Loss

Damping intervenes in any musical system, and, in the simplest case, may be modelled through the addition of a ``linear loss" term to a given system. In the case of the simple harmonic oscillator, one may add damping as

$$\frac{d^2 u}{dt^2} = -\omega_{0}^2 u -2\sigma \frac{du}{dt}$$ (3.61)

where $\sigma\geq 0$ is the damping parameter for the system. The characteristic equation for (3.61), obtained through Laplace transformation, or through the insertion of a test solution $u = e^{st}$, is

$$s^2+2\sigma s +\omega_{0}^2 = 0$$ (3.62)

which has solutions

$$s_{\pm} = -\sigma \pm\sqrt{\sigma^2-\omega_{0}^2}$$ (3.63)

If damping is small, i.e., if $\sigma < \omega_{0}$, the roots above simplify to

$$s_{\pm} = -\sigma\pm j\omega_{1}\qquad{\rm with}\qquad \omega_{1} = \sqrt{\omega_{0}^2-\sigma^2}$$ (3.64)

In this case, the solution (3.5) may be generalized to

$$u(t) = e^{-\sigma t}\left(A\cos(\omega_{1}t)+B\cos(\omega_{1}t)\right) = C_{1}e^{-\sigma t}\cos(\omega_{1}t+\phi_{1})$$ (3.65)

where $A$ and $B$ are defined by

$$A = u_{0}\qquad B = \frac{v_{0}+\sigma u_{0}}{\omega_{1}}$$ (3.66)

and $C_{1}$ and $\phi_{1}$ are defined as,

$$C_{1} = \sqrt{A^2+B^2}\qquad \phi_{1} = \tan^{-1}(-\frac{B}{A})$$ (3.67)

It should be clear that the bound

$$\vert u(t)\vert \leq C_{1}e^{-\sigma t}\leq C_{1}$$ (3.68)

holds for all $t\geq 0$.

The case of large damping, i.e., $\sigma \geq \omega_{0}$ is of mainly academic interest in musical acoustics. The main point here is that for $\sigma>\omega_{0}$, the solution will consist of two exponentially damped terms, and for $\sigma=\omega_{0}$, degeneracy of the roots of the characteristic solution leads to some limited solution growth.

The energetic analysis of (3.61) is a simple extension of that for system (3.1). One now has, after again multiplying through by $\frac{du}{dt}$,

$$\frac{d{\mathfrak{H}}}{dt} = -\sigma\left(\frac{du}{dt}\right)^2\... ...thfrak{H}}(t_{2})\leq{\mathfrak{H}}(0)\quad{\rm for}\quad t_{1}\geq t_{2}\geq 0$$ (3.69)

where ${\mathfrak{H}}$ is defined as before. In other words, the energy is a positive and monotonically decreasing function of time. This leads to the bound

$$\vert u(t)\vert\leq C_{0}$$ (3.70)

which is identical to that obtained in the lossless case.

The bounds (3.68) and (3.70) obtained through frequency domain and energetic analysis, respectively, are distinct. Notice, in particular, that bound (3.70) is insensitive to the addition of loss, but, at the same time, is more general than bound (3.68), which requires the assumption of small damping (i.e., $\sigma < \omega_{0}$). It is possible to reconcile this difference by extending the energetic analysis somewhat in the following way.

First, note that although ${\mathfrak{H}}(t)$ will be monotonically decreasing in the case of linear loss, by (3.69), the specifics of its rate of decay are unclear. To this end, define the function $\bar{{\mathfrak{H}}}$ by

$$\bar{{\mathfrak{H}}} = {\mathfrak{H}}+\sigma u\frac{du}{dt} = \fr... ...{2}\left(\frac{du}{dt}\right)^2+\frac{\omega_{0}^2}{2}u^2+\sigma u\frac{du}{dt}$$ (3.71)

Note that $\bar{{\mathfrak{H}}}$ approaches ${\mathfrak{H}}$ in the limit as $\sigma\rightarrow 0$. It is simple to show that

$$\frac{d\bar{{\mathfrak{H}}}}{dt} = -\sigma\bar{{\mathfrak{H}}}\qquad\rightarrow\qquad\bar{{\mathfrak{H}}}(t) = e^{-\sigma t}\bar{{\mathfrak{H}}}(0)$$ (3.72)

Thus $\bar{{\mathfrak{H}}}$ decays exponentially, at decay rate $\sigma$.

On the other hand, $\bar{{\mathfrak{H}}}$, in contrast to ${\mathfrak{H}}$, is not necessarily a non-negative function of the state $u$ and $du/dt$. This non-negativity property is crucial in obtaining bounds such as (3.14). To this end, it is worth determining the conditions under which $\bar{{\mathfrak{H}}}$ is non-negative. First note that

$$u\frac{du}{dt}\geq -\frac{1}{2}\left(\alpha u^2 +\frac{1}{\alpha}\left(\frac{du}{dt}\right)^2\right)$$ (3.73)

for any real constant $\alpha >0$. Choosing $\alpha = \sigma$ leads to

$$\bar{{\mathfrak{H}}}\geq \frac{\omega_{0}^2-\sigma^2}{2}u^2$$ (3.74)

which is clearly non-negative under the condition of low loss $\sigma < \omega_{0}$ which was assumed in the frequency domain analysis earlier in this section. Under this condition, one has, from the above inequality, that

$$\vert u(t)\vert\leq \frac{1}{\omega_{1}}\sqrt{2\bar{{\mathfrak{H}... ...frac{1}{\omega_{1}}\sqrt{2\bar{{\mathfrak{H}}(0)}}\qquad{\rm for}\qquad t\geq 0$$ (3.75)

The bounding quantity on the right of the above inequality is in fact identical to $C_{1}$, and thus the bound above is identical to (3.68) obtained using frequency domain analysis.

In general, true energy functions such as ${\mathfrak{H}}$ for physical systems are non-negative, and the introduction of modified energetic quantities such as $\bar{{\mathfrak{H}}}$, for which non-negativity conditions must be determined, might seem to be a needless complication. On the other hand, for numerical approximations, as will be seen shortly, the discrete equivalent of the system energy is not necessarily non-negative, and the determination of non-negativity conditions leads to numerical stability conditions. Thus the simple analysis above serves as a useful example, in miniature, of what will follow in this book. Even in the discrete case, it will sometimes be useful to introduce modified energetic quantities such as $\bar{{\mathfrak{H}}}$, especially when dealing with certain types of non-centered finite difference schemes.