| In this thesis,we consider wave-current interactions of small amplitude two-dimensional steady periodic equatorial water waves,which propagate over a flat bed and beneath a free surface.The mathematical study of equatorial fluids has attracted a great deal of attention in the past decade because its further study is crucial for predicting large-scale natural phenomena.In addition,since the equator acts as a natural waveguide,the waves in the equatorial region are mainly banded,so we can approximate the equatorial gravitational water waves to two dimensions.On this basis,it is assumed that the flow has piecewise linear vorticity function.The existence of vorticity makes the strict mathematical study of water waves extremely difficult.In the study of water waves,the existence of solutions of water waves with constant vorticity has not been solved until the seminal work of Constantin and Strauss in 2004,and the case of discontinuous vorticity was also dealt with by these two professors in 2011.In particular,we consider the case of fixed mean-depth.Fixed mean-depth is a physical perspective proposed by Henry.When the mass flux is fixed,the mean-depth of the solution of the water wave problem changes.In order to further extend the scope of analysis under the condition of fixed mean-depth,after analyzing the existence of laminar flow solutions,we derive the dispersion relation of small amplitude waves caused by the discontinuous linear vorticity distribution under the condition of fixed mean-depth.From the perspective of natural science,dispersion relation is very important because it is related to wavelength,wave velocity,fixed mean-depth,the location of the jump in vorticity and vorticity distribution itself.On the other hand,the dispersion relationship is also very important from the perspective of analysis,because it provides conditions for laminar bifurcation.The additional benefit of dispersion relations is that resonance analysis of water waves can be performed. |
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