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Unformatted text preview: Orignally written for Phys374, Prof. Ted Jacobson Department of Physics, University of Maryland Gravity waves on water Waves on the surface of water can arise from the restoring force of gravity or of surface tension, or a combination. For wavelengths longer than a couple of centimeters surface tension can be neglected, and the waves are called gravity waves . Short wavelength surface waves are called capillary waves . Dimensional analysis told us that the speed of gravity waves with wavelength λ much shorter than the depth of the body of water but still long enough to ignore surface tension must be proportional to √ gλ , while those with wavelength much longer than the depth have speed proportional to √ gh . In this supplement we’ll derive the speed for the general case. Actually we’ll do more: we’ll find the motion of the water, as a function of time and depth, for small amplitude waves. Incompressible flow Mass conservation is expressed by the continuity equation, ∂ t ρ + ∇· ( ρ v ) = 0. If, like water (but unlike, say, a gas), the fluid hardly compresses, then the mass density ρ is nearly constant in space and time, so the continuity equation reduces to ∇· v = 0. By Gauss’s theorem this implies the vanishing of the fux integral R ∂ V v · dS for any volume V . That is, the net volume flow into or out of any region is zero. Equivalently, the flow is volume-preserving. Fluid equation of motion The mass and volume of a portion of fluid is constant as it is carried along in an incompressible flow. Newton’s law F = m a thus applies to each infinitesimal “fluid element” of volume δV , which has a fixed mass δm = ρδV . 1 The acceleration of a fluid element with trajectory r ( t ) is the rate of change of its velocity v ( r ( t ) ,t ) as it is carried along by the flow. This receives contributions both from any explicit time dependence in v and from the fact that the flow may carry it into a location where the velocity field is different. That is, the acceleration is the convective derivative of the velocity, a = ∂ t v + ( v · ∇ ) v . (1) 1 For compressible flows one must use the more general form of Newton’s second law, F = d p /dt , where p is the momentum of a fluid element, which can change both because of acceleration and because of mass flow. See § 11.3,6 of the textbook for details. 1 (See section 5.5 of the textbook for a discussion of the convective derivative.) The force on a fluid element arises from internal forces of pressure p and viscosity μ , as well as external forces such as gravity. The pressure force is-∇ pδV (cf. section 5.2 of the textbook), and the viscous force is μ ∇ 2 v δV (cf. section 11.6 of the textbook, although I’m not convinced that the derivation there is really valid). The gravitational force is- δm ∇ Φ, where Φ is the gravitational potential per unit mass....
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