Math 575-Lecture 25 Deep water waves, dispersion relation The full 2D equations describing the water are given by the Euler equations 1 @ u + u@ u + v@ u = − @ p; t x y ρ x 1 @ v + u@ v + v@ v = − @ p − g; t x y ρ y @xu + @yv = 0: We aim to find the equations for the surface waves. Assume that the water is deep and the flows are irrotational: u = rφ. Then, the Euler equations are reduced to the Bernoulli equations 1 @ φ + juj2 + (p/ρ + gy) = 0; t 2 and ∆φ = 0: At the surface y = h(x; t), we have v = ht + uhx; at y = h(x; t): and if we impose p = 0, 1 @ φ + (u2 + v2) + gh = 0; at y = h(x; t): t 2 Since the water is deep, h = h0 + η; with = η=h0 1. This is called the small amplitude assumption. By throwing away a totally time dependent constant in φ, we are able to keep the leading order effects: 2 v(x; h0 + η) = ht + O( ); 2 @tφ(x; h0 + η) + gη + O( ) = 0 Further, by Taylor expansion, we find to leading order, @yφ(x; h0; t) = v(x; h0) = ηt;@tφ(x; h0; t) + gη = 0 1 The deep water is then described by @tφ(x; t) + gη = 0; y = h0 ηt = @yφ(x; t); y = h0; φxx + φyy = 0: Inserting the ansatz η = A sin(kx − !t), we find the dispersion relation !2 = gjkj: The energy is tranported with group velocity d! v = g dk If k ∼ 0, the wavelength is big, the approximation we did p = 0 at y = h is reasonable and the equations we have are good. Such water waves are called the gravity waves. If k ∼ 1, p = 0 at y = h is no longer good. The surface tension matters . The first equation should be revised to T @ φ(x; t) + gη − η = 0: t ρ xx These waves are called `capillary waves'. This is the case when a drop falls onto the surface of lake. The dispersion relation is then !2 = gjkj + T k2/ρ. Shallow water equations Consider the water above the ground y = 0. The free surface is given by y = h(x; t). h0 is the typical value of h. We are not assuming that the amplitude of the wave is small. Instead, we assume h0=L 1 where L is the typical length of the domain (or wavelength). This is the shallowness assumption. Consider the Euler equations again: 1 @ u + u@ u + v@ u = − @ p; t x y ρ x 1 @ v + u@ v + v@ v = − @ p − g; t x y ρ y @xu + @yv = 0: 2 0 0 0 0 As before, we scale x; y by x = x=L and y = y=h0. u = u=U and v = v=(Uh0=L), 0 p t = t=(L=U), U = gh0. We assume we are considering the regime where all prime numbers are O(1). By finding the leading order terms, we have 1 @ u + u@ u + v@ u = − @ p; t x y ρ x 1 0 = − @ p − g; ρ y @xu + @yv = 0: By the second equation: p = p0 − ρg(y − h(x; t)) Hence, the first equation is reduced to @tu + u@xu + v@yu = −ghx Consider that u(x; y; 0) = u0(x), i.e. it does not depend on y. Then, according to the above equation, u(x; y; t) = u(x; t) for all time. Hence, @tu + u@xu = −ghx By the incompressibility condition, v = −@xuy; because v = 0 for y = 0. Lastly, consider a fluid particle y = h(x; t). It then implies v = ht + uhx for y = h(x; t), or ht + uhx + @xuh(x; t) = 0. We then have the Shallow water equations ut + uux + ghx = 0; ht + uhx + uxh(x; t) = 0: In conservation law form 2 ut + (u =2 + gh)x = 0; ht + (uh)x = 0: or (ρht) + (ρhu)x = 0; g (ρh2) + (ρhu2) + ( (ρh)2) = 0: t x 2ρ x 3 p Letting c = gh, we can rewrite the equations into (@t + (u + c)@x)(u + 2c) = 0; (@t + (u − c)@x)(u − 2c) = 0: Such kind of nonlinear hyperbolic systems can develop discontinuity, called `shock' even though the initial value is smooth. 4.