The Fourier series is introduced as a very useful way to represent any periodic signal using a sum of sinusoidal (“pure”) signals. A display of the amplitudes of each sinusoid as a function of the frequency of that sinusoid is a spectrum and allows analysis in the frequency domain. Each sinusoidal signal of such a complex signal is referred to as a partial, and all those except for the lowest-frequency term are referred to as overtones. For periodic signals, the frequencies of the sinusoids will be integer multiples of the lowest frequency; that is, they are harmonics. Pitch is a perceived quantity related to frequency, and it may have a complicated relationship to the actual frequencies present in terms of the series. For periodic signals, changes in the relative phase of the partials do not change the perception of sounds that are not too loud.

One-dimensional traveling wave pulses are defined and generalized to sinusoidal traveling waves. The mathematical description of sinusoidal waves is considered in depth. It is shown that any traveling wave can be considered the sum of standing waves, and vice versa. Longitudinal polarization is introduced and contrasted with the previous examples involving transverse polarization. The effects of dispersion, where the wave speed may depend on frequency, on wave propagation are briefly discussed.

The distinction is made between linear and nonlinear physics problems. Whereas the linear problems can be solved as a sum of simpler problems, nonlinear situations cannot be treated this way. That has implications for the solutions for the bowed and blown (wind) instruments, where the driving force is nonlinear, so the resonant modes cannot be treated individually. The stick-slip mechanism for bowed instruments is used as an example where friction provides the nonlinear force. The driving terms for reed instruments are also shown to be nonlinear and can be, in part, understood by thinking about negative resistance. The nonlinear coupling between the modes has implications for the overall tuning and for the frequencies of the overtones.

The historical connections between physics and music are discussed, and the questions “What is music?” and “What is physics?” are addressed. The area of overlap is presented as the topic for the remainder of the book. The use of numbers having limited accuracy, identified with units and possibly other qualifiers, to specify quantities measured by physicists is presented. Scientific notation and unit prefixes are reviewed.

Sound in one-dimensional systems, pipes, is considered in the context of modeling of the physics of wind instruments. Acoustic impedances are defined, which are used to characterize a system’s response to a force. The convenience of using complex numbers, which have real and imaginary parts, to describe impedance for waves, which have an amplitude and phase, is presented. Reflection and transmission of sound at points where a pipe’s impedance changes are considered, along with how these lead to resonances. The differences between sound propagation in cylindrical, conical, and Bessel horn-shaped pipes are presented. A model pipe with periodic holes is used to model the finger holes found in woodwinds. Such a pipe exhibits a critical frequency below which the impedance is imaginary, resulting in reflection, and above which is real-valued, allowing sound to propagate. An explicit example is shown using simple calculations for a fife, illustrating that the critical frequency becomes important for the upper range of woodwinds. A solution method for more advanced pipe models is presented. One more advanced model is that used for the human vocal tract, which can be modeled with pipes and acts as a time-dependent filter.

This chapter discusses two different approaches to describing fluid flow: a Lagrangian approach (following a fluid element as it moves) and a Eulerian approach (watching fluid pass through a fixed volume in space). Understanding each of these descriptions of flow is needed to fully understand the dynamics of fluids. This chapter is devoted to diving into the differences between the two descriptions of fluid motion. Understanding this chapter will help tremendously in the understanding of the upcoming chapters when the Navier–Stokes equations and energy equation are discussed. This chapter will introduce the material derivative. It is extremely important to understand this derivative before the Navier–Stokes equations themselves are tackled.

In the previous chapter, the forces acting on a moving fluid element were exhaustively studied. Using Newtons second law of motion, the Navier–Stokes equations for both compressible and incompressible flows were obtained. This chapter uses an alternative approach to developing the Navier–Stokes equations. Namely, by starting from a Eulerian description (as opposed to a Lagrangian description), the integral form and conservation form of the Navier–Stokes equations are developed. The continuity and Navier–Stokes equations in its various forms are tabulated and reviewed in this chapter. This chapter ends by solving some very simple, yet common, problems involving the incompressible Navier–Stokes equations.

This chapter develops the Navier–Stokes equations using a Lagrangian description. In doing so, the concept of a stress tensor and its role in the overall force balance on a fluid element is discussed. In addition, the various terms in the stress tensor as well as the individual force terms in the Navier–Stokes equations are investigated. The chapter ends with a discussion on the incompressible Navier–Stokes equations.