Abstract
We have already encountered several deficiencies in the theories presented in ChapterĀ 4 that indicate the limitations of the linear approximation. In this chapter we will consider three specific nonlinear models, namely, unsteady one-dimensional gas dynamics, two-dimensional steady gas dynamics and shallow water theory.
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Notes
- 1.
This is an example of what is often known as the kinematic wave equation, namely,
$$\displaystyle{\frac{\partial u} {\partial t} + \frac{\partial } {\partial x}(f(u)) = 0}$$for some function f.
- 2.
For brevity, we will sometimes refer to these two families of characteristics as the positive and negative characteristics, respectively.
- 3.
It is interesting to relate this assumption to that of irrotationality (ExerciseĀ 5.8).
- 4.
We can be more precise about how small these quantities are; if we define \((\eta -h)/h = O(\varepsilon )\), then u will be \(O(\varepsilon \sqrt{gh})\).
- 5.
The step leading toĀ (5.44) is yet another example of the Fredholm Alternative.
- 6.
Scott Russell famously observed āa great wave of elevationā while riding his horse along the towpath of a canal in 1845 ([49]).
- 7.
We now use notationĀ (5.51) rather than x 0ā=āRlā(Ae it) in order to avoid confusion in evaluating the nonlinear terms when it is important to note that Rlā(A 2) ā (RlāA)2.
- 8.
Note that if we had introduced the variable T 1ā=āÉ t in the solution of our model equationĀ (5.47), we would merely have found that \(\partial A/\partial T_{1} = 0\).
- 9.
Note that we are, at this stage of the book, unable to make any comparable statement about the relatively trivial (5.1).
- 10.
As mentioned before, solutions containing discontinuities or shocks will be considered in detail in ChapterĀ 6
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Ockendon, H., Ockendon, J.R. (2015). Nonlinear Waves in Fluids. 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_5
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