To supplement the investigation of tubular bell acoustics, I carried out an experiment on a tubular bell at St Mary, Caterham, Surrey aimed at identifying the physical modes of vibration. I am grateful to Brian Rochester for arranging access to the tower and helping with the experiment.
The tube used was no. 7 at Caterham, chosen because it was the longest tube which could be accessed throughout its length (the no. 8 tube passes through the floor). The tube, as is common with tubular bells, was suspended from a loop of rope passing through a hole in both sides of the tube near the top. Here is a picture of the bells in the tower. The no. 7 bell is second from the right:
In a tubular bell suspended in this way, both ends are free to vibrate. As a result, all the modes of vibration have an antinode (a point of maximum vibration) at both ends of the tube, and a varying number of nodes (stationary points) along the length of the tube. The idea behind the experiment is that when the tube is struck at a particular point along its length, modes with an antinode at the striking point are maximally stimulated, and modes with a node at the striking point are not stimulated.
The experiment used the same technique as that used to identify modes of vibration in bells of traditional shape, and in hemispherical bells. The no. 7 tube at Caterham is 2.45m in length. It was marked in chalk at 4cm intervals giving 62 strike points in total, including the top and bottom of the tube. The spacing and number of strike points were chosen in the hope that modes of vibration with up to 16 nodes and antinodes along the length (i.e. four measurement points per cycle) would be visible. The tube was struck with a hammer at each point while recording the sound on a laptop. The resulting recording was then split into 62 individual sound files.
The overall spectrum of the bell is as follows. At least 16 partial frequencies could be identified in the sound. The amplitude of the lowest two partials (at 55hz and 158hz) was barely above the noise level, as can be seen from the spectrum:
Nearly all the partials had a doublet, and in the higher frequency partials the doublet split was very wide. Nodal patterns along the length of the tube were used to confirm that the two halves of each doublet pair were the same mode of vibration.
The 62 files taken were analysed using similar software used for analysing vibration patterns up the waist of a conventionally-shaped bell. A fourier transform of each file was calculated, and the amplitude of the transform at the frequencies of interest taken. Where there was a doublet, the amplitudes of the pair were averaged. The partial amplitudes from each file were normalised by dividing by the total amplitude for all frequencies in the transform, to compensate for varying strengths of hammer blows.
The results were quite satisfactory. Apart from the two lowest frequency partials, where as already noted the amplitude was barely above the noise level, the nodal and antinodal patterns for each partial were clearly visible along the length of the tube, and the number of nodes corresponded with theory. In the plots below, the horizontal axis is the relative distance along the tube, and the vertical axis is the relative amplitude of the partial. The name of the mode is the same as used in the investigation of tubular bell acoustics. The frequency or frequencies of each mode are given in the heading of the plot.
| Mode | Nodes | Nodal pattern |
| P1 | Unclear |  |
| P2 | Unclear, possibly three |  |
| P3 | 4 |  |
| P4 | 5 |  |
| P5 | 6 |  |
| P6 | 7 |  |
| P7 | 8 |  |
| P8 | 9 |  |
| P9 | 10 |  |
| P10 | 11 |  |
| P11 | 12 |  |
| P12 | 13 |  |
| P13 | 14 |  |
| P14 | 15 |  |
| P15 | 16 |  |
| P16 | 17 |  |
For the transverse modes of vibration of a tube with free ends, a symmetry argument shows that the minimum number of nodes along the length of the tube is two. The theory given in the Physics of Musical instruments (ref 1) predicts that each successive mode has one more node. If we assume that the two lowest frequency modes observed in this experiment have two and three nodes respectively, then the experiment results match the theory.
The tube is not uniform, because it has a hole through both walls near the upper end from which it is suspended. The hole is approximately 10.3cm below the top of the tube which is a fraction 0.042 of the tube length. It is possible that this lack of uniformity gives rise to the doublets observed in most of the partials.
The highest frequency mode with 17 nodes shows a clear nodal pattern, despite having less than four strike points per cycle.
References
- The Physics of Musical Instruments, Neville H Fletcher and Thomas D Rossing, 2nd Edition, Springer 1998