Sound Traveling in One-Dimensional Space

Abstract

Because of the harmonic structure of waves, sound traveling in an acoustic tube or vibration of a string is a simple system providing an informative model to describe sound generation in musical instruments. This chapter deals with sound waves traveling in a space of one-dimensional from the point of view of the periodic structure in the time domain. According to the $z-$transform of the impulse response, the difference is well expressed by a pair of boundary conditions referred to as open–open and open–closed conditions. Allowing the coefficient of reflection to be represented as a complex number rather than solely real, the harmonic structure then varies. In addition, introducing a pulse sequence instead of a single number for the coefficient of reflection, the spectral characteristics are found to change considerably. Not only the transfer functions but also the driving-point impedance may be derived through geometrical modeling in the time domain. For one-dimensional systems, the phase response has a minimum phase with a propagation phase delay. The propagation phase delay is determined by the poles and zeros of the transfer function. Interestingly, the propagation phase delay can be estimated by the difference in the number of poles and zeros, even after the propagation phase is discarded. The geometrical view of the radiation impedance helps in providing an understanding of the schematics of the radiation condition from vibration.