Editing Capillary waves
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====Defining the problem==== | ====Defining the problem==== | ||
Three contributions to the energy are involved: the [[surface tension]], gravity, and | Three contributions to the energy are involved: the [[surface tension]], gravity, and hydrodynamics. The parts due to surface tension (again the derivatives are taken to be small) and gravity are exactly as above. | ||
The new contribution involves the [[kinetic energy]] of the fluid: | The new contribution involves the [[kinetic energy]] of the fluid: | ||
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where <math>v</math> is the module of the velocity field <math>\vec{v}</math>. | where <math>v</math> is the module of the velocity field <math>\vec{v}</math>. | ||
(Again, we are neglecting the flow of the gas above for simplicity.) | (Again, we are neglecting the flow of the gas above for simplicity.) | ||
====Wave solutions==== | ====Wave solutions==== | ||
Let us suppose the surface of the liquid is described by a traveling plane wave: | Let us suppose the surface of the liquid is described by a traveling plane wave: |