8.3 Solutions
357
Given function is y = A
√
x + vt
∂ y
∂ x
=
A
2
√
x + vt
∂ 2 y
∂ x 2 = −
A
4(x + vt) 3/2
(2)
∂ y
∂t
=
Av
2
√
(x + vt)
∂ 2 y
∂t 2 = −
A
4
v 2
(x + vt) 3/2
(3)
Equation (1) is satisfied with the use of (2) and (3).
8.2 The wave equation is
∂ 2 y
∂ x 2 =
μ
F
∂ 2 y
∂t 2
(1)
y = 2A sin
nπ x
L
cos(2πft) (standing wave)
∂ y
∂ x
=
2Anπ
L
cos
nπ x
L
cos(2πft)
∂ 2 y
∂ x 2 = −
2An 2 π 2
L 2
sin
nπ x
L
cos(2πft) = −
n 2 π 2 y
L 2
(2)
∂ y
∂t
= −4π f A sin
nπ x
L
sin(2πft)
∂ 2 y
∂t 2 = −8π
2 f
2 A sin
nπ x
L
cos(2πft) = −4π
2 f
2 y
but
F
μ
= v = f λ
∴
μ
F
∂ 2 y
∂t 2 = −
4π 2 f 2 y
v 2
= −
4π 2 y
λ 2 = −
n 2 π 2 y
L 2
(3)
∵ L =
nλ
2
Thus
μ
F
∂ 2 y
∂t 2 =
∂ 2 y
∂ x 2
8.3 Let the string AB of length L be plucked at the point C, distant d from the end
A and be raised through height h, Fig. 8.4.