Non-dimensionalized Form

Non-dimensionalization is an extremely useful means of simplifying physical systems, especially in preparation for the application of numerical simulation techniques. As a first step, one may introduce the dimensionless coordinate $x' = x/L$, for some constant $L$ with dimensions of length. In the general case of the wave equation defined over an arbitrary spatial domain ${\mathcal D}$, this constant is arbitrary, but over a finite spatial domain, such as $x\in[0,x_{0}]$, it may be taken to be the length $L = x_{0}$, thus yielding a system defined over the domain $x'\in[0,1]$. The wave equation (6.1) then becomes, in primed coordinates,

$$u_{tt} = \gamma^{2}u_{x'x'}$$ (6.2)

where $\gamma = c/L$ is a constant with dimensions of frequency. Such a description is useful in relating the wave equation to the simple harmonic oscillator, as described in Chapter 3. This is also a good example of a reduction in the number of parameters necessary to specify a system, which is of particular importance in physical modeling sound synthesis--it is redundant to specify both $c$ and $L$, and the single parameter $\gamma$ suffices. The fewer in number the parameters, the easier it will be for the musician working with a synthesis algorithm to navigate the space of possible timbres. In general, forms in this book will be presented, whenever possible, in such a spatially-nondimensionalized form, and the primed notation will be suppressed.

In the standard numerical analysis literature, it is commonplace to see a further temporal nondimensionalization, through the introduction of a temporal variable $t' = t/T$, for a characteristic time constant $T$. A judicious choice of $T = \sqrt{L/c}$ leads to the form

$$u_{t't'} = u_{x'x'}$$ (6.3)

Though this form is apparently simpler than (6.2) above, in practice, especially when programming synthesis routines, there is no real advantage to such a further step (i.e., an extra parameter, namely the time step, will be re-introduced during the discretization procedure). In addition, the crucial frequency domain behaviour of such a fully nondimensionalized system is slightly obscured through the introduction of such temporal scaling. For this reason, only ``half-way" spatial nondimensionalization will be employed in this book.