Miscellaneous Exercises 359
wheregis the acceleration of gravity andUis mean depth. The tidal
motion of the sea is represented by the boundary condition
u(a,t)=U+hcos(ωt).
Find a bounded solution of the partial differential equation that satisfies
the boundary condition by setting
u(x,t)=U+y(x)cos(ωt).
(See Lamb,Hydrodynamics, pp. 275–276.)27.Is there any combination of parameters for which the solution of Exer-
cise 26 does not exist in the form suggested?
28.If the estuary of Exercise 26 has uniform width but variable depth
h=Ux/a, then the equation foruis
∂
∂x
(
x∂u
∂x)
=
a
gU∂^2 u
∂t^2 ,^0 <x<a,^0 <t,
subject to the same boundary condition as in Exercise 26. Find a
bounded solution in the form suggested. (See Eq. (1) of Section 5.8.)29.The equation for radially symmetric waves inn-dimensional space is
1
rn−^1∂
∂r(
rn−^1 ∂u
∂r)
=^1
c^2∂^2 u
∂t^2
whereris distance to the origin. Find product solutions of this equation
that are bounded at the origin.30.Show that the equation of Exercise 29 has solutions of the form
u(r,t)=α(r)φ(r−ct)
forn=1andn=3. [See “A simple proof that the world is three-
dimensional” by Tom Morley,SIAM Review, 27 (1985): 69–71.]31.A certain kind of chemical reactor contains particles of a solid catalyst
and a liquid that reacts with a gas bubbled through it. M. Chidambaran
[“Catalyst mixing in bubble column slurry reactors,”Canadian Journal
of Chemical Engineering, 67 (1989): 503–506] uses the following problem
to model the catalyst concentrationCin a cylindrical reactor:
Dr^1 r∂∂r(
r∂∂Cr)
+Dz∂(^2) C
∂z^2 +U
∂C
∂z=^0 ,^0 <r<R,^0 <z<L,
∂C
∂r(R,z)=^0 ,^0 <z<L.