MATH 656 Supplementary Homework Spring 2010

MATH 656 Supplementary Homework
5. Write Maxwell’s equations
~t = ∇ × B
~ t = −∇ × E
in the form of a symmetric hyperbolic system. Is it strictly hyperbolic? What does
the energy inequality amount to in this case?
6. The equations of linear elasticity in R3 can be written in the form
ρ~utt = µ∆~u + (λ + µ)D(div ~u)
where ~u = [u1 , u2 , u3 ]T = ~u(x, t) is the displacement vector. Here ρ is the density and
λ, µ are the Lamé parameters, which we assume are constant. Show that there exist
solutions in the form
~u = ∇φ + ∇ × ψ
~ are solutions of scalar wave equations
where the functions φ, ψ
φtt = c2p ∆φ
~tt = c2 ∆ψ
~ this means each component satisfies the scalar wave equation.) Find
(In the case of ψ
the wave speeds cp , cs as part of your solution.
7. Transform the second order equation vtt − vxx + q(x)v = 0 into an equivalent first
order system in two different ways.
i) Introduce new variables u1 = v, u2 = vt , u3 = vx
ii) Introduce new variables u1 = vx + α(x)v + vt , u2 = vx + α(x)v − vt for some
well chosen function α(x).
What are the characteristic speeds in each case?