next link uses a waveguide system that can be analysed more readily, to illustrate the behaviour in quantitative form.
To see why the reflection from points of critical curvature accounts for what we heard in the pencil-tapping experiment, we have to think about vibration damping. A saw is made of high-carbon steel, which has very low vibration damping. But we were holding the saw blade with our fingers, and at the other end the steel blade was rivetted to a wooden handle. The handle and your fingers both give efficient ways to dissipate any vibrational energy in the saw. So when you tap it without the S-bend, you hear a dull thud or a rattle from the handle. But now bend into the S shape. A vibration mode like Fig. 2 is created, confined to the middle region of the saw blade. Now the vibrational energy can’t reach your fingers or the handle, and the sound you hear is characterised by the very low intrinsic damping of the steel. The result is the long-ringing note we heard.
This idea of vibration being confined to a region of a curved shell through the stiffening effects of curvature will allow us to understand the physics behind our next instrument, the Caribbean steel pan. Steel pans are made from mild steel, traditionally using an old oil drum. The maker shapes the top with the hammer, and creates a pattern of curvature which not only gives a musical note, but somehow allows many different notes to be played on the same piece of steel. It is a truly remarkable piece of vibration engineering.
The first step in the process of turning an oildrum into a steel pan is to “sink” the top. Mild steel is quite ductile, so by working hard enough with a hammer it is possible to turn the flat top of the drum into a bowl shape. By this stage, the ductility of the steel is likely to have been used up, though the process of strain hardening. To restore some ductility for the next step, the steel needs to be annealed, in other words heated to a suitable temperature and allowed to cool slowly. The traditional way to do this involves a bonfire containing old tyres: the carbon from the burning tyres helps with controlling the properties of the steel.
The next step is to mark out where the various notes are to be on the pan: this depends on the type and register of the particular pan being made. Now the clever part begins. Each note area is worked with the hammer, starting out on the reverse side in order to undo the bowl curvature from the sinking stage. The result is that each note is a flattish area, surrounded by curved steel in the rest of the bowl. That is the secret: it is a kind of two-dimensional version of the musical saw, and it can work in a similar way to allow some modes of vibration to be confined to single patches on the pan surface.
This confinement effect is dramatically illustrated in the set of holographic images in Fig. 4. The approximately circular area at the bottom of the pan is the note $A_3$, with nominal frequency 220 Hz. In the first picture we can see that there is indeed a confined mode close to this frequency. But there is more: another mode at roughly double that frequency, with a single node line, another at the next doubling of frequency, with two node lines, and in between a mode with a frequency close to 2.5 times the fundamental, and musically consonant with it. The response near the double-octave frequency, in the lower right pane of Fig. 4, shows activity in two other areas as well. This is no mistake: these are the notes $A_4$ and $A_5$, tuned in their own patches, sufficiently accurately that they all resonate with each other at a frequency around 880 Hz.
217 Hz 
441 Hz 
540 Hz 
886 Hz
A similar pattern is seen in Fig. 5, with a sequence of confined modes involving the $A_4$ area. But by the highest frequency illustrated here, at 1822 Hz, something different has happened. We now see vibration spread over much of the surface of the pan, rather than being confined to one area or a small number of related areas. This is exactly what we would expect from the description, and the theory presented in section 3.5.1. The bowl curvature presents a barrier to bending waves, but as you go higher in frequency and shorter in wavelength, that barrier becomes less effective. The situation in the pan is, in this respect, different from the musical saw or the idealised beam on an elastic foundation. In those problems, curvature or foundation stiffness increased inexorably with distance from the neutral point. So at any frequency there would eventually be a point where internal reflection had to occur. But in the pan, the bowl curvature is fixed. Once the wavelength is short enough to overcome that barrier, the confinement effect is lost.
424 Hz 
886 Hz 
1035 Hz 
1822 Hz
So the pan-maker has not only managed to tune one confined mode to the desired frequency in each note region, he has also tuned up to three other modes to harmonically-related frequencies. It seems remarkable that the process offers enough degrees of freedom to achieve this. The area and aspect ratio of the note region give two variables to play with, but to tune four modes in a single patch requires subtle adjustments to the detailed topography of the patch.
Based on what we have heard with the marimba and the church bells, we would expect this set of harmonically-related frequencies to allow a clear sense of musical pitch from each note. We can confirm this with a simulated example, in Sound 2. Based on the measured frequencies of a typical steel pan note, and choosing appropriate Q-factors and impact time for the excitation, the result is quite recognisably like the familiar sound of a steel band.