Sinusoidal Solution

It is well-known that the solution to (3.1), if $u(t)$ is constrained to be real-valued, and if $\omega_{0}\neq 0$, has the form

$$u(t) = A\cos(\omega_{0}t)+B\sin(\omega_{0}t)$$ (3.3)

where clearly, one must have

$$A = u_{0}\qquad B = v_{0}/\omega_{0}$$ (3.4)

Another way of writing (3.3) is as

$$u(t) = C_{0}\cos(\omega_{0}t+\phi_{0})$$ (3.5)

where

$$C_{0} = \sqrt{A^2+B^2}\qquad \phi_{0} = \tan^{-1}(-B/A)$$ (3.6)

$C_{0}$ is the amplitude of the sinusoid, and $\phi_{0}$ is the initial phase.

From the sinusoidal form (3.5) above, it is easy enough to deduce that

$$\vert u(t)\vert = C_{0}\vert\cos(\omega_{0}t+\phi_{0})\vert\leq C_{0}\qquad{\rm for}\qquad t\geq 0$$ (3.7)

which serves as a convenient bound on the size of the solution for all $t$, purely in terms of the initial conditions and the system parameter $\omega _{0}$. Such bounds, in a numerical sound synthesis setting, are extremely useful, especially if a signal such as $u$ is to be represented in a limited precision audio format. Note however, that the bound above is obtained only through a priori knowledge of the form of the solution itself (i.e., it is a sinusoid). As will be shown in the next section, it is not really necessary to make such an assumption.

The use of sinusoidal, or more generally complex exponential solutions to a given system in order to derive bounds on the size of the solution was developed, in the discrete setting, into a framework for determining numerical stability of a simulation algorithm, also known as von Neumann analysis [210,102]. Much more will be said about this from Chapter 5 onwards.