1. USSC3002 Oscillations and Waves Lecture 12 Water Waves Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg http://www.math.nus/~matwml Tel (65) 6874-2749 1

2. VELOCITY POTENTIAL 2 We assume that the water has constant density and that its velocityis irrotational Theorem 1. If u is irrotational on a domain then there exists a velocity potential function such that Proof Choose any path and Stokes Thm 

3. INCOMPRESSIBILITY 3 Proof Follows from the divergence theorem. We further assume that the water is incompressible. Corollary 1 If u is both irrotational and incompressible Theorem 2. If the flow is incompressible then Corollary 2 If u is both irrotational and incompressible Definition (this operator can operate on real or vector valued functions)

4. MATERIAL DERIVATIVE 4 Lemma Along the flow of any particle, whose position is x = x(t), the rate of change of any (real or vector valued) function H(t,x) is given by the material derivative or total derivative defined by Theorem 3 Every fluid satisfies the equation Proof This follows directly from the chain rule. where body force density andthe stress tensor. is the Proof Follows from Newton’s 2 nd Law.

5. NAVIER STOKES EQUATIONS 5 Corollary 4. Newt. fluids satisfy the Navier-Stokes eq. Definition A Newtonian fluid is one satisfying where viscosity coeffs, = identity matrix, p = pressure, = strain tensor. Corollary 5. The incompressible Navier-Stokes eq. are Stokes assumptions where= kinematic viscosity coefficient.

6. BERNOULLI’s EQUATION 6 where V is the potential for the body force Corollary 6. An irrotational flow of an incompressible inviscid Newtonian fluid for which the body force is conservative satisfies Bernoulli’s equation Proof Corollaries 1 and 2  The inviscid assumption  by corollary 5. hence the left side is independent of x and therefore is a function of time. It is customary to absorb C(t) into

7. TIDAL WAVES 7 are also called long waves in shallow water. Their wavelengths are much longer than the water depth so we may ignore the vertical component of acceleration. We will assume that water is inviscid. Since corollary 5  hence undisturbed free (horizontal) surface and Letbe the be the elevation of the water above the point Therefore Henceandare independent of depth.

8. TIDAL WAVES IN A STRAIGHT CHANNEL 8 Fig 1 shows a wave moving in thedirection in a channel whose cross-section area A and surface breadth b vary slowly so that is independent of Then Since the net influx through and  equals the increase of water in the region between these planes planes and so above eqn  rectangle A=bh 

9. SURFACE WAVES 9 If the velocity is small and gravity is the only body force thenand Bernoulli’s equation  and since a particle on the free surface stays there therefore hence also in a long rectangular tank these and boundary cond 

10. TUTORIAL 12. 10 Problem 1. Locate a statement of Stokes theorem and use it to give a detailed proof of Theorem on vufoil 2. and use it to show that cor 4 implies cor 5. Problem 2. Prove corollaries 1 and 2 on vufoil 3. Problem 3. Derive the identity Problem 4. Compute the speed of a tsunami travelling in a single direction for sea depths: 40m, 400m, 4km. Problem 5. (Extra) Compute u for problem 4 if and wavelength = 10h. Hint Useto computefrom Problem 6. (Extra) Show that the group velocity for surface water waves equals c/2.