Lecture notes on linear wave theory. Lectures given at the ..., Study Guides, Projects, Research of Optics

Download Lecture notes on linear wave theory. Lectures given at the ... and more Study Guides, Projects, Research Optics in PDF only on Docsity! Lecture notes on linear wave theory. Lectures given at the summer school on: WATER WAVES and OCEAN CURRENTS. Nordfjordeid 21-29 june 2004. Kristian B Dysthe Department of Mathematics University of Bergen Norway June 2, 2004 1 Introduction. To give an introduction to linear wave theory for surface waves lasting for a few hours is a nearly impossible challenge. There is no time for mathematical details, yet the theory is mathematical in its nature. The notes are probably going to contain more details than the lectures. Still they are rather sketchy. Consequently I shall have to rely on my listeners’ ability to fill in the details that are left out. Excellent books for further reading are for example the following: • G.B.Whitham : Linear and Nonlinear Waves. John Wiley & Sons, 1974. • J. Lighthill : Waves in Fluids. Cambridge University Press, 1978. • G.D. Crapper : Introduction to Water Waves. Ellis Horwood Limited 1984. 1 2 Basic equations. We start by assuming that our fluid is of homogeneous density ρ , and also ideal and incompressible. Consequently the continuity equation is simply ∇ · v = 0 (1) We shall also assume that vorticity has no major place in wave propaga- tion. This, however, calls for a comment. From the theory of an ideal and homogeneous fluid we recall that the vorticity ∇× v is a property associ- ated with the fluid elements. It is carried along by the fluid motion. This implies that if a particular fluid element had zero vorticity initially, it will always have zero vorticity. The main property of a wave is its ability to transport information, energy and momentum over considerable distances without transport of matter. Thus the velocity field associated with the wave is irrotational and given by a velocity potential, ϕ , which according to the equation (1) above satisfies the Laplace equation ∇2ϕ = 0 (2) The boundary conditions for (2) at the bottom is simply ∂ϕ ∂z = 0 for z = −h (3) where we use the oceanographic convention: the z-axis pointing vertically upwards with z = 0 at the equilibrium surface. The actual surface is located at z = η(t, x, y) and the kinematic surface condition states that a ”fluid particle” at the surface at any given time is always at the surface: d dt (z − η(t, x, y)) = 0 ⇐⇒ dη dt = ∂ϕ ∂z (4) where we use the notation dA dt ≡ ∂A ∂t +∇ϕ · ∇A There is also a dynamical condition at the free surface. Above (i.e. for z > η) there is an atmospherical pressure pa which is taken to be constant. 2 the lower boundary condition is then readily found to be B = C cosh [k(z + h)] cosh(kh) Inserted into the linearized surface boundary conditions (9) one obtains the equations iωA+ k tanh(kh)C = 0 and (g + k2T ρ )A− iωC = 0 The condition for existence of a plane wave solution (i.e. a nonzero solution A and C) then becomes ω2 = (gk + T ρ k3) tanh(kh) (10) This relation between the frequency ω and the wave number k is called the dispersion relation. In real form the plane wave solution can now be written η = a cos(k · x− Ω(k)t) (11) and ϕ = Ω(k) k a cosh [k(z + h)] sinh(kh) sin(k · x− Ω(k)t) (12) where Ω(k) is a solution of the dispersion relation (10), and a is the real amplitude of the wave. Before looking at more general solutions of the linearized equations we consider how the dispersion relation (10) can be simplified in some special parameter domains. 3.2 Waves of different wavelengths. We rewrite (10) as ω2 = gk(1 + ( k k0 )2) tanh(kh) where k0 = √ gρ T is a characteristic wave number (corresponding to a wavelengh of 1.71cm for pure water). The different parameter regimes of the two numbers kh and k/k0 serves to distinguish between different wavetypes: 5 • kh >> 1 deep water waves. • kh << 1 shallow water waves. • k/k0 << 1 gravity waves. • k/k0 >> 1 capillary waves. For deep water gravity waves the dispersion relation is simplified to ω2 = gk (13) and for shallow water gravity waves one have ω2 = ghk2 (14) A graph of the dispersion relation for gravity waves is shown in Figure (1). The approximations (13) and (14) are also shown. The wave pattern (crests and troughs) moves with the phase velocity vph = ω/k. For waves on deep water we have v2 ph = g k + gk k2 0 from which it is found that vph has a minimum of √ 2(Tg/ρ)1/4 (' 23cm/s, pure water) for k = k0. 3.3 Motion of the fluid particles. The motion of a fluid particle due to the wave is found by integrating the equation of motion dr dt = ∇ϕ (15) where r =(x(t), y(t), z(t)) gives the position of the fluid particle, and the right hand side is evaluated at that point. This is of course a nonlinear differential equation even though we shall use the linearized solution (12) for ϕ. We expect the motion to consist of a periodic ocillation s(t) and possibly a slow translatory motion of the average position, or guiding center R(t) = (X(t), Y (t), Z(t)). We write r = R + s . To lowest significant order in the wave steepness we evaluate ∇ϕ at the guiding center and neglect 6 the variation of R during a wave period. The equation for s now becomes (taking the x-axis parallel to k) ds dt = Ω(A cos θ, 0, B sin θ) where θ = k ·R−Ωt and A = a cosh [k(Z + h)] sinh(kh) and B = a sinh [k(Z + h)] sinh(kh) which is readily integrated to give s = (−A sin θ, 0, B cos θ) The trajectory of the ocillating movement is found by eliminating θ giving[ x−X A ]2 + [ z − Z B ]2 = 1 which is an ellipse with half axis A (horizontal) and B (vertical). It is readily seen that B = a (the amplitude) for a surface particle and zero for a bottom particle. For a wave on deep water we have the simplification A = B = aekZ implying that the trajectories are all circles with radius a at the surface, and decreasing exponentially downwards (see Figure (2) ). The equation for the slow motion of the guiding center is obtained by going to the next order in wave steepness. This seems a bit strange since we still use the expression (12) which is derived from linear wave theory. It is not difficult, however, to show that by taking into account the next order of approximation for ϕ one does not change the result for the guiding center. Developing the right hand side of (15) in powers of s, and averaging over one wave period (keeping R constant), we obtain dR dt = 〈s·∇∇ϕ〉 = 1 2 kΩa2 cosh [2k(Z + h)] sinh2(kh) This is a slow horizontal drift of the fluid particles in the wave direction called Stokes drift. The total mass transport M due to the wave is found by integrating the Stokes drift from the bottom to the surface M = ∫ 0 −h ρ dR dt dz = ρΩa2k 2k tanh(kh) (16) For deep water we have the simplifications dR dt = k k Ωa2e2kz and M = k 2k ρΩa2 7 By comparing this with the expression (16) it is seen that the average mo- mentum is equal to the total mass transport. These expressions for the average energy and momentum remain valid as an approximation even if the amplitude of the wavetrain is slowly varying. If L is a characteristic length for a significant variation of the amplitude (corresponding to 1/ |∆k| in the example above) then (20) and (21) are correct to the order (kL)−2. By a slow variation it is understood that the amplitude has a very small relative variation during a wave period. Since both E and P are quadratic in the amplitude, the transport ve- locity for these quantities is also the group velocity as already anticipated. 5 The initial value problem. The linear superposition of elementary waves can be used to solve initial value problems. In the following we consider such a problem for a gravity wave on deep water travelling along a channel (one dimensional propaga- tion). Let the fluid be at rest initially (i.e. ϕ = 0 at t = 0) with a perturbed surface. We take the initial perturbation for η to be an impuls function. If that problem can be solved (as it was by Cauchy and Poisson in 1816) the solution for a more general initial value can be found by convolution. The initial conditions are now η(0, x) = δ(x) , ∂η ∂t (0, x) = 0 Consider now a general solution of the linearized equations η(t, x) = ∫ ∞ −∞ R1(k)ei(kx−Ω(k)t)dk + ∫ ∞ −∞ R2(k)ei(kx+Ω(k)t) written as Fourier integrals. The first integral represents a wave moving to the right, and the second a wave moving to the left (corresponding to the two solutions ±Ω(k) of the dispersion relation). From the initial conditions we obtain ∂η ∂t (0, x) = ∫ ∞ −∞ −iΩ(k)(R1(k)−R2(k))eikxdk = 0 and η(0, x) = ∫ ∞ −∞ (R1(k) +R2(k))eikxdk = δ(x) implying that R1 = R2 = 1 4π 10 Thus the solution becomes η(t, x) = 1 π ∫ ∞ 0 cos(kx) cos(Ω(k)t)dk (22) which is an exact, nice and compact solution. The content of it is , however, far from easy to see. Next we show how some important information can be extracted from it. 5.1 Asymptotic solution of the initial value problem. In the above initial value problem we consentrate on the waves moving to the right. The challenge is then to find an asymptotic approximation (far from the initial impuls) to the Fourier integral representing these waves η(t, x) = 1 4π ∫ ∞ −∞ ei(kx−Ω(k)t)dk = 1 4π ∫ ∞ −∞ eitw(k)dk (23) where w = k x t − Ω(k)t The leading term in an asymptotic development of this integral for large t is known to come from a small area around the points of stationary phase i.e. the solution of the equation dw dk = 0 ⇔ dΩ dk = x t (24) The physical interpretation of this relation is that the main contribution at time t and location x comes from the Fourier component whose group velocity is exactly right for travelling the distance x in the timespan t. Let K(x t ) be the relevant solution of (24) with respect to k. The leading term of the asymptotic expansion of the integral (23) is (using the socalled stationary phase method and observing that w′′ > 0 for a gravity wave) η(t, x) ' 1√ 2πtw′′(K) cos [ tw(K) + π 4 ] = A cos θ (25) where A is a slowly varying amplitude and θ is the wave phase. For deep water waves (i.e. Ω = √ g |k|) an explicite solution K(x t ) of equation (24) can be found as K = g( t 2x )2 11 giving (show this) θ = −gt 2 4x and A = t 2 √ g πx3 The solution (25) represent a slowly varying wavetrain that locally looks like a plane wave with a local wave number K and frequency Ω(K) where ∂θ ∂x = K and ∂θ ∂t = −Ω(K) Show that K and Ω thus defined satisfies the dispersion relation. It is also straight forward to show directly that the relative variations in the quantities K,Ω and A over a wave period is small as long as Ωt >> 1. Out of all this emerges the following picture: Initially a compact region is disturbed. The Fourier spectrum of the initial disturbance is rather broad- band. The waves corresponding to each Fourier component starts moving. Its part of the energy is transported with the corresponding group veloc- ity. At first all these waves add up to some rather involved pattern. After a while, since they move with different velocities, an ordering takes place and increases with time: The longest waves in front and the shortest in the rear. In fact if after a long time one observes the train going by, the local frequency is increasing linearily with time since Ω = −∂θ ∂t = gt 2x (26) Although we developed these results under rather special conditions (one dimensional propagation and an impulsive initial condition) they can readily be generalized. For two dimensional propagation a corresponding version of the stationary phase method can be used and the condition for stationary phase is then ∂Ω ∂k = x t with solutions K = g ( t 2r )2 x r for deep water waves where r = |x| . The asymptotic wave phase θ now becomes θ = −gt 2 4r with K =∇θ and Ω = −∂θ ∂t An example of the effects discussed above is the common experience when throwing a stone into a pond. The circular symmetric wavetrain resulting 12 where E is the mean energy density in the rest frame (see equation (20)). This shows that the total mean energy is not invariant under a Gallilei transformation. It is also seen that the quantity N = E/Ω called action density is invariant since N = E Ω = Eu Ω + U · k (29) The averaged momentum density is seen from (21) to be invariant P = k E Ω = kN 6.1 Refraction In reality a current is hardly uniform in space and time, also depth h varies with the horizontal dimensions. However, in many cases the time and length scales associated with these quantities are much larger than those of the wave. This can be expressed as k >> ∣∣∣∣1h∇h ∣∣∣∣ , k >> ∣∣∣∣ 1 U ∇U ∣∣∣∣ and ω >> ∣∣∣∣ 1 U ∂U ∂t ∣∣∣∣ It is then rather natural to assume that locally (in time and space) the wave properties are the same as that of a corresponding plane wave under uniform conditions. Formally this can be shown to be true by an asymptotic expan- sion in a small parameter made from the ratio of wave and current scales (or depth scales). The lowest order result from such an exercise is the socalled geometric optics approximation or ray theory. Since the medium through which the wave is propagating is slowly varying in space (and possibly in time) the amplitude, wave number and (possibly) the frequency are varying too. Starting with a ”locally plane” wave represented by η = Re(a(εx,εt)eiθ) where θ is the wave phase and a is a slowly varying amplitude (the slow variation is made explicite by a small parameter ε ). The local frequency and wave vector are then (as earlier) defined by ω(εx,εt) = −∂θ ∂t and k(εx, εt)=∇θ (30) and are also assumed to be slowly varying. Making a similar anzats for the potential ϕ and inserting this into the linearized equations one obtain to the 15 zero order in ε just the dispersion relation (10). To the first order in ε one obtains an equation governing the variation of the amplitude which can be written (after a lot of work!) ∂N ∂t +∇ · (∂ω ∂k N) = 0 (31) where N = E/Ω is the action density and E is given in terms of the ampli- tude by the expression (20). This has the form of a conservation equation and tell us that action is conserved and the action density is transported with the group velocity, like some ”wave fluid” density. Now the stream lines of this wave fluid are the rays, which can also be thought of as trajectories of wave packets. They are defined through the equation dx dt = ∂ω ∂k (32) To integrate (32) we need a similar equation for k . It follows from the relations (30) that ∂k ∂t = ∇∂θ ∂t = −∇ω Through the dispersion relation ω is a function of k . It may also be an explicite function of time and space through the physical parameters entering the dispersion relation, i.e. the depth h and the current velocity U . The right hand side of the equation above can now be developed as − ∂2θ ∂xi∂t = ∑ j ∂ω ∂kj ∂kj ∂xi + ∂ω ∂xi = ∑ j ∂ω ∂kj ∂ki ∂xj + ∂ω ∂xi where the last equality follows from the fact that k is a gradient vector. We now get the equation dk dt = −∂ω ∂x (33) where d dt = ∂ ∂t + ∂ω ∂k ·∇ . The two equations (32) and (33) can be considered to be dynamical relations for the motion of wave groups. In fact they have the canonical form of the Hamiltonian equations with ω corresponding to the Hamiltonian and k to the momentum variable. This correspondence is not very surprising considering the wave-particle correspondence first sug- gested by Louis de Broglie. The conservation equation for action (31) is the corresponding evolution equation for the amplitude. Together these three equations constitute the ”geometrical optics” relations for surface gravity waves. In the following we give some examples. 16 6.2 Examples. 6.2.1 Swell approaching beach. The frequency change of swell (that we considered earlier) is a slow process. When considering the transition of swell from deep water to shallow water we may consider the frequency of the incoming waves to be constant. We shall also neglect any background current. The frequency during the transition is also constant because we have dω dt = ∂ω ∂t + dk dt · ∂ω ∂x + dx dt · ∂ω ∂k = ∂ω ∂t = 0 which follows from the equations (32) and (33). Consequently it is the wave vector that must change in order to satisfy the dispersion relation with a changing depth. Consider a straight beach parallel to the y-axis, with a depth that depends on the x-coordinate only. From (33) we then have dky dt = 0 thus ky is a constant. Since we now have a steady state problem equation (31) can be integrated to give ∂ω ∂kx E = ωkx 2k2 (1 + 2kh sinh(2kh) )E = constant where the constant is equal to the component of the energy flux in the x-direction. Using the subscript 0 to denote the wave properties of the incoming wave at deep water we obtain from the dispersion relation and the equations above E E0 = kxk 2 0 kx0k2 (1 + 2kh sinh(2kh) ) and k k0 = coth(kh) = cos(ψ0) cos(ψ) where ψ is the angle between k and the x-axis. Explicite solutions can be found when the wave has arrived at shallow water i.e. kh << 1. In this 17 whose variation is taken to be Gaussian (see bottom of the figures). In Figure (8) the current maximum is higher than the critical one (i.e. vg0/2), such that reflection occurs at a level x = xc. In Figure (9) the current maximum is subcritical so that the wave is able to pass through the current region. 6.2.3 Ship waves. It is wellknown that a ship moving at a constant velocity generate a V- shaped wave pattern that is stationary in the reference frame moving with the ship. It is somewhat less wellknown that the half-angle of the V is 19.5◦ and that the waves at the edge of the wave pattern (caustic) is moving at an angle of 35.3◦ to that of the ship, regardless of the ship form and speed. This was explained by Lord Kelvin more than a hundred years ago. The explanation starts from two observations • In the reference frame of the ship the wave pattern is stationary i.e. the frequency is zero. • The ship is the wave source, therefore (still in the ship-frame) the wave energy at a point of the pattern must have travelled along a straight line from the ship. In the ship-frame the water has a uniform velocity −U and the condition of zero frequency (assuming deep water) becomes ω = √ gk − k ·U = √ gk − Uk cosψ = 0 (34) The group velocity is then (with the x-axis parallel to the ship track, see Figure (10)) vg = 1 2 √ g k k k −U = (−1 2 √ g k cosψ + U, 1 2 √ g k sinψ) Simple trigonometry, and using equation (34) give us tan θ = vg sinψ U − vg cosψ = sinψ cosψ 1 + sin2 ψ = tanψ 1 + 2 tan2 ψ (35) where the angles θ and ψ are explained in Figure (10a). Solving for tanψ we obtain the two real solutions tanψ = 1 4 tan θ (1± √ 1− 8 tan2 θ) 20 corresponding to the two wave-systems in the ”Kelvin” wake (see Figure (10b)). These different waves meet in cusps along the perifery of the wave pattern. As seen from the formula above there are no solutions for |tan θ| > 2 √ 2 i.e. |θ| > 19.3◦ . Further, the angle that the cuspwaves at the perifery make with the ship track, is given by tan−1( 1√ 2 ) ' 35.3◦. Figure (11) shows a picture of a ship wake pattern, where only the transversal wave system is in evidence. Which parts of the pattern are amplified and which are suppressed, depend on the form and speed of the ship. 21