In this chapter, the student learns how to perform certain classes of definite integrals using contour integration methods. Although the integration variable is real for most integrals of interest, such as the inverse Fourier transform, analysis of the integral is extended to complex values of the integration variable and theorems related to integrating around closed contours on the complex plane are used to solve classes of definite integrals. The key theorems include Cauchy’s theorem for integrating so-called analytic functions, Jordan’s lemma, and the residue theorem for the important case where inside a closed contour on the complex plane, the integrand has places called singularities at which the function is not well behaved. Contour integration is used to analyze and derive results for the constitutive laws of a material when the current response depends not just on current forcing but also on the history of the forcing. This topic is called delayed linear response, which is developed at length. Contour integration, when combined with Fourier transforms, provides the solution of various types of initial-value and boundary-value problems in infinite and semi infinite domains.

In the previous chapter we introduced transmission lines, structures consisting of a pair of conductors that guide a high-frequency electromagnetic wave in the space between the conductors from a source to a load. We considered a number of different geometries (parallel plate, twin-lead, co-axial), each with a uniform cross-section, that guide the wave in the $\pm z$-direction, and we discussed the properties of signals carried by the transmission lines, largely in terms of the potential difference $\tilde{V}\left(z\right)$ between the conductors and the current $\tilde{I}\left(z\right)$ flowing in one conductor and returning in the other. For many high-power, high-frequency applications, waveguides are often used to guide electromagnetic waves from source to load. Unlike a transmission line, a waveguide typically consists only of a single conductor, which is hollow, with the wave existing in the interior of the waveguide. Waveguides are often used for frequencies ranging from 10 to 100 GHz, and suffer lower losses in this range than do transmission lines. The typical transverse dimension of a waveguide is a few centimeters, on the order of the wavelength of the wave inside. A few examples of waveguides are illustrated in Fig. 10.1. The parallel plate waveguide shown in Fig. 10.1(a) is not in common usage, but serves as a simple geometry for developing the properties of general waveguides. (Parallel plates differ from most other waveguides, in that they consist of two conductors and can function as transmission lines or waveguides.) Also shown in this figure are (b) rectangular and (c) circular waveguides. While the electromagnetic principles for the circular waveguide are similar to those of the rectangular waveguide, the mathematical functions needed for this geometry are not familiar to most undergraduate students, so we’ll need to introduce these functions. Also, while we devote our attention here to conducting waveguides, be aware that dielectric waveguides, such as optical fibers and photonic crystals can be treated in a similar manner. With the exception of TechNotes 10.1 and 10.2, we will not consider dielectric waveguides in this text. Optical fibers are common conduits for optical signals, and find application in medical instruments and communications systems.

Distinction is made between shallow water and deep water wave theories, depending on the value of the Ursell number. Potential flow theory is applied and the Stokes development is followed and first-order (linear), second-order, and third-order wave theories, in regular and irregular waves, are described. The concepts of phase and group velocities, mass transport, and energy flux are introduced. The application of stretching models to wave crest kinematics is described. At second-order distinction is made between bound (or locked) wave components (that accompany the first-order wave system) and free components (that travel independently). It is emphasized that, from third-order, such phenomena as mutual modifications of the phase velocities, or exchanges of energy, can take place between wave components. These interactions may lead to the occurrence of rogue waves, or to strong runups often observed at midships. The stream function wave model, which encompasses shallow and deep water cases, is presented. Finally the nonlinear Schr¨odinger equation that describes the time and space evolution of the wave envelope is applied to the prediction of the Benjaminis–Feir instability.

We discuss the basic concepts of waves, including phase velocity, dispersion, group velocity. We show how to use the Fourier principle to construct any general wave from the harmonic waves.

Chapter 3 is a general, rather short and partly descriptive introduction to general wave theory, without application of any differential equation. The emphasis is on mechanical waves, e.g., acoustic waves.

Chapter 4 introduces basic differential equations and boundary conditions for gravity waves propagating along a water surface. Assuming low wave amplitudes, equations are linearised. Then a quantitative discussion is given for harmonical (sinusoidal) waves propagating either on deep water, or otherwise on water of constant depth. Phase and group velocities are introduced, and then formulas are derived for the potential energy and the kinetic energy associated with a water wave. A closely related result is an important formula for the wave-power level, which equals the wave’s group velocity multiplied by the wave’s stored – kinetic + potential – energy per unit of sea surface. An additional subject is the wave’s momentum density. A section concerns real sea waves. Further, circular waves are mathematically described. Two sections of the chapter concern mathematical tools to be applied in Chapters 5–8 of the book. A final section considers water waves analysed in the time domain.