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[7] Choi, W., & Camassa, R., 1996. “Weakly nonlinear internal waves in a twofluid system,” Journal of Fluid Mechanics, vol. 313, pp. 83–103. [8] Choi, W., & Camassa, R., 1999. “Fully nonlinear internal waves in a twofluid system,” Journal of Fluid Mechanics, vol. 396, pp. 1–36. [9] Davis, R. E., & Acrivos, A., 1967. “Solitary internal waves in deep water,” Journal of Fluid Mechanics, vol. 29, pp. 593–607. [10] Driscoll, T., Schwarz-Christoffel toolbox for Matlab, http://www.math.udel.edu/˜driscoll/software. [11] Green, A. E. & Naghdi, P. M., 1976. “A derivation of equations for wave propagation in water of variable depth,” Journal of Fluid Mechanics, vol. 78, pp. 237–246. [12] Guazzeli, E., Rey, V. & Belzons, M., 1992. “Higher order Bragg reflection of gravity surface waves by periodic beds,” Journal of Fluid Mechanics, vol. 245, pp. 301–317. [13] Jo, T.-C. & Choi, W., 2002. “Dynamics of strongly nonlinear internal solitary waves in shallow water,” Studies in Applied Mathematics, vol. 109, pp. 205– 227. [14] Joseph, R. I., 1977. “Solitary waves in finite depth fluid,” Journal of Physics A, vol. 10, pp. L225–L227. [15] Keener, J. P., 2000. Principles of Applied Mathematics: Transformation and Approximation, Westview Press. 99
[16] Koop, C. G. & Butler, G., 1981. “An investigation of internal solitary waves in a two-fluid system,” Journal of Fluid Mechanics, vol. 112, pp. 225–251. [17] Kubota, T., Ko, D., & Dobbs, L., 1978. “Propagation of weakly non linear internal waves in a stratified fluid of finite depth,” AIAA Journal Hydrodynamics, vol. 12, pp. 157–165. [18] Lamb, H., 1932. Hydrodynamics, Dover. [19] Matsuno, Y., 1992. “Nonlinear evolutions of surface gravity waves on fluid of finite depth,” Physical Review Letters, vol. 69, pp. 609–611. [20] Matsuno, Y., 1993. “A unified theory of nonlinear wave propagation in twofluid systems,” Journal of the Physical Society of Japan, vol. 62, pp. 1902– 1916. [21] Muñoz, J. C. & Nachbin, A., 2004. “Dispersive wave attenuation due to orographic forcing,” SIAM Journal of Applied Mathematics, vol. 64, Issue 3, pp. 977–1001. [22] Muñoz, J. C. & Nachbin, A., 2005. “Stiff microscale forcing and solitary wave refocusing,” SIAM Multiscale Modeling and Simulation, vol. 3, issue 3, pp. 680–705. [23] Muñoz, J. C. & Nachbin, A., 2006. “Improved Boussinesq-type equations for highly-variable depths,” IMA Journal of Applied Mathematics, vol. 71, pp. 600–633. [24] Nachbin, A., 2003. “A terrain-following Boussinesq system,” SIAM Journal on Applied Mathematics, vol. 63, pp. 905–922. 100
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INSTITUTO NACIONAL DE MATEMÁTICA P
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Acknowledgements First, I would lik
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4.1 Hierarchy of one-dimensional mo
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Resumo É obtido um modelo reduzido
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when the steepening of a given wave
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in particular that of solitary wave
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Chapter 2 Derivation of the reduced
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demands that η t + u i η x = w i
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function f (x, z, t), let its assoc
Page 19 and 20:
Substitution of w 1 into Eq. (2.2)
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i. e. u 1z =βw 1x . Hence ( ) u (0
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fluid layer. 2.2 Connecting the upp
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Furthermore, [√ ( √ )] βφt x,
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The problem in conformal coordinate
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problem (2.22). Therefore, P x =−
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assumption was made on the wave amp
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Its linearization around the undist
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c 0 chosen, there will be a right-
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and for similar reasons η t +η x
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3. For system (2.28) a similar unid
Page 41 and 42:
Chapter 3 A higher-order reduced mo
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expression above as Let us express
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and substituting in the asymptotic
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and it is easy to take x-derivative
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which together with η tt = ( (1−
Page 51 and 52:
For the weakly nonlinear model of o
Page 53 and 54: Again, ω2 k 2 large. → 0 as k→
Page 55 and 56: 2.6 2.4 2.2 ω/k 2 1.8 1.6 1.4 1.2
Page 57 and 58: Chapter 4 Numerical results 4.1 Hie
Page 59 and 60: Linear Weakly nonlinear Strongly no
Page 61 and 62: is to approximate theξ-derivative
Page 63 and 64: inary axis (where the eigenvalues o
Page 65 and 66: 2.4 2.2 RK4 AM3 AM4 2 1.8 1.6 |R(z)
Page 67 and 68: Also, recall that K1=E(η n , u n 1
Page 69 and 70: • T is the convolution matrix for
Page 71 and 72: The AM4 scheme used is: η n+1 j V
Page 73 and 74: 0.6 0.4 0.2 η 0 −0.2 40 35 30 25
Page 75 and 76: 0.6 0.5 0.4 0.3 η 0.2 0.1 0 −0.1
Page 77 and 78: where c 1 and c 2 are two functions
Page 79 and 80: 0.5 0.4 0.3 0.2 η 0.1 0 −0.1 −
Page 81 and 82: 0.5 0.4 0.3 0.2 η 0.1 0 −0.1 −
Page 83 and 84: 0.5 0.4 0.3 0.2 η 0.1 0 −0.1 −
Page 85 and 86: 0.6 0.5 0.4 0.3 η 0.2 0.1 0 −0.1
Page 87 and 88: 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0
Page 89 and 90: −η 0.1 0.08 0.06 0.04 0.02 0 60
Page 91 and 92: Conclusions and future work In the
Page 93 and 94: Appendix A Approximation for the ho
Page 95 and 96: thenω 0 satisfies the Neumann prob
Page 97 and 98: The inverseT ofT has a symbol−i t
Page 99 and 100: Setξ=πξ/l. In the new coordinate
Page 101 and 102: Takingξ-derivatives, φ ξ (ξ,ζ)
Page 103: Bibliography [1] Artiles, W. & Nach
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