Abstract
The archetypal hyperbolic equation is the wave equation in one spatial dimension. It governs phenomena such as the propagation of longitudinal waves in pipes and the free transverse vibrations of a taut string. Its relative simplicity lends itself to investigation in terms of exact solutions of initial and boundary-value problems. The main result presented in this chapter is the so-called d’Alembert solution, expressed within any convex domain as the superposition of two waves traveling in opposite directions with the same speed. Some further applications are explored.
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Notes
- 1.
As discussed in Sect. 7.1.3, this applies to strong signals as well.
- 2.
A subset of \({\mathbb R}^n\) is convex if whenever two points belong to the set so does the entire line segment comprised by them. The requirement of convexity is sufficient for the validity of the statement. If the domain is not convex, we may have different functions in different parts of the domain, and still satisfy the differential equation.
- 3.
- 4.
See [3], p. 42.
- 5.
Clearly, we are tacitly invoking an argument of uniqueness, which we have not pursued.
- 6.
See [6], p. 50, a text written by one of the great Russian mathematicians of the 20th century.
- 7.
This simplified model is not realistic for the actual Slinky for many reasons. We are only using it as a motivation for a well defined problem in linear one-dimensional elasticity.
- 8.
References
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Knobel R (2000) An introduction to the mathematical theory of waves. American Mathematical Society, Providence
Sneddon IN (1957) Elements of partial differential equations. McGraw-Hill, Republished by Dover, 2006
Sobolev SL (1989) Partial differential equations of mathematical physics. Dover, New York
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Epstein, M. (2017). The One-Dimensional Wave Equation. In: Partial Differential Equations. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-55212-5_8
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DOI: https://doi.org/10.1007/978-3-319-55212-5_8
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