Appendix: All Solutions to the Problems
947
iωσ u − j − ∇ ·
∇u
μ
= 0
( 1 )
If μ is a constant, then
u − iωσ μu = −μj
(2)
The natural boundary condition is
∇u · n| S = 0, or |∇u|| S = 0
( 3 )
10.22 Find the Euler equations of the functional J [u, v]
=
˝
V (|∇u|
2
+ |∇v|
2
+ αu
2
+ βv
2
− γ u
2 v)dV .
Solution The integrand of the functional is
F = |∇u| 2 + |∇v| 2 + αu 2 + βv 2 − γ u 2 v = ∇u · ∇u + ∇v · ∇v + αu 2 + βv 2 − γ u 2 v
The various partial derivatives are
∂ F
∂u
= 2αu − 2γ uv,
∂ F
∂∇u
= 2∇u,
∂ F
∂v
= 2βv − γ u
2
,
∂ F
∂∇v
= 2∇v
The Euler equations of the functional are
u = αu − γ uv
v = βv −
1
2
γ u
2
10.23 There is the Lagrange function in the analysis of a complex sound intensity
field
L =
G 0
ρc 2 e
ψ
(∇φ)
2
k 2 +
(∇ψ)
2
4k 2 − 1
where, G 0 is the auto-spectra of sound pressure; ρ is the density of the medium; c is
the sound velocity; k is wave number, k = ω/c, here ω is the circle frequency. Write
out its Euler equations and natural boundary conditions.
Solution Calculating the various partial derivatives, we have
∂ L
∂ψ
=
G 0
ρc 2 e ψ
(∇ϕ) 2
k 2 +
(∇ψ) 2
4k 2 − 1
,
∂ L
∂∇ϕ
=
2G 0
ρc 2 k 2 e ψ ∇ϕ,
∂ L
∂∇ψ
=
G 0
2ρc 2 k 2 e ψ ∇ψ
The Euler equations of the functional are