Problems
33
Problems
1. Suppose that
a 2 U
a 2 u
a 2 u
-+a--+b-=O
at 2
axat
ax 2
is a hyperbolic partial differential equation and that a and bare constants.
Show that this equation can betransfonned to a decoupled pair of first-order
wave equations. What are the propagation speeds of the solutions to these
first-order wave equations?
2. Show that when (1.11) is hyperbolic, it can be transformed to the alternative
canonical form
UH - uT/" + L(;, 1/, u, u,,) = 0,
where L(;, 1/, u, us , u,,) is once again a linear function of u,
and u"
with coefficients that may depcnd on ; and 1/. (Hint: start with (1.12) and
define new independent variables equal to ; + 1/ and ; - 1/.)
3. If gravity and density stratification are neglected , the two-dimensional Euler equations for inviscid isentropic ftow reduce to a system of four equations in the unknowns (u , w, p, p). Linearize this system about a basic
state with constant (uo, wo, Po. po) and show that the linearized system is
hyperbolic. (Hint: transform the perturbation thennodynamic variables to
p' /(cPo) and o' - p' /c
2 , where c
2
= ap/ap is the square of the speed of
sound in the basic state.)
4. The pressure and density changes in compressible isentropic ftow satisfy
the relation
dp
1 dp
= c 2 di·
s
dt
(a) Derive the preceding relationship.
(b) Show that the preceding relationship is approximated as
dp =0
dt
in the incompressible system, as
dp
pdp
dt = 7idi
in the anelastic system, and as
dp
dt
p 1 dß
c(dt
in the pseudo-incompressible system (where the tilde denotes the steady
reference field).