Velocity excitations and impulse responses of strings - Aspects of continuous and discrete models

Abstract

This paper discusses aspects of the second order hyperbolic partial differential equation associated with the ideal lossless string under tension and it's relationship to two discrete models. These models are finite differencing in the time domain and digital waveguide models. It is known from the theory of partial differential operators that in general one has to expect the string to accumulate displacement as response to impulsive excitations. Discrete models should be expected to display comparable behavior. As a result it is shown that impulsive propagations can be interpreted as the difference of step functions and hence how the impulsive response can be seen as one case of the general integrating behavior of the string. Impulsive propagations come about in situations of time-symmetry whereas step-function occur as a result of time-asymmetry. The difference between the physical stability of the wave equation, which allows for unbounded growth in displacement, and computational stability, that requires bounded growth, is derived.