Lecture Notes on Wave DispersionInstructor: Alexander M. BalkDepartment of Mathematics, University of Utah155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090JWB 304; [email protected], (801)-581-7512November 21, 20111 DispersionDispersion means that different waves (waves with different wave vectors)propagate with different velocities.In particular, the propagation speed of the surface gravity wavesc =Ωkk=rgkdepends on the wave number k, i.e. on the wave length λ.For gravity waves on ocean surface typical wave length ranges from about0.1m to 100m and longer.λ = 0.1m → T = 0.25s and c = 0.4m/s,λ = 100m → T = 8s and c = 13m/s.We see significant dispersion: The wave propagation speed varies substantiallyover the range of wavelengths of interest. This drastically contrasts the soundwaves, which are described by the wave equation∂2ψ∂t2= c2∇2ψ.Its general solution is also a superposition of wavesψ = Aei(k·x−ωt), where ω = ±Ωk,but the dispersion law Ωk= ck is linear in k, and these waves propagate withconstant speed, independent of the wave number.1Such waves are often called non-dispersive. In multidimensional situations,these waves are sometimes called semi-dispersive: Their velocity is indepen-dent of k = |k|, but depends on the direction of the vector k: The velocityis parallel to the wave vector k. The dispersion law Ωk= a · k (with someconstant vector a) corresponds to completely non-dispersive waves and arisesfrom an equation∂ψ∂t= a ·∇ψ.Let us consider some examples of different dispersive waves.• Linear Klein-Gordon equation∂2φ∂t2− α2∂2φ∂x2+ β2φ = 0.Again, we can find solutions of this equation by the metho d of separationof variables. Since the equation is invariant with respect to translationsin x and in t, we already know the form of the solutionsφ = Aei(kx−ωt)which are waves. These functions are indeed solutions pr ovidedω = ±Ωk, where Ωk=qα2k2+ β2.The function Ωkis the dispersion law. We see that the waves are disper-sive if β 6= 0. When β = 0, the linear Klein-Gordon equation becomesthe wave equation.• Linear Kortveg-deVries equation∂φ∂t+ α∂φ∂x= β∂3φ∂x3.Its general solution is a superposition of wavesφ = Aei(kx−ωt)with frequency ω defined by the dispersion lawω = Ωk= αk + βk3.2 Group velocityThe importance of waves stems from the fact that they transport the energyand information over long distances almost without changing the medium.Recall electromagnetic waves, sound waves , and sea waves. The speedc =λt=Ωkk— which is called the phase speed — is not the speed with which the energyand information are transported. Waves transport the energy and informationwith the group velocityC =∂Ω∂k.2On the graph of the dispers ion law Ωkwe can easily see the group andphase velocities. For the waves with the wave number k0, the group velocityC is the slope of the tangent to curve ω = Ωkat k0; the phase velocity c isthe slope of the straight line passing through the origin and the point on thegraph with abscissa k0.For example, in the case of the linear Klein-Gordon equation, the curveΩkvs. k has this shapekωk0ω=Ωk ω=αkWe see thatas k0→ 0 : C(k0) → 0, c(k0) → ∞.So, the group and the phase velocities can be quite different.Which speed do we see?We through a s tone into a pond and therebygenerate a great variety of surface waves (with various wave length). Yet aftersome time if we look into certain place on the pond we see quite regular waveswith some specific wave length.Suppose we observe waves at (x, t), i.e. at some place (x − dx, x + dx) atsome time (t − δt, t + δt). The coordinate x is counted from the point wherethe stone was thrown, and time t is counted from the in s tant when the stonewas thrown. We assum e a good separation of scales:x ≫ dx ≫ λ =2πk, t ≫ δt ≫ T =2πΩk.What waves will we see at (x, t)?A naive approach to answer this question goes like this. The waves propa-gate with different velocities; at (x, t) we should see waves with phase velocityc = x/t. Since c is determined by the wave number k, we can find their wavelength: λc=2πx2gt2. This reasoning turns out to be wrong: In fact, at (x, t) wewill see waves 4 times longer.For simplicity consider a 1D water surface (2D fluid). The general shapeof the water surface represents a superposition of wavesη(x, t) =ZAkei(kx−Ωkt)dk +ZBkei(kx+Ωkt)dk.Here the wave number k can be negative; −∞ < k < +∞. If x and t arelarge, then a small variation of k would lead to a large variation in the phaseθ = kx − Ωkt. To be specific, here we concentrate on th e first integral; thesecond integral is considered similar. So, while we integrate over k, the terms3with different k will cancel each other... ...except for the point k0of stationaryphase:dθdkk0= 0 ⇔xt=dΩdkk0.Thus, at (x, t), we will see waves with such wave number k that the groupvelocityC(k) =dΩdkequals x/t. Their wavelength λC=8πx2gt2is 4 times larger than the wavelengthλcbased on the phase speed.Let us continue with the example of throwing a stone into a pond. Supposeafter throwing a stone, we fix our gaze at a particular crest and follow it. Thenwe would see that the distance to the neighboring crest is continually changing.Moreover, if we continue to f ollow the crest, we sudd en ly lose sight of it. JamesLighthill (in Waves in Fluids) writes: “It seems an optical illusion at first; butthen the next crest, too, disappears! Meanwhile, crests are coming alongthick and fast behin d. Indeed, at the inside edge of the concentric pattern,new crests are appearing from nowhere... The sud denness with which sizablecrests near the outside of the pattern disappear rules out gradual attenuationas the mechanism.” The true explanation is in the fact that for the waves ondeep water, the phase velocity is twice bigger than the group velocity. We willsee later that the energy travels with the group velocity.The original disturb ance pr oduces waves in some range of wave numbersk1< k < k2. The typical wavelength is related to the size of the stone: λ1=2π/k1is several times larger than the s ize of the stone, while λ2= 2π/k2isseveral times smaller than the size of the stone. So, the energy is concentratedin the spatial intervalC1t < x < C2t [C1= C(k1), C2= C(k2)].The individual waves — traveling fast, with the phase velocity c(k) = 2C(k)— disappear at the outside