1. Gerald B. Whitham Linear and Non-Linear Waves
&
Linear and Non-Linear Waves
Introduction and General Outline
1.1 The Two Main Classes of Wave Motion
Pusan National University
Department of Naval Architecture & Ocean Engineering
Kim Min Il

2. Gerald B. Whitham Linear and Non-Linear Waves
&
Linear and Non-Linear Waves
Introduction and General Outline
1.1 The Two Main Classes of Wave Motion
Pusan National University
Department of Naval Architecture & Ocean Engineering
Kim Min Il

3. Gerald Beresford Whitham
( )
Whitham, a pioneer in the area of nonlinear waves whose research focused on fluid dynamics and the study of wave phenomena, including sonic booms, supersonic flow and shock-wave dynamics, and ocean waves.
B.S. in 1948, M.S. in 1949, and Ph.D. in 1953 from the University of Manchester under the direction of Sir James Lighthill.
He was a Faculty Member in the Department of Mathematics at the Massachusetts Institute of Technology during 1959–1962.
He came to Caltech in 1961 as a visiting professor of applied mathematics, was a professor of aeronautics and mathematics from 1962 to 1967, a professor of applied mathematics from 1967 to 1983, and the Powell Professor until his retirement in 1998.
Whitham received the Norbert Wiener Prize in Applied Mathematics in 1980, jointly awarded by the Society for Industrial and Applied Mathematics (SIAM) and the American Mathematical Society (AMS).
Whitham, who was instrumental in setting up Caltech's applied mathematics program in 1962, served as the executive officer for applied mathematics from 1971 to 1980.
In 1955, together with G. B. Whitham, Lighthill set out the first comprehensive theory of kinematic waves(an application of the method of characteristics), with a multitude of applications, prime among them fluid flow and traffic flow.
Sir James Lighthill
Born December 1927
Halifax, West Yorkshire
Died January 2014
(aged 86)
National USA
Known for Wave action
Whitham equation
Averaged Lagrangian
Filed Applied mathematics

4. Gerald B. Whitham Linear and Non-Linear Waves
&
Linear and Non-Linear Waves
Introduction and General Outline
1.1 The Two Main Classes of Wave Motion

5. Introduction and General Outline
CHAPTER 1
Introduction and General Outline
Wave motion is one of the broadest scientific subjects and unusual in that it can be studied at any technical level.
On the other hand they are also intensively studied by specialists, and almost any filed of science or engineering involves some questions of wavy motion.
There has been a correspondingly rich development of mathematical concepts and techniques to understand the phenomena from the theoretical standpoint and to solve the problems that arise.
This book is an account of the underlying mathematical theory with emphasis on the unifying ideas and the main points that illuminate the behavior of waves.
But one could then fill volumes with solutions and techniques for specific problem. This is not the purpose of the book.
The study of nonlinear waves started over a hundred years ago with the pioneering work of Stokes (1847) and Riemann (1858), and it has proceeded at an accelerating pace, with considerable development in recent years. The purpose here is to give a unified treatment of this body of material.

6. Introduction and General Outline
CHAPTER 1
Introduction and General Outline
The mathematical ideas are liberally interspersed with discussion of specific cases and specific physical fields.
Many of these topics are related to some branch of fluid mechanics, or to examples such as traffic flow which are treated in analogous fashion.
But the account is not written specifically for fluid dynamicists. The ideas are presented in general, and topics for application or motivation are chosen with a general reader in mind.
It is assumed that flood waves in rivers, waves in glaciers, traffic flow, sonic boom, blast waves, ocean waves from storms, and so on, are of universal interest.
The book is divided into two parts, the first on hyperbolic waves and the second on dispersive waves. The distinction will be explained in the next section.

7. 1.1 The Two Main Classes of Wave Motion
CHAPTER 1
1.1 The Two Main Classes of Wave Motion
There appears to be no single precise definition of what exactly constitutes a wave.
Various restrictive definitions can be given, but to cover the whole range of wave phenomena it seems preferable to be guided by the intuitive view that a wave is any recognizable signal that is transferred from on part of the medium to another with a recognizable velocity of propagation.
Nevertheless, one can distinguish two main classes. The first is formulated mathematically in terms of hyperbolic partial differential equations, and such waves will be referred to as hyperbolic.
The second class cannot be characterized as easily, but since it starts from the simplest cases of dispersive waves in linear problems, we shall refer to the whole class as dispersive and slowly build up a more complete picture.

8. 1.1 The Two Main Classes of Wave Motion
CHAPTER 1
1.1 The Two Main Classes of Wave Motion
The prototype for hyperbolic waves is often taken to be the wavy equation
Is, in fact, the simplest of all. As will be seen, there is a precise definition for hyperbolic equations which depends only on the form of the equations and is independent of whether explicit solutions can be obtained or not.
On the other hand, the prototype for dispersive waves is based on a type of solution rather than a type of equation. A linear dispersive system is any system which admits solutions of the form.
The first part of this book is devoted to hyperbolic waves and the second to dispersive waves.
The theory of hyperbolic waves enters again into the study of dispersive waves in various curious wavy, so the second part is no entirely independent of the first.
The remainder of this chapter is an outline of the various themes, most of which are taken up in detail in the remainder of the book.

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