8.3 Solutions
367
Wave velocity v =
ω
k
=
471
62.8
= 7.5 m/s
λ =
2π
k
=
2π
62.8
= 0.1 m
Distance between nodes =
λ
2
=
0.1
2
= 0.05 m
8.24 y = A sin(kx + ωt)
(i) y = 8.2 × 10 −2 sin(22x + 100t) (negative x-direction)
(ii) y = 8.2 × 10 −2 sin(100t − 22x) (positive x-direction)
(iii) λ =
2π
k
=
2π
22
= 0.2856 m
T =
2π
ω
=
2π
100
= 0.0628 m
v =
ω
k
=
100
22
= 4.545 m/s
(iv) y = 8.2 × 10
−2
× sin(22 × 3.2 + 100 × 2.5)
= 8.2 × 10
−2
× sin(51 × 2π) = 0
8.25
F
μ
1/2
=
M LT −2
M L −1
1/2
=
LT
−1
= [v]
8.26 Let the travelling wave be represented by
y = A sin(kx − ωt)
Then
∂ y
∂ x
= k A cos(kx − ωt)
(1)
∂ y
∂t
= −ω A cos(kx − ωt)
= −vk A cos(kx − ωt) = −v
∂ y
∂ x
(2)
Combining (1) and (2),
∂ y
∂ x
= −
∂ y
∂t
/v.
8.27 (a) Let a long string of linear density μ be stretched by a force F. Assume that
the damping is negligible. Take the x-axis in the direction of the undisplaced string and y-axis in the direction perpendicular to it. If θ is the
angle between the tangent to the string and the x-axis, the tension in the
horizontal direction (x-axis) would be T cos θ and in the vertical direction
(y-axis) it would be T sin θ . Assuming that θ is very small, cos θ 1 and
consequently the x-component of the tension remains constant. We are
therefore concerned only with the y-component of the tension.
367
Wave velocity v =
ω
k
=
471
62.8
= 7.5 m/s
λ =
2π
k
=
2π
62.8
= 0.1 m
Distance between nodes =
λ
2
=
0.1
2
= 0.05 m
8.24 y = A sin(kx + ωt)
(i) y = 8.2 × 10 −2 sin(22x + 100t) (negative x-direction)
(ii) y = 8.2 × 10 −2 sin(100t − 22x) (positive x-direction)
(iii) λ =
2π
k
=
2π
22
= 0.2856 m
T =
2π
ω
=
2π
100
= 0.0628 m
v =
ω
k
=
100
22
= 4.545 m/s
(iv) y = 8.2 × 10
−2
× sin(22 × 3.2 + 100 × 2.5)
= 8.2 × 10
−2
× sin(51 × 2π) = 0
8.25
F
μ
1/2
=
M LT −2
M L −1
1/2
=
LT
−1
= [v]
8.26 Let the travelling wave be represented by
y = A sin(kx − ωt)
Then
∂ y
∂ x
= k A cos(kx − ωt)
(1)
∂ y
∂t
= −ω A cos(kx − ωt)
= −vk A cos(kx − ωt) = −v
∂ y
∂ x
(2)
Combining (1) and (2),
∂ y
∂ x
= −
∂ y
∂t
/v.
8.27 (a) Let a long string of linear density μ be stretched by a force F. Assume that
the damping is negligible. Take the x-axis in the direction of the undisplaced string and y-axis in the direction perpendicular to it. If θ is the
angle between the tangent to the string and the x-axis, the tension in the
horizontal direction (x-axis) would be T cos θ and in the vertical direction
(y-axis) it would be T sin θ . Assuming that θ is very small, cos θ 1 and
consequently the x-component of the tension remains constant. We are
therefore concerned only with the y-component of the tension.