Appendix: All Solutions to the Problems
857
where, n is the unit vector of outward normal direction of Γ . Therefore the natural
boundary condition can also be written as
∂u
∂n
+ σ u + p
Γ
= 0
5.18 Let the known function p(x, y) ∈ C
1
(D), q(x, y), f (x, y) ∈ C(D), σ (Γ ) ∈
C(Γ ), Γ is the boundary of D. Write the Euler equation and the natural boundary
condition of the mixed type functional
J [u] =
¨
D
[ p(x, y)(u
2
x + u
2
y ) + q(x, y)u
2
− 2 f (x, y)u]dxdy +
Γ
σ (Γ )u
2 dΓ
Solution The Euler equation of the functional is
∂( pu x )
∂ x
+
∂( pu y )
∂ y
− qu = − f (x, y)
The natural boundary condition is
σ u + pu x
dy
dΓ
− pu y
dx
dΓ
Γ
= 0
or
[σ u + p(u x n x + u y n y )]
Γ = 0
According to Eq. (1.3.24), there is
∂u
∂n
= ∇u · n =
∂u
∂ x
i +
∂u
∂ x
j
· (n x i + n y j ) = u x n x + u y n y
where, n is the unit vector of outward normal direction of Γ . Therefore the natural
boundary condition can also be written as
p
∂u
∂n
+ σ u
Γ
= 0
5.19 Find the Euler equation and the corresponding boundary condition of the
acoustic field functional J =
1
2
˝
V (|∇ p|
2
− k
2 p
2
− 2iωρqp)dV +
1
2
˜
S
iωρ
Z
p
2 dS.
Solution Putting F =
1
2
(|∇ p|
2
− k
2 p
2
− 2iωρqp), G =
iωρ
2Z
p
2 , the various partial
derivatives are