Physical Scattering Junctions

All scattering matrices arising from the connection of scalar
physical waveguides have the same basic structure which is an
efficient subset of the set of all lossless scattering matrices in
the sense of (26). To fix the setting, consider the parallel
junction of lossless acoustic tubes, each with real, positive,
scalar wave admittance . The pressure at the junction is the
same for all waveguides at the junction point, and the
velocities from each branch sum to zero. These physical constraints
yield scattering matrices of the form

where and are defined as follows:

(39) |

(40) |

It is easy to verify that the matrix
satisfies the condition
of losslessness (26). In the particular case of
equal-impedance waveguides, we have

We notice that the elements of each row of matrix
in (38) and (41) are *allpass complementary*,
i.e.,
, for all . Since
the scattering matrix
for equal-impedance waveguides is
symmetric, the elements of any row or column of matrix
are
allpass complementary. Since
remains unchanged in the
normalized case, the elements of each row and column are also power
complementary.

The equal-impedance case is particularly important when is a power of two. In this case, all the multiplications can be replaced by right-shifts in fixed-point arithmetic, and considerable savings in circuit area can be achieved in a VLSI implementation. This case has been explored in the design of efficient digital waveguide meshes [91,122,125].

The scattering matrix for a connection of normalized physical waveguides
is given by (32). It can be more explicitly written as

where , and the power-complementary property (37) can be verified for both rows and columns.

In the expressions (38)-(42), the elements of the matrices are real, since the one-variable waveguides intersecting at the junction have real, positive impedances.

Normalized -way scattering junctions represented by real matrices can be implemented by CORDIC processors [38]. Since the scattering matrix is orthogonal (real and unitary), it corresponds to a rotation in dimensions. Any such rotation can be obtained by planar rotations [34], with each planar rotation being realizable using a CORDIC processor. This formulation can be useful for VLSI implementations.

The junction represented by the expressions (38)
to (42) is formally a *parallel* junction. The
corresponding *series* junction can be obtained by taking the *dual* of the parallel junction, i.e., by replacing admittance with
impedance and interchanging pressure and velocity. In the series
junction, the velocity is the same in all waveguides at the
junction, and the pressures (or forces) sum to zero.

Parallel junctions occur naturally in the case of intersecting acoustic tubes, while series junctions arise at the intersection of transversely vibrating strings, where the transverse velocity must be the same for all strings. For waves in strings, force waves play the role of pressure waves in acoustic tubes. Note that the wave equation (1) for strings is usually written in terms of transverse displacement. However, choosing velocity and force as Kirchhoff variables unifies the treatment of vibrating strings with that of acoustic tubes.

- Parallel Junction of Multivariable Complex Waveguides
- Loaded Junctions
- Example: Bridge Coupling of Piano Strings

Download wgj.pdf

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

[Automatic-links disclaimer]