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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 [email protected] http://www.math.nus/~matwml Tel (65) 6874-2749 1 VELOCITY POTENTIAL We assume that the water has constant density and that its velocity u (u1 , u2 , u3 ) is irrotational u3 u2 j k i x2 x3 u1 u3 curl u u det x1 x2 x3 x3 x1 0 u2 u1 u1 u2 u3 x 1 x2 Theorem 1. If u is irrotational on a domain R then there exists a velocity potential function : R such that p and Proof Choose any x path ( p, x ) Stokes Thm u grad x ( x) u dy y ( p , x ) x 2 1 2 3 INCOMPRESSIBILITY We further assume that the water is incompressible. Theorem 2. If the flow is incompressible then div u u u1 x1 u2 x2 u3 x3 0 Proof Follows from the divergence theorem. Corollary 1 If u is both irrotational and incompressible u , 0. 1 x1 Definition u u u 2 x2 u 2 x2 (this operator can operate on real or vector valued functions) Corollary 2 If u is both irrotational and incompressible ( 12 u u) u u 3 MATERIAL DERIVATIVE 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 DH Dt H t u H Proof This follows directly from the chain rule. Theorem 3 Every fluid satisfies the equation Du F where F F (t , x) is the Dt body force density and (t , x) the stress tensor. Proof Follows from Newton’s 2nd Law. 4 NAVIER STOKES EQUATIONS Definition A Newtonian fluid is one satisfying Stokes assumptions pI ( u ) I 2 D where I = identity matrix, p = pressure, , viscosity coeffs, D 12 (u (u)T ) = strain tensor. Corollary 4. Newt. fluids satisfy the Navier-Stokes eq. Du Dt F p ( ( u)) (2 D) Corollary 5. The incompressible Navier-Stokes eq. are F 1 p u, u 0 where = kinematic viscosity coefficient. Du Dt 5 BERNOULLI’s EQUATION Corollary 6. An irrotational flow of an incompressible inviscid Newtonian fluid for which the body force is conservative satisfies Bernoulli’s equation p u u V 1 2 t C(t ) where V is the potential for the body force F V . Proof Corollaries 1 and 2 ( p 12 u u V t ) 1 p u u F u t u by corollary 5. The inviscid assumption 0 hence the left side is independent of x and therefore is a function of time. It is customary to absorb C(t) into . 6 TIDAL WAVES 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 F3 g D u3 p 1 corollary 5 0 D t F3 p g x3 hence p g x3 constant. Let x3 0 be the undisturbed free (horizontal) surface and (t, x1, x2 ) be the elevation of the water above the point ( x1 , x2 ,0) Therefore p(t , x) pair g ( (t , x1 , x2 ) x3 ). x1 Hence and g independent of depth. D u1 Dt D u2 Dt g x2 are 7 TIDAL WAVES IN A STRAIGHT CHANNEL Fig 1 shows a wave moving in the x1 direction in a channel whose cross-section area A and surface breadth b vary slowly so that u1 is independent of x2 D u1 u1 u1 u1 u1 u1 Then g x1 D t t u1 x1 u2 x2 u3 x3 t x3 Since the net influx through x1 planes x1 s and x1 s s, equals the increase of water in s the region between these planes s and | b | A x ( Au ) b t ( Ag ) b so above eqn x x t 1 2 1 rectangle A=bh 2 1 2 x1 2 2 1 c2 t 2 h , c gh 8 SURFACE WAVES If the velocity is small and gravity is the only body force then u u 0 and V g therefore Bernoulli’s equation t 1 g free surface and since a particle on the free surface stays there 0 D Dt hence also ( (t , x1 , x2 ) x3 ) x1 t 1 g x1 t u3 1 g fs 1 g x1 fs t 1 x1 u t x1 2 t 2 g fs 2 x2 1 u1 g t u x3 u3 fs 0 in a long rectangular tank these and boundary cond 2 h g 2 2 2 C cosh ( x3 h) cos ( x1 ct ), c 2 tanh 9 TUTORIAL 12. Problem 1. Locate a statement of Stokes theorem and use it to give a detailed proof of Theorem on vufoil 2. Problem 2. Prove corollaries 1 and 2 on vufoil 3. Problem 3. Derive the identity (2 D) u ( u ) and use it to show that cor 4 implies cor 5. 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 cos (t k x1 ) and wavelength = 10h. Hint Use u 0 to compute u3 from u1 . Problem 6. (Extra) Show that the group velocity for surface water waves equals c/2 . 10