644 Solutions to exercises
Section 14.3
4.f(x,t) 1 = 1 Ae
B(x− 2 t)6.For separation constant λ
2
;
λ
2
1 = 1 0: f(x,y) 1 = 1 (a 1 + 1 bx)(c 1 + 1 dy)
λ
2
1 > 1 0: f(x,y) 1 = 1 [ae
λx1 + 1 be
−λx][c 1 cos 1 λy 1
- 1 d 1 sin 1 λy]
7.f(x,y) 1 = 1 Ae
(cx−y 2 c)Section 14.4
- (i)E
1,1
1 = 12 ,E
1,2
1 = 1 E
2,1
1 = 15 ,
E
2,2
1 = 18 ,E
1,3
1 = 1 E
3,1
1 = 110 ,
E
2,3
1 = 1 E
3,2
1 = 113 ,
E
1,4
1 = 1 E
4,1
1 = 117 ,E
3,3
1 = 118
- (i)
(ii)Degeneracy ifa 1 = 1 b 1 = 1 c:
1 if p,q,r all equal (e.g. E
2,2,2
)
3 if two equal (e.g. E
1,1,2
,E
1,2,1
,E
2,1,1
)
6 if all different (e.g. E
1,2,3
,E
1,3,2
,E
2,3,1
,
E
2,1,3
,E
3,1,2
,E
3,2,1
)
Section 14.5
- (i)E
0,1
1 = 1 5.7831,E
±1,1
1 = 1 14.6819,
E
±2,1
1 = 1 26.3744,E
0,2
1 = 1 30.4715,
E
±3,1
1 = 1 40.7070,E
±1,2
1 = 1 49.2186
Section 14.6
- (i)ψ
1,0,0
1 = 1 (Z
3
2 π)
122
e
−Zr(ii)E 1 = 1 −Z
2
22
- (i)ψ
2,1,0
1 = 1 (Z
5
232 π)
122
re
−Zr 221 cos 1 θ
(ii)E 1 = 1 −Z
2
28
- (iii) , n 1 = 1 1, 1 2, 1 3,1=,
l 1 = 1 0, 1 1, 1 2,1=, where the allowed
values ofx
n,lare the zeros of the
spherical Bessel functionj
l(x).
(iv)
E
h
ma
10
2
2
8
,
=
ψ
100
1
2
,,
sin )
=
π
π
a
ra
r
(
E
x
ma
nlnl,
,
=
22
2
2
A
E
h
m
p
a
q
b
r
c
pqr,,=++
22
2
2
2
2
2
8
×
×
22
b
qy
bc
rz
c
sin sin
ππ
ψ
pqrxyz
a
px
a
,,
(, ,) sin=
2 π
fxy Ae
Bx y(, )
()
=
+
22Section 14.7
- (i)
(ii)
Chapter 15
Section 15.2
- (i)
Section 15.3
- (i)cos 1 tx 1 = 1 j
0
(t)P
0
(x) 1 − 15 j
2
(t)P
2
(x) 1
- 19 j
4
(t)P
4
(x) 1 −1-
(ii)sin 1 tx 1 = 13 j
1
(t)P
1
(x) 1 − 17 j
3
(t)P
3
(x) 1
- 111 j
5
(t)P
5
(x) 1 −1-
Section 15.4
- (ii)
- (ii)
- (ii)
- (ii)
- (ii)
x
l
2
1
4
4
πcos
1
6
2
cos
6 πx
l
ll x
l
22
2
6
41
2
2
−
π
π
2
cos
1
5
3
- siin
5 πx
l
81
3
3
2
33
lx
l
x
l
πππ
sin sin
π
2
222
3
4
1
2
2
3
3
−−+−
cos cosxxxcos
2
1
2
2
3
3
4
4
sin sinxxxxsin sin
−+−+
7
7
−+
cos x
1
2
2
1
3
3
5
5
+−+
π
cos cosxxxcos
V
Q
R
=,=Qqr −
2
0
3
2
22
4
31
πε
(cos )θ
1
4
1
2
5
16
01 2
PP P++
c
aaa
1
135
3
357
= +++
uxy
ya
ba
(,) xa
sinh( )
sinh( )
= sin( )
3
3
3
π
π
π
uxy A n ya n xa
nn( , )= sinh( ) sin( )
=
∑
ππ
1
∞
T
x
l
=− 3 Dtl
22
sin exp [ ]
π
π /
yxt
x
ll
(,) sin cos= 3
ππvt