Bars and Linear Strings

In this chapter, the finite difference methods discussed in the last chapter with regard to the 1D wave equation are extended to the more musically interesting case of bars and stiff strings. Linear, or low-amplitude vibration is characteristic of a variety of instruments, including xylophones and marimbas, and, to a lesser extent, string instruments such as acoustic guitars and pianos, which can, under certain conditions, exhibit nonlinear effects (this topic will be discussed in detail in Chapter 8). Linear time-invariant systems such as these, and the associated numerical methods are still amenable to frequency domain analysis, which can be very revealing with regard to effects such as dispersion (leading to perceived inharmonicity). In the numerical setting, frequency domain analysis is useful in obtaining simple necessary stability conditions, and in dispersion which results from discretization error; energy analysis allows the determination of suitable numerical boundary conditions.

In §7.1, the idealized thin bar model is introduced--while too simple to use for sound synthesis purposes, many important ideas relating to discretization may be dealt with here in a simplified manner. The more realistic stiff string model, incorporating stiffness, tension, and loss terms, is dealt with in §7.2. Various musical features are discussed in the following sections, namely the coupling to a hammer model, in §7.3, multiple strings in §7.4, and the preparation of strings in §7.5. Finally, in §7.6, the more complex case of bars of variable cross-section is introduced; this is the first instance, in this book, of a system with spatial variation, and as such, von Neumann type stability analysis for finite difference schemes no longer applies directly, though energetic methods remain viable.

References for this chapter include: [133,17,45,44,104,105,9,187,18,234,47,135,70,245,121]