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Collisions are an innate part of the function of many musical instruments. Due to the nonlinear nature of contact forces, special care has to be taken in the construction of numerical schemes for simulation and sound synthesis. Finite difference schemes and other time-stepping algorithms used for musical instrument modelling purposes are normally arrived at by discretising a Newtonian description of the system. However because impact forces are nonanalytic functions of the phase space variables, algorithm stability can rarely be established this way. This paper presents a systematic approach to deriving energy conserving schemes for frictionless impact modelling. The proposed numerical formulations follow from discretising Hamilton's equations of motion, generally leading to an implicit system of nonlinear equations that can be solved with Newton's method. The approach is first outlined for point mass collisions and then extended to distributed settings, such as vibrating strings and beams colliding with rigid obstacles. Stability and other relevant properties of the proposed approach are discussed and further demonstrated with simulation examples. The methodology is exemplified through a case study on tanpura string vibration, with the results confirming the main findings of previous studies on the role of the bridge in sound generation with this type of string instrument. (Vasileios Chatziioannou), m.vanwalstijn@qub.ac.uk (Maarten van Walstijn)

SummaryIn this paper, we present new methods for constructing and analysing formulations of locally reacting surfaces that can be used in finite difference time domain (FDTD) simulations of acoustic spaces. Novel FDTD formulations of frequency-independent and frequency-dependent impedance boundaries are proposed for 2D and 3D acoustic systems, including a full treatment of corners and boundary edges. The proposed boundary formulations are designed for virtual acoustics applications using the standard leapfrog scheme based on the rectilinear grid, and apply to FDTD as well as Kirchhoff variable digital waveguide mesh (K-DWM) methods. In addition, new analytic evaluation methods that accurately predict the reflectance of numerical boundary formulations are proposed.The results obtained from numerical experiments and numerical boundary analysis (NBA) are analysed in time and frequency domains in terms of the pressure wave reflectance for different angles of incidence and various impedances. The results show that the proposed boundary formulations structurally adhere well to the theoretical reflectance. In particular, both reflectance magnitude and phase are closely approximated even at high angles of incidence and low impedances. Furthermore, excellent agreement was found between the numerical boundary analysis and the experimental results, validating both as tools for researching FDTD boundary formulations.

Methods of measuring the acoustic behavior of tubular systems can be broadly characterized as steady state measurements, where the measured signals are analyzed in terms of infinite duration sinusoids, and reflectometry measurements which exploit causality to separate the forward and backward going waves in a duct. This paper sets out a multiple microphone reflectometry technique which performs wave separation by using time domain convolution to track the forward and backward going waves in a cylindrical source tube. The current work uses two calibration runs ͑one for forward going waves and one for backward going waves͒ to measure the time domain transfer functions for each pair of microphones. These time domain transfer functions encode the time delay, frequency dependent losses and microphone gain ratios for travel between microphones. This approach is applied to the measurement of wave separation, bore profile and input impedance. The work differs from existing frequency domain methods in that it combines the information of multiple microphones within a time domain algorithm, and differs from existing time domain methods in its inclusion of the effect of losses and gain ratios in intermicrophone transfer functions.

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