vibration, and the subsequent more stable after-sound vibration. He describes the initial vibration as the ‘prompt sound, vertical polarisation’ and the second ‘after sound horizontal polarisation’.

……the vertical polarization is the primary one excited by the hammer, and so begins its life at a much higher amplitude than the horizontal one. However, since the bridge, which is attached to the soundboard, “gives” much more easily in the vertical than in the horizontal direction, the decay of the vertical mode is also much more rapid. As a result, the relatively slight amount of horizontal vibration becomes, after a while, dominant.¹⁵²

All piano strings vibrate in the vertical plane immediately after being struck by the hammer, and rapidly move into an elliptical circular pattern of vibration for the majority of a long note’s duration.

The drop in level [of the initial decay of a piano string] would appear as a straight line if the decay of the sound were of a type called exponential, which is what a physicist would expect from a linear system such as the string and the soundboard. Instead, it is clear that the curve breaks into two portions of quite different decay rates. The initial portion, called “prompt sound,” drops (in this case) at a rate of about 8 dB/sec; the final one, called “aftersound,” at less than one- quarter that rate. As we shall see, the prompt sound is simply related to the theoretical decay rate determined by the string’s coupling to the soundboard; whereas the after sound , which gives the piano its perceived sustaining power, represents the “miracle.”¹⁵³

Agreeing with Weinrich’s definition of two vibration polarities, professor Robert Anderssen of the CSIRO produced mathematical evidence of the prominent vertical string vibration caused by the Stuart bridge agraffe. In his The Challenge For the Piano Maker¹⁵⁴ Anderssen bases his research on the findings of Gabriel We inreich¹⁵⁵

, Richard Dain¹⁵⁶ , and Fletcher and Rossing¹⁵⁷ , stating it is the non- linear component of piano string vibration which provides the evidence. Anderssen made the observation that immediately after the string is struck by the hammer, the transition of the string from its vertically coupled rest point of the Stuart agraffe into its vertical oscillation is more efficient with less energy loss than and the transition from the horizontal pinned coupling of the modern piano string to its vertical oscillation.

Since the horizontal zig –zag clamp induces an energy exchange into a polarization orthogonal to the original polarization of the initial disturbance, as well as back into the vertical polarization, we would anticipate that the sustain would be different from that of the vertical zig-zag clamping on the Stuart & Sons pianos. In fact, as is clear from the [non-linear string equation], this complex energy exchange will affect the spacing between the eigen frequencies, but in different ways, depending on the nature of the clamping. This prediction is consistent with the perceived stronger sustain of the notes on the Stuart & Sons piano as compared with traditional grand pianos [the modern piano].¹⁵⁸

^{152     2}Weinreich. The Coupled Motion Of Piano Strings.
^{153     3}Weinreich. The Coupled Motion Of Piano Strings.
^{154     2}Anderssen, The Challenge For the Piano Maker,73.
^{155     4}Weinreich. The Coupled Motion Of Piano Strings.
^{156     2}Dain, 22.
^{157     3} Fletcher & Rossing
^{158     3}Anderssen, The Challenge For the Piano Maker,73.