Acoustics of bell plates

Bell plates (sold commercially as Belleplates) are metal plates that, when struck, ring with a strong initial transient, followed by a nearly pure decaying sinusoidal tone. They are played like handbells but are rather cheaper and less loud. One might say that their sound is sweeter but less interesting than that of handbells.  The photo shows three plates: the first two are in the standard shape used for bell plates, the third is an equilateral triangle with a tang at one corner. The beaters used to make the sound files below are shown: a soft fabric beater, a soft rubber mallet and a hard rubber mallet. These three shapes all play well, that is to say they all ring for tens of seconds when struck. However, as we shall see below, only a limited class of shapes works for bell plates: in general, polygons with a handle at one corner just go 'clunk'.

Larger bell plate, standard shape soft mallet fabric beater hard mallet	Smaller bell plate, standard shape Smaller bell plate	Equilateral bell plate with second partial without second partial

The shape of bell plates is important to the sound. We demonstrate this by progressively slicing strips off the end of a plate, recording the sound for each shape. We then slice vertical strips off the corners until we regain the initial shape. The soft rubber mallet is used in all sound files.

The orginal bell plate shape: it rings well	5 mm removed: the pitch has risen, but it now rings for a shorter time	10 mm removed: higher pitch and shorter ring	15 mm removed
20 mm removed	25 mm removed: the ring is so short that it is hard to identify the pitch	30 mm removed	35 mm removed: the ring is not only short but very soft.

A slice off both corners: a slight increase in pitch	Second slice from corners	Third slice from corners
Here we're back to the original proportions	Fifth slice from corners	Sixth slice from corners

So why is the shape critical?

Consider the two simplest modes of vibration of a rectangular plate. These are named according to the numbers of nodes parallel to the short and long axes respectively: (0,2) and (2,0). Their Chladni patterns are shown below. The pale lines are nodes: lines of no displacement. areas on either side of a nodal line move in opposite directions. Because it is easier to bend a regtangle about its short axis than about its long, the (0,2) mode has the lower frequency. (What happens for a square, when the (0,2) and (2,0) mode have the same frequency? If you add these two vibrations together in phase, you get the mode shown in the third photograph.)

         

These (2.0) and (0,2) modes are sketched below. Cross sections along the lines a-a' and b-b' are sketched to show the shape at two instants separated by half a cycle.

The lower half of the figure shows the analogous modes of a bell plate. The (2,0) mode (left) is the more important. If we imagine cutting away the corners of a rectangular plate, we move the nodal lines closer towards the centre. Eventually they intersect. This is the shape used for bell plates.

Now at a nodal line, the position of the plate isn't changing. So a node is the place to hold a plate if you want it to ring. However, on either side of the node, the plate is moving in opposite directions, so the plate is rotating locally. (See the cross sections in the sketches above.) If you hold a plate with your fingers at a simple nodal line, as we do in an example below, you damp the sound a little.

At the intersection of two nodal lines, there is no rotation: both displacement and slope are zero. This is the place to put the handle, which becomes an extended node itself. Let's see how this works:

	A. Here I am holding it at the normal position, which is an extended region formed from the fusion of the two nodes of the (2,0) mode.
	B. Here I'm holding it at a point on one of the nodes of the (2,0) mode. Still the same mode, so it has the same pitch. But because it is a simple node, rather than the intersection of two nodes, the plate is rotating locally about this node. This rotation transfers energy to my fingers, so it rings for a shorter time.
	C. This point is an antinode of the (2,0) mode but a node of the (0,2), so of course it is the latter mode that sounds: about one tone lower than the (2,0).
	D. This point is at the intersection of a node of the (0,2) and a node of the (0,2). So, by striking at a point that is not a node of either, it is possible to sound the two modes together. (In B above, I also held it at this point, but struck at a node of the (0,2), so only the (2,0) sounded). The finger and thumb damp both modes, so the interference between the two modes is not very clear. So...
	E. Here the plate is suspended by a thread through a hole drilled at the intersection of the two nodes. The local rotation of each mode transfers rather little energy to the thread, so the plate rings in both modes. The transients are clearer too. Notice that the interaction of several different modes that are not harmonically related makes the sound more like that of a bell.

How does the sound depend on direction?

Here I strike the plate so that, at first, the microphone is on a line normal to the plane of the plate. Then I rotate it so that the microphone lies in the plane of the plate. In the latter situation, the plate radiates approximately as a dipole, so the sound is weaker.