Introduction

The digital waveguide (DW) method has been used for many years to provide highly efficient algorithms for musical sound synthesis based on physical models [437,450,399]. For a much longer time, finite-difference time-domain (FDTD) schemes have been used to simulate more general situations, usually at higher cost [554,395,74,77,45,400]. In recent years, there has been interest in relating these methods to each other [123] and in combining them for more general simulations. For example, modular hybrid methods have been devised which interconnect DW and FDTD simulations by means of a KW converter [224,227]. The basic idea of the KW-converter adaptor is to convert the ``Kirchoff variables'' of the FDTD, such as string displacement, velocity, etc., to ``wave variables'' of the DW. The W variables are regarded as the traveling-wave components of the K variables.

In this appendix, we present an alternative to the KW converter. Instead of converting K variables to W variables, or vice versa, in the time domain, conversion formulas are derived with respect to the current state as a function of spatial coordinates. As a result, it becomes simple to convert any instantaneous state configuration from FDTD to DW form, or vice versa. Thus, instead of providing the necessary time-domain filter to implement a KW converter converting traveling-wave components to physical displacement of a vibrating string, say, one may alternatively set the displacement variables instantaneously to the values corresponding to a given set of traveling-wave components in the string model. Another benefit of the formulation is an exact physical interpretation of arbitrary initial conditions and excitations in the K-variable FDTD method. Since the DW formulation is exact in principle (though bandlimited), while the FDTD is approximate, even in principle, it can be argued that the true physical interpretation of the FDTD method is that given by the DW method. Since both methods generate the same evolution of state from a common starting point, they may only differ in computational expense, numerical sensitivity, and in the details of supplying initial conditions and boundary conditions.

The wave equation for the ideal vibrating string, reviewed in §C.1, can be written as

$$Ky''= \epsilon {\ddot y},$$

where the following notation is used:

$$\begin{array}{rclrcl} K& \isdef & \mbox{string tension} & \qquad y & \isdef & y(t,x) \\ \epsilon & \isdef & \mbox{linear mass density} & {\dot y}& \isdef & \frac{\partial}{\partial t}y(t,x) \nonumber \\ y & \isdef & \mbox{string displacement} & y'& \isdef & \frac{\partial}{\partial x}y(t,x) \nonumber \end{array}$$

In the following two subsections, we briefly recall finite difference and digital waveguide models for the ideal vibrating string.