Suppose the medium is not homogeneous (gravity waves impinging on a beach,

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1 Slowly vaying media: Ray theoy Suppose the medium is not homogeneous (gavity waves impinging on a beach, i.e. a vaying depth). Then a pue plane wave whose popeties ae constant in space and time is not a pope desciption of the wave field. Howeve, if the changes in the bacgound occu on scales that ae long and slow compaed to the wavelength and peiod of the wave, a plane wave solution may be locally appopiate. (Fig. 2.1) This means: λ << L m whee L m is the length scale ove which the medium changes. Conside the local plane wave φ = iθ( x,t) (x,t) a(x,t)e a vaies on the scale L m while θ vaies on the scale λ. θ 1 1 a 1 = 0( ); = 0( ) λ a Lm φ = ae iθ λ θ+0( ) Lm with θ = x wt Define the local wavenumbe and the local fequency as: = θ t ω= θ x Fom these definitions it follows that: = 0 the local wave numbe is iotational. Consevation of cests in a slowly vaying medium. Suppose we go fom point to point ove the cuve C 1. 1

2 C2 C1 Figue by MIT OpenCouseWae. Figue 1 slowly vaying wave fonts The numbe of wave cess we pass along C 1 is n c1 = 1 d s = 1 d s The numbe of wave cests we pass along C 2 is c 1 s (ω, ) n c2 = 1 d s = 1 d s c 2 efoe fo plane waves ω = Ω( w ) only, now ω=ω(, x,t). ae slowly vaying functions of space/time, the dispesion elation is explicitly dependent on space/time. Now we can intoduce the goup velocity in anothe way x = Ω, x + Ω i x,t i i x = = Ω, x +c i gi x Whee we use the summation convention ove epeated indices, and c gi = Ω i by definition i = 1, 2, 3 = x,y,z 2

3 c g = Ω goup velocity The diffeence is: n c1 n c2 = 1 π [( ) d s ] = 1 π [( + ) d s ] = 1 c 1 c 2 c 1 c 2 c total d s = x nd ˆ 0 ˆ n = unit vecto nomal to C We have used Stoes theoem elating the line integal of the tangential component of to the aea integal of its cul ove the aea bounded by the closed contou C. The incease of phase is the same on C 1 and C 2. This means the numbe of cests along C 1 is the same as the numbe of cests along C 2, that is the numbe of cess inside the aea is conseved. Cests ae neithe ceated no destoyed inside. The cests have no ends, so the numbe of cests within a wave goup will be the same fo all time. This is obviously tue only fo slowly vaying plane waves. Fom the definition of and w it follows: + ω=0 (1) We have seen that the numbe of cests we coss fom to is the same along any path connecting and. Then: n = 1 n = 1 d s = 1 d s ω d s = 1 (ω() ω()] This says that the ate of change of the numbe of wave cests between and is equal to the fequency of cest inflow at minus the fequency cest outflow at. 3

4 Cests ae neithe ceated no destoyed in the smoothly vaying function φ. The numbe in any local egion inceases o deceases solely due to the aival of pe-existing cests at, not to the ceation o destuction of existing cests. We now intoduce the dynamics by asseting that the wavenumbe and fequency must be elated by a dispesion elation in the same way as fo a plane wave. Since by eq. (1) i = we have = Ω c g i o + c g ω= Ω, x (1) equation fo ω Similaly fom (1) i x + Ω x,t + Ω x, t i j j = 0 i + Ω j = Ω j + c g = Ω,t (2) The ay equation gives the velocity at which the wave pacet, o wave goup, moves: o c g = dx dt o c gx = dx dt ; c gy = dy dt in two dimensions. Then the ay path in the (x,y) plane is 4

5 dy dx = c gy c gx d dt = + c g c g = d x dt (I) + c g ω= Ω t, x + c g = Ω,t (III) Ω = Ω (, x,t) has an explicit paametic dependence on ( x,t), fo instance when waves ente in wate of changing depth. The ay equations give the evolution of the local wavenumbe and the local fequency ω as we move along the ay, i.e. we move with the wave pacet at the local goup velocity (II) c g. Is a plane wave a paticula solution of the ay theoy fomulation? Suppose the medium is homogeneous, no changes in ( x,t) ω=ω( ) only Solution: plane wave φ=ae i( x ωt) whee (,ω) Initial condition s x 2 0 do not change but ae constant in space φ( x ) = ae i x and The ay equation (III) gives = 0: Ω 0 gives (t = 0) neve changes along the ay and emains equal to (t=0).. ω=ω( ) gives ω at t = 0 5

6 s = 0 ; Ω = 0 eq. (II) gives = 0 ω = ω(t=0) The fequency neve changes along the ay. Thus the plane wave solution in a homogeneous medium is entiely consistent with the ay theoy fomulation. 6

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