D’Alembert’s solution

Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

Series	Geophysical References Series
Title	Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author	Enders A. Robinson and Sven Treitel
Chapter	1
DOI	http://dx.doi.org/10.1190/1.9781560801610
ISBN	9781560801481
Store	SEG Online Store

Jean-le-Rond d’Alembert was born in Paris on 16 November 1717, and died there on 29 October 1783. He enunciated the principle known by his name (d’Alembert, 1743). D’Alembert’s principle allows the reduction of a dynamic problem into a static problem. This feat is accomplished by introducing a fictitious force equal in magnitude to the product of the mass of the body and its acceleration but in a direction opposite to the direction of acceleration. The result can be inferred from Newton’s third law of motion, the full consequences of which had not been realized previously. This principle enabled mathematicians to obtain the differential equations of motion for any rigid system.

Geophysicists remember d’Alembert as the one who first found and solved the wave equation (Robinson and Clark, 1987b). The 1D wave equation is

${\begin{aligned}{\frac {{\partial }^{\rm {2}}u\left(x,t\right)}{\partial x^{\rm {2}}}}{\rm {=}}{\frac {\rm {1}}{v^{\rm {2}}}}{\frac {{\partial }^{\rm {2}}u\left(x,t\right)}{\partial t^{\rm {2}}}},\end{aligned}}$

(2)

where v is a constant. He was able to show that

${\begin{aligned}u\left(x,t\right){\rm {=}}f\left(x{\rm {+}}vt\right){\rm {+}}g\left(x-vt\right)\end{aligned}}$

(3)

is the general solution of the 1D wave equation, where $f\left(\cdot \right)$ and $g\left(\cdot \right)$ are arbitrary functions. This solution is known as the d’Alembert formula. The letter f is the standard notation for a mathematical function, and this usage will always be so. However, we already have used the letter f for the frequency, and that can lead to confusion. An alternative would be to denote frequency by the Greek letter v (lowercase nu), which is done in Sheriff’s Encyclopedic Dictionary of Applied Geophysics (Sheriff, 2002). However, the Greek nu looks too much like the letter v, which is used for velocity. Because the meaning should be clear from the context, in this book we will use the letter f both as a symbol for a mathematical function and as a symbol to denote frequency. A rose is a rose is a rose. In other words, things are what they are.

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Table of Contents (book)

Also in this chapter

• Introduction
• Wavefronts and raypaths
• One-dimensional waves
• Sinusoidal waves
• Phase velocity
• Wave pulses
• Geometric seismology
• The speed of light
• Huygens’ principle
• Reflection and refraction
• Ray theory
• Fermat’s principle
• Fermat’s principle and reflection and refraction
• Diffraction
• Analogy
• Appendix A: Exercise

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