8.3 Solutions
377
For large waves, first term in (6) is important:
v p =
g
k
=
gλ
2π
=
9.8 × 1.0
2π
= 1.25 m/s
v g =
1
2
v p = 0.625 m/s
8.43 E 2 = c 2 p 2 + m 2 c 4
¯
h
2
ω
2
= c
2 ¯
h
2 k
2
+ m
2 c
4
∴ ω =
c 2 k 2 +
m 2 c 4
¯
h 2
v p =
ω
k
=
c 2 +
m 2 c 4
¯
h 2 k 2
v g =
dω
dk
=
c 2 k
c 2 k 2 +
m 2 c 4
¯
h
2
=
c 2
c 2 +
m 2 c 4
¯
h
2 k 2
∴ v p v g = c
2
8.44 v p =
gλ
2π
+
2π S
ρλ
(1)
Substituting v p = 30 cm/s, g = 980 cm/s 2 , S = 75 dynes/cm and ρ =
1 g/cm 3 , on simplification (1) reduces to the quadratic equation in λ:
λ
2
− 5.767λ + 1.153 = 0
The two roots are λ 1 = 5.56 cm and λ 2 = 0.207 cm.
In determining surface tension it is preferable to use the shorter wavelength
because the surface effect will dominate over gravity:
8.45 v p =
ω
k
=
g
k
v g =
dω
dk
=
1
2
g
k
∴ v g =
1
2
v p
377
For large waves, first term in (6) is important:
v p =
g
k
=
gλ
2π
=
9.8 × 1.0
2π
= 1.25 m/s
v g =
1
2
v p = 0.625 m/s
8.43 E 2 = c 2 p 2 + m 2 c 4
¯
h
2
ω
2
= c
2 ¯
h
2 k
2
+ m
2 c
4
∴ ω =
c 2 k 2 +
m 2 c 4
¯
h 2
v p =
ω
k
=
c 2 +
m 2 c 4
¯
h 2 k 2
v g =
dω
dk
=
c 2 k
c 2 k 2 +
m 2 c 4
¯
h
2
=
c 2
c 2 +
m 2 c 4
¯
h
2 k 2
∴ v p v g = c
2
8.44 v p =
gλ
2π
+
2π S
ρλ
(1)
Substituting v p = 30 cm/s, g = 980 cm/s 2 , S = 75 dynes/cm and ρ =
1 g/cm 3 , on simplification (1) reduces to the quadratic equation in λ:
λ
2
− 5.767λ + 1.153 = 0
The two roots are λ 1 = 5.56 cm and λ 2 = 0.207 cm.
In determining surface tension it is preferable to use the shorter wavelength
because the surface effect will dominate over gravity:
8.45 v p =
ω
k
=
g
k
v g =
dω
dk
=
1
2
g
k
∴ v g =
1
2
v p