Abstract
Looking back at the models derived in the last chapter, we see that many of them comprise linear partial differential equations in time and at least one space variable, together with linear boundary conditions. Moreover, in most of the equations, many of the terms have constant coefficients. We therefore start this chapter by reviewing the mathematical methodologies that are available for the analysis of such models.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Separation of variables can sometimes be applied to variable-coefficient equations, as will be seen later.
- 2.
The negative sign is taken in the exponent for reasons that will become apparent later.
- 3.
- 4.
They can also be observed by throwing a stone into a large deep pond.
- 5.
This transform can be derived formally from the Fourier transform (4.17) as described in [43].
- 6.
Note that with a conventional power series asymptotic expansion \(\phi \sim \sum _{0}^{\infty }\varepsilon ^{n}\phi _{n}\), we can define the terms recursively via \(\phi _{0} =\lim _{\varepsilon \rightarrow 0}\phi\), \(\phi _{1} =\lim _{\varepsilon \rightarrow 0}(\phi -\phi _{0})/\varepsilon\), etc. Equally for an expansion \(\phi \sim e^{u/\varepsilon }\sum _{n=0}^{\infty }A_{n}\varepsilon ^{n}\), where u is real, we can define \(u =\lim _{\varepsilon \rightarrow 0}\varepsilon \log \phi\). However, in this case with u real, no such definition is possible because u now measures the oscillations in ϕ rather than its magnitude. The best that can be done is to say that \(\vert \nabla u\vert ^{2} =\lim _{\varepsilon \rightarrow 0}(-\varepsilon ^{2}\nabla ^{2}\phi /\phi )\).
- 7.
Note that equation (4.44) can be thought of as “conservation of wave number” if ω k is interpreted as “wave number flux”.
- 8.
Note that had ϕ been written as Rl (Φ e i ω t), the first term would have been unacceptable.
- 9.
The term used depends on whether we are referring to light waves or sound waves.
- 10.
The amplitude of the reflected wave would be different if the boundary condition was of the form α Φ +β(∂ Φ∕∂ n) = 0.
- 11.
A quasi-periodic solution consists of a sum of periodic terms with non-commensurate periods.
- 12.
In optics, it is traditional to work with the refractive index which is proportional to 1∕c and is negative when \(\theta _{r}\) is negative.
- 13.
Note that we are implicitly assuming that the initial data is dependent only on x and not on X. If the initial data was just a function of X, \(\varepsilon\) could be removed from the problem by rescaling x and t, and the method of multiple scales could then tell us about the far field as \(x,t \rightarrow \infty \).
- 14.
For a definition of a regular perturbation, see Hinch [26].
- 15.
The contribution to the integral corresponding to complex k 2 can be shown to be negligible as \(x,y \rightarrow \infty \).
- 16.
This is a very useful engineering technique in which the fluid is divided into regions, possibly large ones, and estimates are made for the global changes in mass, momentum and energy in these regions in terms of their boundary values.
- 17.
The logarithm in (4.67) is a Green’s function for this problem [43].
- 18.
More generally, the lift on an arbitrary thin wing is ρ 0 U Γ∕β, where Γ is the circulation around the wing in the incompressible case.
- 19.
This function is a Riemann function for this problem [43].
- 20.
Of course, if we are interested in waves sufficiently close to a sharp corner of some boundary, the method we are about to describe will never be useful, because then L can be arbitrarily small.
- 21.
In spatial dimension n ≥ 2, the characteristic surfaces or wavefronts of a hyperbolic system can be defined by generalising the ideas of Section 4.1. They are the only manifolds of dimension (n − 1) across which jumps in the variables can occur and these jumps can be localised along bicharacteristic curves or “rays” in the characteristic manifold as described in [43].
- 22.
To prove this orthogonality result, put z = ω n r∕c in (†) and then multiply by 2r 2 J′0(ω n r∕c) and integrate from r = 0 to a, integrating by parts once.
- 23.
This is an example of the Born approximation in which the solution of Helmholtz’ equation in a medium that is only weakly inhomogeneous is expanded in such a power series.
References
Acheson, D. J. (1990). Elementary fluid dynamics. Oxford: Oxford University Press.
Arscott, F. M. (1964). Periodic differential equations. An introduction to Mathieu, Lamé and allied functions. Elmsford, NY: Pergamon.
Billingham, J., & King, A. C. (2000). Wave motion. Cambridge: Cambridge University Press.
Born, M., & Wolf, E. (1980). Principles of optics (6th edn.). Elmsford, NY: Pergamon.
Chapman, S. J., Lawry, J. M. H., Ockendon, J. R., & Tew, R. H. (1999). On the theory of complex rays. SIAM Review, 41, 417–509.
Courant, R., & Hilbert, D. (1962). Methods of mathematical physics (Vol. I). New York: Interscience.
Drazin, P. G., & Reid, W. H. (1981). Hydrodynamic stability. Cambridge: Cambridge University Press.
Hinch, E. J. (1991). Perturbation methods. Cambridge: Cambridge University Press.
Hodges, C. H., & Woodhouse, J. (1986). Theories of noise and vibration transmission in complex structures. Reports on Progress in Physics, 49, 107–170.
Kaouri, K., Allwright, D. J., Chapman, C. J. & Ockendon, J. R. (2008). Singularities of wavefields and sonic boom. Wave Motion, 45, 217–237.
Kevorkian, J., & Cole, J. D. (1981). Perturbation methods in applied mathematics. New York: Springer-Verlag.
Lighthill, M. J. (1978). Waves in fluids. Cambridge: Cambridge University Press.
Mouhot, C., & Villani C. (2010). Landau damping. Journal of Mathematical Physics, 51, 015204.
Ockendon, J., Howison, S., Lacey, A., & Movchan, A. (1999). Applied partial differential equations. Oxford: Oxford University Press.
Ockendon, J. R., & Tew, R. H. (2012). Thin layer solutions of the Helmholtz and related equations. SIAM Review, 54, 3–51.
Santosa, F., & Symes, W. W. (1991). A dispersive effective medium for wave propagation in periodic composites. SIAM Journal on Applied Mathematics, 51, 984–1005.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer Science+Business Media New York
About this chapter
Cite this chapter
Ockendon, H., Ockendon, J.R. (2015). Theories for Linear Waves. In: Waves and Compressible Flow. Texts in Applied Mathematics, vol 47 . Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3381-5_4
Download citation
DOI: https://doi.org/10.1007/978-1-4939-3381-5_4
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-3379-2
Online ISBN: 978-1-4939-3381-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)