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Question 1 (Unit 12) - 25 marks Two incompressible, inviscid fluids occupy the regions - < 2 < 0 and 0 < 2 < H, respectively, where 2 is measured vertically upwards. The upper fluid has density P1 and is bounded above by a fixed ceiling. The lower fluid has density P2, where P2 > P1, and is bounded below by a fixed floor at z = -h. In this question, you are asked to investigate the propagation of small-amplitude waves at the interface between the two fluids. The velocity potentials are denoted by 01 (x, z, for the upper fluid and for the lower fluid. Under the assumptions made on page 10 of Unit 12, the following results may be derived. (You are not asked to derive them.) 2201 2201 = (1) 061 062 at 2 = 0 (to first order), 8z 8z (2) P1 Ot² + 8z = at2 8z at 0 (to first order), 2 = (3) where g is the magnitude of the acceleration due to gravity. (a) (i) Write down a condition on 02 similar to that given in Equation (1) for 01. [1] (ii) Write down conditions on 01 at 2 = H and on O2 at 2 = h, with an explanation. [3] (iii) Explain the physical basis for Equation (2). [1] (b) To represent simple sinusoidal waves, the velocity potentials are taken to be $1(x,z,t)= Z1(z)cos(kx-wt), = $2(x,2z,t)=Z2(z)cos(kx-wt), = - for some functions Z1(z) and Z2(z). Show that the propagation of such waves is feasible provided that the (scaled) wave number k and angular frequency w satisfy the wave condition Hence find the wave condition relating k and the wave speed C. [16] (c) Show that when the wavelength is small compared to both the distances h and H between the interface and the boundaries, the waves are dispersive. [4] Hint: Note that coth x 1 as x 80.

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