8.3 Solutions
363
8.14 (a) y = A ln(x + vt)
∂ y
∂ x
=
A
x + vt
,
∂ 2 y
∂ x 2 = −
A
(x + vt) 2
∂ y
∂t
=
Av
x + vt
,
∂ 2 y
∂t 2 = −
Av 2
(x + vt) 2
∴
1
v 2
∂ 2 y
∂t 2 = −
A
(x + vt) 2 =
∂ 2 y
∂ x 2
Thus the wave equation is satisfied.
(b) y = A cos(x + vt)
∂ y
∂ x
= −A sin(x + vt)
∂ 2 y
∂ x 2 = −A cos(x + vt)
∂ y
∂t
= −v A sin(x + vt)
∂ 2 y
∂t 2 = −v
2 A cos(x + vt)
∴
1
v 2
∂ 2 y
∂t 2 = −A cos(x + vt) =
∂ 2 y
∂ x 2
Thus the wave equation is satisfied.
8.15 (a) By prob. (8.3)
y =
∞
n=1
a n sin
nπ x
L
cos
nπvt
L
(1)
a n =
2hL 2
n 2 π 2 d(L − d)
sin
nπ d
L
(2)
Here d =
L
3
and (2) becomes
a n =
9h
n 2 π 2 sin
nπ
3
(3)
Inserting (3) in (1)
∴ y =
3 5/2 h
2π 2
sin
π x
L
cos
πvt
L
+
1
4
sin
2π x
L
cos
2πvt
L
−
1
16
sin
4π x
L
cos
4vt
L
. . .
(4)
(b) For n = 3, 6 or 9, the sine term in (3) becomes zero. Therefore, the third,
sixth and ninth harmonics will be absent.