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Two Coupled Strings

Figure C.30: Two strings terminated at a common bridge impedance.
\includegraphics{eps/f_two_strings}


Two Ideal Strings Coupled at an Impedance

A diagram for the two-string case is shown in Fig. C.30. This situation is a special case of the loaded waveguide junction, Eq.$ \,$(C.98), with the number of waveguides being $ N=2$, and the junction load being the transverse driving-point impedance $ R_b(s)$ where the string drives the bridge. If the bridge is passive, then its impedance $ R_b(s)$ is a positive real function (see §C.11.2). For a direct derivation, we need only observe that (1) the string velocities of each string endpoint must each be equal to the velocity of the bridge, or $ v_1 =
v_2 = v_b$, and (2) the sum of forces of both strings equals the force applied to the bridge: $ f_b = f_1 + f_2$. The bridge impedance relates the force and velocity of the bridge via $ F_b(s) = R_b(s)
V_b(s)$. Expanding into traveling wave components in the Laplace domain, we have
$\displaystyle R_b(s) V_b(s)$ $\displaystyle =$ $\displaystyle F_b(s) = F_1(s) + F_2(s)$ (C.108)
  $\displaystyle =$ $\displaystyle [F^+_1(s) + F^-_1(s)] + [F^+_2(s) + F^-_2(s)]$ (C.109)
  $\displaystyle =$ $\displaystyle R_1 \{V^+_1(s) - [V_b(s) - V^+_1(s)] \}$ (C.110)
  $\displaystyle \,+\,$ $\displaystyle R_2 \{V^+_2(s) - [V_b(s) - V^+_2(s)]\}$ (C.111)

or

$\displaystyle V_b(s) = H_b(s) [ R_1 V^+_1(s) + R_2 V^+_2(s) ]
$

where $ R_i$ is the wave impedance of string $ i$, and

$\displaystyle H_b(s)\isdef \frac{2}{R_b(s) + R_1 + R_2}
$

Thus, in the time domain, the incoming velocity waves are scaled by their respective wave impedances, summed together, and filtered according to the transfer function $ H_b(s) = 2/[R_b(s) + R_1 + R_2]$ to obtain the velocity of the bridge $ v_b(t)$. Given the filter output $ v_b(t)$, the outgoing traveling velocity waves are given by
$\displaystyle v^-_1(t)$ $\displaystyle =$ $\displaystyle v_b(t) - v^+_1(t)$ (C.112)
$\displaystyle v^-_2(t)$ $\displaystyle =$ $\displaystyle v_b(t) - v^+_2(t) \ $ (C.113)

Thus, the incoming waves are subtracted from the bridge velocity to get the outgoing waves. Since $ V^-_2(s) = H_b(s) R_1 V^+_1(s) = H_b(s) F^+_1(s)$ when $ V^+_2(s) =
0$, and vice versa exchanging strings $ 1$ and $ 2$, $ H_b$ may be interpreted as the transmission admittance filter associated with the bridge coupling. It can also be interpreted as the bridge admittance transfer function from every string, since its output is the bridge velocity resulting from the sum of incident traveling force waves. A general coupling matrix contains a filter transfer function in each entry of the matrix. For $ N$ strings, each conveying a single type of wave (e.g., horizontally polarized), the general linear coupling matrix would have $ N^2$ transfer-function entries. In the present formulation, only one transmission filter is needed, and it is shared by all the strings meeting at the bridge. It is easy to show that the shared transmission filter for two coupled strings generalizes to $ N$ strings coupled at a common bridge impedance: From (C.98), we have

$\displaystyle V_b(s) = H_b(s) \sum_{i=1}^N R_i V^{+}_i(s)
$

where

$\displaystyle H_b(s) = \frac{2}{R_b(s) + \sum_{i=1}^N R_i}
$

Thus, $ H_b(s)$ is the shared portion of the bridge filtering, leaving only a scaling according to relative impedance to be done in each branch. The above sequence of operations is formally similar to the one multiply scattering junction frequently used in digital lattice filters [297]. In this context, it would be better termed the ``one-filter scattering termination.'' When the two strings are identical (as would be appropriate in a model for coupled piano strings), the computation of bridge velocity simplifies to

$\displaystyle V_b(s) = H_b(s) [V^+_1(s) + V^+_2(s)] \protect$ (C.114)

where $ H_b(s)\isdef 2/[2 + R_b(s)/R]$ is the velocity transmission filter. In this case, the incoming velocities are simply summed and fed to the transmission filter which produces the bridge velocity at its output. A commuted simulation diagram appears in Fig. C.31.
Figure C.31: General linear coupling of two equal-impedance strings using a common bridge filter.
\includegraphics{eps/fcouplednompy}
Note that a yielding bridge introduces losses into all attached strings. Therefore, in a maximally simplified implementation, all string loop filters (labeled LPF$ _1$ and LPF$ _2$ in Fig.C.31) may be eliminated, resulting in only one filter--the transmission filter--serving to provide all losses in a coupled-string simulation. If that transmission filter has no multiplies, then neither does the entire multi-string simulation.

Coupled Strings Eigenanalysis

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, time-invariant coupling of two strings can be written, in the frequency domain, as

$\displaystyle \left[\begin{array}{c} V_1^-(s) \\ [2pt] V_2^-(s) \end{array}\rig...
... \left[\begin{array}{c} V_1^+(s) \\ [2pt] V_2^+(s) \end{array}\right]. \protect$ (C.115)

Filling in the elements of this coupling matrix $ \mathbf{H}_c$ from the results of §C.13.1, we obtain

$\displaystyle \mathbf{H}_c(s) = \left[\begin{array}{cc} 1-H_b(s) & -H_b(s) \\ [2pt] -H_b(s) & 1-H_b(s) \end{array}\right]
$

where

$\displaystyle H_b(s) = \frac{2}{2+\tilde{R}_b}.
$

Here $ \tilde{R}_b\isdef R_b/R$ is the bridge impedance divided by the string impedance. Treating $ \mathbf{H}_c(s)$ as a constant complex matrix for each fixed $ s$, the eigenvectors are foundC.10to be

$\displaystyle \underline{e}_1 = \left[\begin{array}{c} 1 \\ [2pt] 1 \end{array}...
...uad
\underline{e}_2 = \left[\begin{array}{c} 1 \\ [2pt] -1 \end{array}\right],
$

respectively, and the eigenvalues are

$\displaystyle \lambda_1(s) = 1 - 2H_b(s),
\qquad
\lambda_2 = 1.
$

Note that only one eigenvalue depends on $ s=j\omega$, and neither eigenvector is a function of $ s$. We conclude that ``in-phase vibrations'' see a longer effective string length, lengthened by the phase delay of

$\displaystyle 1-2H_b = \frac{\tilde{R}_b(s)-2}{\tilde{R}_b(s)+2} = \frac{R_b(s)-2R}{R_b(s)+2R}
$

which is the reflectance seen from two in-phase strings each having impedance $ R$. This makes physical sense because the in-phase vibrations will move the bridge in the vertical direction, causing more rapid decay of the in-phase mode. We similarly conclude that the ``anti-phase 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 anti-phase 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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