Analytical solutions of shallow-water wave equations

Aydın, Baran
Analytical solutions for the linear and nonlinear shallow-water wave equations are developed for evolution and runup of tsunamis –long waves– over one- and two-dimensional bathymetries. In one-dimensional case, the nonlinear equations are solved for a plane beach using the hodograph transformation with eigenfunction expansion or integral transform methods under different initial conditions, i.e., earthquake-generated waves, wind set-down relaxation, and landslide-generated waves. In two-dimensional case, the linear shallow-water wave equation is solved for a flat ocean bottom for initial waves having finite-crest length. Analytical verification of source focusing is presented. The role of focusing in unexpectedly high tsunami runup observations for the 17 July 1998 Papua New Guinea and 17 July 2006 Java Island, Indonesia tsunamis are investigated. Analytical models developed here can serve as benchmark solutions for numerical studies.


Analytical modeling of nonlinear evolution of long waves
Aydın, Baran; Kanoğlu, Utku (2015-06-22)
We present an initial-boundary value problem formulation for the solution of the nonlinear shallow-water wave (NSW) equations. We transform the nonlinear equations into a linear problem by using the Carrier-Greenspan transformation. Then, we obtain the solution through the separation of variables method rather than integral transform techniques, which is the usual practice (Carrier et al., J Fluid Mech 2003; Kanoglu, J Fluid Mech 2004). This formulation allows the use of any physically realistic initial wav...
Effective-mass Klein-Gordon-Yukawa problem for bound and scattering states
Arda, Altug; Sever, Ramazan (2011-09-01)
Bound and scattering state solutions of the effective-mass Klein-Gordon equation are obtained for the Yukawa potential with any angular momentum l. Energy eigenvalues, normalized wave functions, and scattering phase shifts are calculated as well as for the constant mass case. Bound state solutions of the Coulomb potential are also studied as a limiting case. Analytical and numerical results are compared with the ones obtained before. (C) 2011 American Institute of Physics. [doi:10.1063/1.3641246]
Analytical solutions for evolution and runup of longwaves over a sloping beach
Ceylan, Nihal; Kanoğlu, Utku; Department of Engineering Sciences (2019)
The initial value problem of the linear evolution and runup of long waves on a plane beach is analyzed analytically. The shallow water-wave equations are solved by integral transform and eigenvalue expansion methodologies. The results from linear solutions are compared with the solution of the nonlinear shallow water-wave equations confirming the runup invariance, i.e. nonlinear and linear theories produce same maximum runup. Then, existing analytical nonlinear solution for shoreline motion is implemented f...
Yıldırım, Raif Orhan (1994-10-01)
Elastic wave propagation through an area discontinuity of two dissimilar, bonded, semi-infinite circular rods is investigated analytically. In particular, the variations of the coefficients of stress reflection and transmission are determined in terms of the nondimensional cross sectional area and mechanical impedance parameters. The coefficients of energy flux reflection and transmission are also included. Then, the case is generalized to include a rigid mass attached at the discontinuity.
On general form of the Tanh method and its application to nonlinear partial differential equations
Hamidoğlu, Ali (American Institute of Mathematical Sciences (AIMS), 2016-6)
The tanh method is used to compute travelling waves solutions of one-dimensional nonlinear wave and evolution equations. The technique is based on seeking travelling wave solutions in the form of a finite series in tanh. In this article, we introduce a new general form of tanh transformation and solve well-known nonlinear partial differential equations in which tanh method becomes weaker in the sense of obtaining general form of solutions.
Citation Formats
B. Aydın, “Analytical solutions of shallow-water wave equations,” Ph.D. - Doctoral Program, Middle East Technical University, 2011.