18.325 :: Waves and Imaging, Fall 2009 :: Laurent DemanetHomework problems, Dec 03 version1. Consider the wave equation with c = 1. In this exercise we prove that waves cannot propagateat a speed faster than c = 1. Consider that the initial conditions u0and u1are zero inside ofthe ball B(x0, t0) of center x0and radius t0. We aim to show that u(x, t) = 0 inside the coneC =[0≤t≤t0B(x0, t0− t)(a) Find the conserved energy E(t) for this wave equation. It is an integral of some quantityover space.(b) Consider this integral over x ∈ B(x0, t0− t) instead of the whole space, i.e., focus on theenergy inside this ball at time t. Call this restricted energy e(t). What is e(0)?(c) Find an expression for de/dt as an integral over the boundary ∂B(x0, t0− t), i.e., overthe sphere of center x0and radius t0− t. [Hints: for any smooth family of volumes V (t),ddtZV (t)f(x, t)dx =Z∂V (t)f(x, t)dSx+ZV (t)∂f∂t(x, t)dx.Then use the wave equation, and integrate by parts.](d) Perform the proper majoration to find that de/dt ≤ 0. [Hint: 2ab ≤ a2+ b2. Thinkabout how a normal derivative relates to the gradient.](e) Argue that e(t) = 0 for all times 0 ≤ t ≤ t0.2. (October version). Solve the Klein-Gordon equation∂2u∂t2− ∆u + m2u = 0,by the method of characteristics, i.e., find an adequate change of the independent variablesthat simplifies this equation to the point that you can solve it explicitly. (Klein-Gordon isthe equation that governs particles of spin 0 in quantum field theory.) (December update:It may be hard to solve the equation exactly using the method of characteristics. The latterseems to be better suited to finding approximate solutions. Instead, it is fine if you use theFourier transform to find the general solution.)3. Calculate the Fourier transforms in x of the Green’s functions g(x, t) (for the wave equation)and φ(x, ω) (for the Helmholtz equation). You may encounter distributions that requirecareful treatment; be as descriptive as possible.4. Generalize the reflection and transmission coefficients to the case of plane waves in two spatialdimensions, and a linear interface. Formulate your result as a function of the angle of incidenceθ that the wave number makes with the direction normal to the interface.5. This problem involves a bibliographical search. Provide one proof that some scattering seriesakin to the Born series seen in class, converges under a weak scattering assumption. Youmay start with volume 1 of ”Methods of modern mathematical physics” by Reed and Simon.Added hint: you can also look up the book by Colton and Kress, ”Inverse acoustic andelectromagnetic scattering theory”.16. This problem is not for the faint of heart, but if you need a challenge, here it is. In class wesaw a perturbative result concerning the accuracy of the Born approximation under a weakscattering assumption. Strengthen this result in some direction of your choosing, perhaps byworking with different norms of the reflectivity V (x). You are allowed to make smoothnessassumptions on the incident field.7. The treatment of reverse-time migration seen in class involves data u(r, t) for an interval intime t, and along the surface z = 0 in r. Consider instead the snapshot setup, where t is fixed,and there are receivers everywhere in the domain of interest. (So we have full knowledge ofthe wavefield at some time t.) Repeat the analysis of the imaging operator, adjoint to theforward operator that forms snapshot data from singly scattered waves. In particular, findwhat the adjoint-state wave equation becomes in this case.8. We saw in class an argument of constrained optimization that fully recovered the adjoint-state method from equating the variations of a Lagrangian to zero. This argument used thetime-dependent wave equation; adapt it to the Helmholtz equation instead.9. (Stationary phase) ConsiderI(β) =Zeiβφ(x)a(x)dx,where φ is C∞and has a single stationary phase point: ∇φ(x∗) = 0. Assume that the Hessianof φ is nondegenerate at x∗. Look up1a formula that gives the leading asymptotic behavior ofI(β) as β → ∞. Make sure you understand and explain what all the quantities mean in thisformula. Then reproduce a proof of this result. (Without proof, counts for a half exercise,with proof, counts for a full exercise).10. We know that(2π)−dZei(x−y)·kdk = δ(x − y),and we’ll try to get a feeling for this formula by stationary phase. The following steps requireto have done the previous question.(a) Why does the phase φ(k) = (x − y) · k NOT satisfy the assumptions of the previousquestion? (Nevertheless, the stationary phase condition x − y = 0 is quite useful as itcorresponds precisely to the locus of singularity of δ(x − y).)(b) A direct application of the theorem will not work, but we can writeZ(2π)−dZei(x−y)·kdkf(x)dy = f(y).Now use the Fourier inversion formula f(x) = (2π)−dReix·ξˆf(ξ)dξ and gather all thephase factors. We’ll consider the stationary phase formula in the pair of variables (x, k),by keeping y and ξ as fixed parameters. Identify the stationary phase point(s) and arguethat the assumptions for validity of the stationary phase formula are now satisfied.(c) Note that we don’t really have a large β here, but that could be fixed by rescalingfrequency k = βk0and consideringˆf compactly supported at large |ξ|. Notwithstanding,use the stationary phase formula in what remains of the integral above, and compare itto the result f (y) that you are supposed to obtain.1On the web or in a book, starting from ”stationary phase” as a keyword. For a book, I like Elias Stein’s ”HarmonicAnalysis”, chapter 82(d) Optional: what does the stationary phase condition tell you, physically?11. In class we saw the equations for the Radon transform and its inverse in two spatial dimen-sions. Write the corresponding equations (Fourier-slice theorem, inversion formula) for theso-called X-ray transform in 3 spatial dimensions, which consists of integrating the functionalong all the straight lines in R3.12. For Kirchhoff modeling we used stationary phase to find the formulas defining the canonicalrelation for the modeling operator F as a map(x, ξ) 7→ ((r, t), (ξr, ω)).Do the same for the Kirchhoff migration operator F∗, and check that you formally obtain theinverse map. (By “formally” I mean that the invertibility conditions are not trivial; discussthem if you