1. Lectures 11-12: Gravity waves Linear equations Plane waves on deep water Waves at an interface Waves on shallower water

2. Water waves The free surface of a liquid in equilibrium in a gravitational field is a plane. If the surface is disturbed, motion will occur in the liquid. This motion will be propagated over the whole surface in the forms of waves, called gravity waves. Let us consider waves on the surface of deep water. We neglect viscosity, as there are no solid boundaries, at which it could cause marked effects; and we also neglect compressibility and surface tension. z x air water 0 The governing equations are

3. Small-amplitude waves The physical parameters characterising the wave-motion are the amplitude of oscillations of fluid particles, a, the wavelength, λ, and the period of oscillations, T. The velocity of a fluid particle,. The significant change of the velocity occurs at a distance λ. The unsteady term can be estimated as. The non-linear term can be estimated as. This means, For the small-amplitude waves, when, the non-linear term is negligibly small.

4. Irrotational motion The linearised Navier-Stokes equation is For the oscillatory motion, the average position of a fluid particle is z=0, the average velocity is 0. The average vorticity must be also 0. As the vorticity is time-independent, it must be 0 at every moment (to make the average value being 0). -- small-amplitude wave motion is an irrotational flow Hence, the velocity field can be represented as. The velocity potential φ is determined by the Laplace equation:

5. Boundary conditions: on the surface ( ζ is the surface elevation above the flat position), pressure is atmospheric. waves on deep water, the fluid velocity is bounded everywhere. 1. We use Bernoulli’s equation to rewrite the first boundary condition in terms of φ. On the surface, Here, we neglect the term containing v 2, as it originates from the non-linear term, which is small for the small- amplitude waves. The velocity potential φ is a technical variable that does not have the physical meaning and is needed only to find the velocity ( ). The velocity does not change if the potential is redefined as follows

6. This allows us to rewrite the Bernoulli’s equation as an equation for the elevation ζ : 2. On the surface, 3. This results in the following boundary condition on the surface 4. Using the Taylor’s series over small ζ, the leading term of the above equation is

7. Finally, we have got the following mathematical problem Equation: Boundary conditions:

8. Plane wave solution Let us seek the solution in the form of a single plane wave, Here, -- the circular frequency (1/ T would be the regular frequency) -- the wavenumber -- the wave speed Substitution into the Laplace equation gives As the velocity is bounded, Illustration: http://www.youtube.com/ watch?v=aKGgsLHN1dc

9. Hence, Or, in terms of velocity,

10. Wave dispersion Applying the boundary condition on the liquid’s surface, we obtain the following dispersion relation The waves of different lengths travel at different speeds (see the video). Video: http://www.youtube.com/watch?v=lWi_KpBy8kU (note that long waves travel faster)http://www.youtube.com/watch?v=lWi_KpBy8kU

11. Phase and group velocities is called the phase velocity, the velocity of travelling of any given phase of a wave. is called the group velocity, this is the velocity of the motion of the wave packet. The red dot moves with the phase velocity, while the green dot moves with the group velocity.

12. Waves at an interface z x 0 ρ2ρ2 ρ1ρ1 Consider gravity waves at an interface between two very deep liquid layers. The density of the lower liquid is ρ 1 and the density of the upper liquid is ρ 2. ρ 2 >ρ 1 Motion is irrotational. The governing equations are (lower liquid) (upper liquid) Boundary conditions: at infinity,, fluids’ velocities are bounded At interface,,

13. We need to re-write the boundary conditions on the interface in terms velocity potential. Applying Bernoulli’s equation at an interface, we have or, redefining φ 1 and φ 2 The pressure is continuous at interface, i.e.

14. The z -component of the velocity is continuous as well, i.e. As a result, we have two following boundary conditions at interface Expanding over powers of small ζ and leaving only the linear terms, we finally obtain

15. Now, seek solution in the form of a plane wave Solving the Laplace equations, we get The boundedness of the velocities leads to and That is,

16. Let us now use the boundary conditions at the interface The resultant dispersion relation The phase and group speeds are The fluid velocity ( A 1 cannot be determined from the linear equations):

17. Water/air interface ρ 2 << ρ 1 dispersion relation and the expressions for the phase and group velocities become as those for the gravity waves on a free surface of deep water

18. Waves on shallow water z -h x 0 The liquid is now bounded by a rigid wall. The fluid velocity is determined by the Laplace equation: Boundary conditions: -- no fluid penetration through the wall -- pressure is atmospheric The boundary condition at a free surface can be rewritten as

19. Using the boundary condition at the wall gives B =0. That is, Solving the Laplace equation, we obtain Seek the velocity potential in the form of a plane wave, Using the boundary condition at an interface produces the dispersion relation Consequently, the phase and group velocities are

20. Deep water and long waves Let us find the dispersion relation and the phase and group velocities for the cases (a) of very deep water ( h / λ >>1) and (b) of long waves ( h / λ <<1). (a) Deep water, h / λ >>1, or kh > >1 (b) Long waves, h / λ <<1, or kh << 1

21. There are two linearly independent solutions of equation (1), and Since equation (1) is linear, any linear combination of solutions (2) is a solution of (1). The general solution can be written as Seeking solution in the exponential form,, gives the auxiliary equation, Appendix: solution of the amplitude equation The roots of the auxiliary equation: (1) This is the linear ordinary differential equation with constant coefficients. (2) Here, A and B are unknown arbitrary constants (to be determined from the boundary conditions). (3)

22. Let us show that the general solution of (1) can be also written in the following form Using the definitions of hyperbolic functions, and re-defining the constants we can prove that (4) is equivalent to (1). Similarly, it can be easily proved that the general solution of (1) can be also written in the forms A 1 and B 1 are the arbitrary constants, different from A and B. (4) or (5) (6)

23. Conclusion: - (3), (4), (5), and (6) are the equivalent forms of the general solution of equation (1); all these forms are the linear combinations of basis solutions (2). All these expressions include two unknown arbitrary constants to be determined from the boundary equations, but the final solution (with determined constants) is unique. - It is recommended to chose such a form of the general solution that can simplify derivations. For instance, (5) should be convenient when one of the boundary conditions is imposed at z =- h, where, which immediately gives one constant, A 3. (3), (4), and (6) can be also used and should produce the same final expression, but intermediate derivations can be lengthier.