By extension we can turn our attention from vibrating hemispheres to vibrating spheres.
I’ve followed the example of Hans Jenny and initially centred my investigation upon bubbles. Now if a bubble is vibrated with a low frequency sine wave it adopts steady states of vibration as shown below (click on each for moving image).
The emergence of these different shapes can be plotted on a graph (figure 76) where the y-axis shows bubble diameter in centimetres, the x-axis frequency in hertz. The thick curved lines represent the progression of each of the bubble symmetries shown above. As with the dishes of water earlier we can see evidence of harmonic behaviour; so for a bubble 12cms in diameter the steady states appear at roughly 12, 18, 24, 30, 36, 42, 48 Hz which would coincide with a harmonic series from 2nd to 8th harmonic whose fundamental is 6Hz.
What about subharmonics (undertones)? What would this graph look like if a bubble did indeed vibrate harmonically? Figure 77 shows the behaviour of an idealised harmonic bubble.
We can recognise the lambdoma superimposed upon the graph, with the overtones proceeding horizontally from left to right and the undertones curving upwards from right to left. So, for instance, when 40 Hz is the generating signal, a bubble 12 cms in diameter will vibrate in the steady state of the 8th harmonic, while a bubble 6 cms in diameter will vibrate in the steady state of the 4th harmonic. All vibrational states are clearly proportionally related.
Returning to our experimental bubble graph (figure 76), it appears that this graph bears a strong if truncated resemblance to the idealised version; which would tend to suggest that our initial proposition was correct: a sphere does vibrate harmonically.
With this in mind, in the bubble video clip in the Overview the music does not just accompany the cymatic visuals but is directly harmonically correlated to the pulsations of the bubbles.