ï~~Sv0
Figure 11: String Loaded with Mass and Spring
4.2.3 Scattering Junction Connection
The next step is to connect the mass and spring
system to a string, as shown in Figure 11. Since
we have decomposed the lumped mass and spring
system into traveling waves, and have picked an
associated wave impedance, Rh, we may join the
system to the string system at a lossless 3-port
scattering junction as shown in Figure 12.
Vh Vh+
M V S V+
Figure 12: Attaching the Wave Digital Hammer
We define vot0 = vh+ and vi_ = vh-, as was shown
in Figure 10, so that traveling wave signals entering the scattering junction are superscripted with
a plus sign, and traveling wave signals leaving the
junction are superscripted with a minus sign. At
the junction point where the mass and spring system connects to the string, the velocity of both
sides of the string, vi and v2, and the driving point
velocity of the spring, vh, must all equal vj, the
junction velocity,
V1 = V2 = VhV =-vi(32)
v1+ +v1 = V2 + V2- = Vh+ + Vh- (33)
In addition, the sum of the forces exerted by the
string halves and the mass and spring at that point
must be zero, since it is a massless point,
fl+f2+fh=fJ=0 (34)
fl+4f- +f2++f2-+fh++f =0 (35)
Combining these two series junction constraints,
(32) and (34), with the wave variable definitions,
(2) and (3), and wave impedance relations, (1), we
can derive the lossless scattering equations for the
interconnection of the wave decomposed mass and
spring with the string halves,
_ 2 (Rovi+ R-Rov2+ -Rhvh+) (36)
Vj-= 2o+ Rh (6
V1- =Vj - Vl (37)
v2-= Vj, - v +(38)
Vh- = Vj - Vh + (39)
These equations say that, as a wave is coming into
a junction, some portion of the wave reflects off
the junction and travels back where it came from,
while the rest of it travels into the junction and
is divided among the other outgoing waves. The
relative proportions of this scattering effect is dependent only on the relative wave impedances of
the interconnected elements.
Figure 13 shows how the mass and spring system
may be attached to a 2D digital waveguide mesh
[Van Duyne & Smith, 1993]. Junctions marked J
are lossless 4-port junctions. The mass and spring
system is attached at the 5-port scattering junction marked S. Equations (36) through (39) must
be modified appropriately to compute the 5-port
junction.
Figure 13: Attaching the Wave Digital Hammer to
a 2D Digital Waveguide Mesh
4.2.4 Making the Felt Nonlinear
Now, to convert our string loaded with a mass and
spring to a string being hit with a felt hammer,
we must cope with the nonlinearity of the spring
"constant". k(Xk) depends on the compression distance, Xk, of the felt. Fortunately we have signals
lying around in the mass spring loop which can give
us this distance directly. In general, fk = kxk or
xk = fk/k for a spring system. If we can find the
force being applied to the spring, we may divide it
by k to obtain the compression distance. But, the
driving point force on the mass and spring system
of Figure 9 is equal to the compression force on the
spring alone. Hence,
Xk = (1/k)(fh- +fh+)
= (Rh/k)(v- -Vh+)
(40)
(41)
ICMC Proceedings 1994
417
Sound Synthesis Techniques
