374
8 Waves
Put kh = x, then
tanh x
x
= 0.96
This gives the solution x = 0.25 or h =
x
k
=
0.25λ
2π
= 0.04λ.
8.37 Surface conditions are modified by the surface tension S. For the capillary
waves
v
2
=
2π S
ρλ
+
gλ
2π
tanh
2π h
λ
(1)
If h >> λ, tanh
2π h
λ
→ 1, and
v
2
=
2π S
ρλ
+
gλ
2π
(2)
Substituting λ = 0.366 cm, ρ = 1.0 g/cm 3 , g = 980 cm/s 2 and v = f λ =
100 × 0.366 = 36.6 cm/s in (2) we find S = 74.7 dynes/cm.
8.38 For capillary waves when h >> λ
v
2
=
2π S
ρλ
+
gλ
2π
(1)
The minimum value of the wavelength λ m can be found out by minimizing (1):
∂(v 2 )
∂λ
= −
2π S
ρλ 2
m
+
g
2π
= 0
λ m = 2π
S
gρ
(2)
Ignoring the second term in the right-hand side of (1) and using (2)
v =
gs
ρ
1/4
For mercury and water
v 1 : v 2 =
S 1
ρ 1
1/4
:
S 2
ρ 2
1/4
=
544
13.56
1/4
:
74
1
1/4
= 0.858 : 1
8 Waves
Put kh = x, then
tanh x
x
= 0.96
This gives the solution x = 0.25 or h =
x
k
=
0.25λ
2π
= 0.04λ.
8.37 Surface conditions are modified by the surface tension S. For the capillary
waves
v
2
=
2π S
ρλ
+
gλ
2π
tanh
2π h
λ
(1)
If h >> λ, tanh
2π h
λ
→ 1, and
v
2
=
2π S
ρλ
+
gλ
2π
(2)
Substituting λ = 0.366 cm, ρ = 1.0 g/cm 3 , g = 980 cm/s 2 and v = f λ =
100 × 0.366 = 36.6 cm/s in (2) we find S = 74.7 dynes/cm.
8.38 For capillary waves when h >> λ
v
2
=
2π S
ρλ
+
gλ
2π
(1)
The minimum value of the wavelength λ m can be found out by minimizing (1):
∂(v 2 )
∂λ
= −
2π S
ρλ 2
m
+
g
2π
= 0
λ m = 2π
S
gρ
(2)
Ignoring the second term in the right-hand side of (1) and using (2)
v =
gs
ρ
1/4
For mercury and water
v 1 : v 2 =
S 1
ρ 1
1/4
:
S 2
ρ 2
1/4
=
544
13.56
1/4
:
74
1
1/4
= 0.858 : 1