数学物理方程及相关分析工具简明教程（英文版）

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Part I Second-Order Linear Partial Differential Equations
　　Chapter 1 The Wave Equation
　　In this chapter， partial differential equations that govern many featured wave propagation phenomena are discussed. We start with modelling the process of string vibration through a partial differential equation， known as the wave equation， or the equation for string vibration. Then the concepts of initial and boundary conditions are introduced. This is followed by the presentation of a number of relevant methods widely used for solving problems governed by the wave equation. Our scope is then extended to cover two-/three-dimensional cases of wave propagation. The chapter concludes with discussion over the properties of solutions for problems governed by the wave equation.
　　The contents in this chapter are summarised in the knowledge map given in Fig. 1.1. Note that the contents in Fig. 1.1 are classified into several groups identified by the shapes of their frames. “Modelling” corresponds to a stage where a physical phenomenon is being represented by means of certain mathematical formulations. “Analysis” corresponds to a stage where certain mathematical tools are introduced so as to analyse the derived formulations (from physics). “Interpretation” corresponds to a stage where the mathematical results obtained through the analysis stage are employed to offer (often more insightful) interpretations to the physical phenomena that are modelled. For developing solid understanding of the course materials， certain mathematical knowledge that is supposed to be covered in the prerequisite courses， e.g. calculus， linear algebra， etc” is required， and these contents are summarised in the “prerequisite” stage. The linkages of the introduced contents with other topics likely to appear for future studies are also mentioned， and these “related topics” are identified in hexagonal frames. The setting here also applies for the knowledge map charted for other chapters in this book.
　　Fig. 1.1 Knowledge map~wave equation
　　1.1 Equation for String Vibration
　　1.1.1 Derivation of the Equation for String Vibration
　　In this section， we consider deriving a mathematical model to describe string vibration. To this end， we refer to a string， as shown in Fig. 1.2， of initial length L and of (linear) mass density p. As for modelling， one often needs to specify certain physically reasonable presumptions from the key features of the process of interest. For the case of string vibration， three assumptions are given as follows.
　　1. The diameter of a string should be far shorter than its length. This agrees with the common understanding of a “string”. But we need to express such a fact in mathematical terms. If we use d to denote the string diameter， then d《L， Hence， the string can be fully represented by one spatial variable， say， jc. As shown in Fig. 1.2， the x-axis is set in parallel with the configuration when the string is straight， and any point on the string should correspond to a value of x G [0， L].
　　2. The string undergoes small and transverse vibration. Here the term “trans-verse” means that each point on the string only moves along the direction that is perpendicular to the x-axis. Thus， a function u(x， t) can be introduced to denote the (transverse) displacement of the string section around x9 away from the x-axis at time t. The term “small vibration” means that the magnitude of the displacement u is far smaller than that of the string length L. Note that “small” is meant in a relative sense. For instance， if the string of interest is from a guitar， then a displacement of 1 dm is quite large. But if the string of interest is a cable upholding a suspension bridge， then a displacement of 1 dm can still be considered as “small”. Mathematically， the non-dimensional quantity of is employed to measure the degree of string vibration. Under small vibration， we write ，indicating that is small in magnitude， or we can simply require
　　(1.1)
　　3. A string resists elongation by a line tension following Hooke，s law. A string is expected to resist elongation，leading to a state of tension. The resisting force， or the line tension denoted by T(x， t)， is supposed to follow Hooke’s law， i.e. the magnitude of the line tension at (jc， t) is proportional to the local increment in the string arc-length denoted by s{x， t).
　　Moreover， a transversely vibrating string may also experience an external driving force，whose distribution can be denoted by F(x， t). Here F(x， t) is measured in unit “Newton per metre”. In Fig. 1.1，the key physical quantities that are needed for modelling string vibration are summarised.
　　For a transversely vibrating string， its behaviour can be dictated through the displacement function u(x， t). Thus， our modelling target is to set up mathematical formulations for m(jc， t). Here the modelling procedure encompasses the use of the above-listed three assumptions and the law of conservation in momentu