4.7 Comments and References 283
u(x, 0 )=f(x), ∂∂uy(x, 0 )=0, 0 <x<1,u( 0 ,y)=0, u( 1 ,y)=0, 0 <y;c. ∂(^2) u
∂x^2
=∂u
∂y
,0<x<1, 0 <y,
u(x, 0 )=f(x),0<x<1,
u( 0 ,y)=0, u( 1 ,y)=0, 0 <y.
- Show that iff 1 ,f 2 ,...all satisfy the periodic boundary conditions
f(−π)=f(π ), f′(−π)=f′(π ),
then so does the functionc 1 f 1 +c 2 f 2 +···,wherethec’s are constants.- Longitudinal waves in a slender rod may be described by this partial differ-
ential equation:
∂^2 u
∂x^2
=∂
(^2) u
∂t^2
− ∂
(^4) u
∂x^2 ∂t^2
Show how to separate the variables.
- Deflections of a thin plate and slow flow of a viscous fluid may both be
described by the biharmonic equation
∂^4 u
∂x^4 +^2
∂^4 u
∂x^2 ∂y^2 +∂^4 u
∂y^4 =^0.
Assume thatu(x,y)=X(x)Y(y)and show that the variables don’t separate.
Show that, under the additional assumptionX′′/X=−λ^2 , a differential
equation forYresults.4.7 Comments and References
While the potential equation describes many physical phenomena, there is one
that makes the solution of the Dirichlet problem very easy to visualize. Sup-
pose a piece of wire is bent into a closed curve or frame. When the frame is
held over a level surface, its projection onto the surface is a plane curveCen-
closing a regionR. If one forms a soap film on the frame, the heightu(x,y)of
the film above the level surface is a function that satisfies the potential equa-
tion approximately, if the effects of gravity are negligible (see Chapter 5). The
height of the frame above the curveCgives the boundary condition onu.For
example, Fig. 2(b) shows the surface corresponding to the problem solved in
Section4.2.AgreatdealofinformationaboutsoapfilmsisinthebookThe
ScienceofSoapFilmsandSoapBubblesby C. Isenberg (see the Bibliography).