Questions About Elastic Waves

verfasst von: Jüri Engelbrecht

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book addresses the modelling of mechanical waves by asking the right questions about them and trying to find suitable answers. The questions follow the analytical sequence from elementary understandings to complicated cases, following a step-by-step path towards increased knowledge. The focus is on waves in elastic solids, although some examples also concern non-conservative cases for the sake of completeness. Special attention is paid to the understanding of the influence of microstructure, nonlinearity and internal variables in continua.

With the help of many mathematical models for describing waves, physical phenomena concerning wave dispersion, nonlinear effects, emergence of solitary waves, scales and hierarchies of waves as well as the governing physical parameters are analysed. Also, the energy balance in waves and non-conservative models with energy influx are discussed. Finally, all answers are interwoven into the canvas of complexity.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction: What is all that about?

Abstract

There are several possibilities to acquire knowledge about the physical world. One can observe the phenomena and use the data of observations for explanations. One can conduct special experiments in order to get more specific data. One can construct theories, in which case modelling provides data about the phenomena. The modelling involves mathematical interpretation very much in sense of Galileo Galilei (1564–1642). But mathematics cannot enter just out of the blue sky.

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Chapter 2. What is wave motion?

Abstract

As surprising as it may sound, there is no simple answer to this question. Better not ask what a wave is, but ask what can be said about a wave, explains J. Pierce [197]. The confusion is caused by the wave motion itself, which can be related to propagating disturbances or oscillations. Nevertheless, let us first present some definitions.

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Chapter 3. How to model waves?

Abstract

In Chap. 2, questions about the essence of waves were asked and answers given. On the other hand, every wave needs a medium to propagate. That is why it is essential to start with a description of media (materials). As mentioned in the Introduction, the focus in this book is on waves in solids. Except some comparative examples, waves in fluids or gases are excluded as well as electromagnetic waves. Also, we shall use the concept of continua. If for some reason or another, a discrete model (lattice model) is the starting point, continualization will be used in order to get to a continuous model.

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Chapter 4. What are internal variables? A. Berezovski answers

Abstract

Internal structures (microstructures) appear in solids at different length scales. Generally speaking, their influence on the macromotion can be understood and measured on the macrolevel. However, “there is no unique answer to the question how the microstructure influence can be accounted for in a continuum mechanical model” [132]. In addition, thermodynamical constraints should be taken into account, which is not obvious in many theories. In the previous section, heterogeneity is linked to the real microstructure, like “cells” in the Mindlin theory. One possibility to bring thermodynamics directly into modelling of dynamical phenomena is provided by the concept of internal variables. The idea of internal variables can be traced back to P. Duhem, P. Bridgman and J. Kestin (see historical overview in [161]). A contemporary presentation of the formalism of internal variables is presented by Maugin [154] and Maugin and Muschik [163].

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Chapter 5. What are evolution equations?

Abstract

Let us first return to Sect. 2.2, where the simplest wave equation (2.1) is presented together with its solution (2.7) under initial conditions (2.3). Figure 2.1 shows the fronts $x + c_{0}t = 0$ and $x - c_{0}t = 0$, and it becomes obvious that the wave equation is actually a two-wave equation: one wave propagating to the right, another one to the left.

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Chapter 6. What physical effects are involved?

Abstract

In the previous chapters, attention was given to deriving complicated wave equations or evolution equations. One should certainly ask: “why equations”? Ian Stewart [230] asks this question and gives the answer that “equations are the lifeblood of mathematics, science and technology”, and adds that “… they reveal deep and beautiful patterns and regularities”.

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Chapter 7. What physical mechanisms govern waves in non-conservative systems?

Abstract

Most mathematical models described in the previous chapters are conservative like their prime example, the classical wave equation. The celebrated KdV equation is also conservative and admits infinitely many conserved quantities [2, 59]:

$$\displaystyle\begin{array}{rcl} \int _{-\infty }^{+\infty }u\mathit{dx} = \mathit{const}.,& &{}\end{array}$$

(7.1)

$$\displaystyle\begin{array}{rcl} \int _{-\infty }^{+\infty }u^{2}\mathit{dx} = \mathit{const}.,& &{}\end{array}$$

(7.2)

$$\displaystyle\begin{array}{rcl} \int _{-\infty }^{+\infty }(u^{3} + \frac{1} {2}u_{x}^{2})\mathit{dx} = \mathit{const}.,\ldots & &{}\end{array}$$

(7.3)

These equations express the conservation of mass, momentum, and energy, respectively. The accuracy of a numerical method can be checked by calculating these conserved quantities at every time step.

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Chapter 8. What is complexity of waves?

Abstract

In the previous chapters, several notions and phenomena characteristic to waves in heterogeneous and nonlinear solids were introduced and analyzed. Internal variables were introduced to describe the fields in microstructured solids, flavoured by nonlinearities. The importance of including nonlinear effects was already stressed earlier by the author [66]. These concepts lead to interesting physical effects, e.g., with respect to dispersion, or to changes in wave profiles, or to interaction processes of waves.

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Backmatter

Titel: Questions About Elastic Waves
verfasst von: Jüri Engelbrecht
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-14791-8
Print ISBN: 978-3-319-14790-1
DOI: https://doi.org/10.1007/978-3-319-14791-8