is driven by transverse changes in the
squared slope of the string:
It turns out that piano strings exhibit audible nonlinear effects, especially in the first three octaves of its pitch range at fortissimo playing levels and beyond [26]. As a result, for highest quality piano synthesis, we need more than what is obtainable from a linearized wave equation such as Eq.(9.30).
As can be seen from a derivation of the wave equation for an ideal
string vibrating in 3D space (§B.6), there is
fundamentally nonlinear coupling between transverse and
longitudinal string vibrations. It turns out that the coupling from
transverse-to-longitudinal is much stronger than vice versa, so that
piano synthesis models can get by with one-way coupling at normal
dynamic playing levels [30,164]. As
elaborated in §B.6 and the references cited there, the
longitudinal displacement
is driven by transverse changes in the
squared slope of the string:
![$\displaystyle \epsilon\ddot{\xi} = ES\xi'' + \frac{1}{2}ES[(y')^2]'
$](img2281.png)
where
![$\displaystyle [(y')^2]' \isdefs \frac{\partial}{\partial x} \left[\frac{\partial}{\partial x}\, y(t,x)\right]^2.
$](img2282.png)
Since longitudinal waves travel an order of magnitude faster than transverse waves, this coupling gives rise to high-frequency inharmonic overtones (corresponding to longitudinal modes of vibration) in the sound. Since the nonlinear coupling is distributed along the length of the string, the longitudinal modes are continuously being excited by the transverse vibration across time
and position 
along the string.
In addition to the excitation of longitudinal modes, the nonlinear transverse-to-longitudinal coupling results in a powerful longitudinal attack pulse, which is the leading component of the initial ``shock noise'' audible in a piano tone. This longitudinal attack pulse hits the bridge well before the first transverse wave and is therefore quite significant perceptually. A detailed simulation of both longitudinal and transverse waves in an ideal string excited by a Gaussian pulse is given in [394].
Another important (i.e., audible) effect due to nonlinear transverse-to-longitudinal coupling is so-called phantom partials. Phantom partials are ongoing intermodulation products from the transverse partials as they transduce (nonlinearly) into longitudinal waves. The term ``phantom partial'' was coined by Conklin [85]. The Web version of [18] includes some illuminating sound examples by Conklin.
