In §
6.12.2, general coupling of horizontal and vertical
planes of
vibration in an ideal string was considered. This
eigenanalysis will now be applied here to obtain formulas for the
damping and
mode tuning caused by the coupling of two
identical strings at a bridge. This is the case that arises in
pianos
[
543].
The general formula for linear, timeinvariant coupling of two
strings can be written, in the
frequency domain, as

(C.115) 
Filling in the elements of this coupling
matrix
from the results of §
C.13.1, we obtain
where
Here
is the bridge
impedance divided by the string
impedance. Treating
as a constant complex matrix for each fixed
,
the
eigenvectors are found
^{C.10}to be
respectively, and the
eigenvalues are
Note that only one eigenvalue depends on
, and neither
eigenvector is a function of
.
We conclude that ``inphase vibrations'' see a longer effective string
length, lengthened by the
phase delay of
which is the
reflectance seen from two inphase strings each having
impedance
. This makes physical sense because the inphase
vibrations will move the bridge in the vertical direction, causing
more rapid decay of the inphase mode.
We similarly conclude that the ``antiphase vibrations'' see no length
correction at all, because the bridge point does not move at all in
this case. In other words, any bridge termination at a point is rigid
with respect to antiphase vibration of the two strings connected to
that point.
The above analysis predicts that, in ``stiffness controlled''
frequency intervals (in which the bridge ``looks like a damped
spring''), the ``initial fast decay'' of a piano note should be a
measurably flatter than the ``aftersound'' which should be exactly in
tune as if the termination were rigid.
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