Abstract
It is not easy to see how a uniform or nearly uniform wave train can realistically emerge from some general initial condition or from a realistic forcing unless the initial condition or the forcing is periodic. That turns out not to be the case, and the ideas we have so far developed about group velocity and energy propagation turn out to be invaluable in getting to the heart of the general question of wave signal propagation. Indeed, it is the very dispersive nature of the wave physics (i.e., the dependence of the phase speed on the wave number) that is responsible for the emergence of locally nearly periodic solutions. This can be seen by examining the solution to the general initial value problem. This was first done by Cauchy in 1816. It was also solved at the same time by Poisson. The problem was considered so difficult at that time that the solution was in response to a prize offering of the Paris Academie (French Academy of Sciences). Now it is a classroom exercise.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Jeffreys J, Jeffreys BS (1962) Methods of mathematical physics. Cambridge University Press, Cambridge, pp 716 (especially Chapters 14 and 17)
Lighthill J (1978) Waves in fluids. Cambridge University Press, Cambridge, pp 504 (especially Chapter 3, Section 3.7)
Morse PM, Feshbach H (1953) Methods of theoretical physics, vol 1. McGraw-Hill, New York, pp 997 (especially Section 4.8)
Stoker JJ (1957) Water waves. Interscience, New York, pp 567
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Pedlosky, J. (2003). The Initial Value Problem. In: Waves in the Ocean and Atmosphere. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05131-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-662-05131-3_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-05564-5
Online ISBN: 978-3-662-05131-3
eBook Packages: Springer Book Archive