Chapter 8. Underpinnings IV: Nonlinearity


This chapter gives an overview of nonlinear phenomena in acoustics and vibration, concentrating on aspects relevant to musical instruments. This is a large and complicated subject, and there is no way that this short account can do full justice to it. The intention is to introduce and illustrate some key concepts, especially ones that have direct relevance to musical applications.

Sections 8.1 and 8.2 describe some common sources of nonlinearity, and introduce the important distinction between “smooth” and “non-smooth” nonlinearities. This makes a difference to what kind of analysis methods can be brought to bear. An important technique that relies on smoothness is the method of “harmonic balance”, introduced here through a simple example.

Section 8.3 introduces the idea of “phase space”, a powerful tool used by the mathematicians to give a geometrical interpretation of nonlinear phenomena. This leads on in section 8.4 to an introduction to the idea of “chaos”. Many nonlinear systems, even ones that look beguiling simple, can exhibit motion that, although fully determined by a mathematical equation, nevertheless strikes an observer as having an unpredictable nature. A particular manifestation of chaotic response is “sensitive dependence”: starting the system off with two very similar initial conditions (but not completely identical ones) produces two versions of the response which after a while become completely different.

Section 8.5 brings us to a topic more obviously relevant to musical instruments. Any instrument capable in principle of producing a sustained note (rather than a transient note as on a guitar, piano or xylophone) must involve nonlinearity, to compensate the energy dissipation that will occur in all physical systems. So all the wind instruments and bowed-string instruments rely for their very existence on nonlinearity. Some examples of this kind of “self-excited vibration” are presented, as a taster for much fuller discussion to come in later chapters in which bowed strings and wind instruments are explored in detail.