8.2 Problems
351
8.38 Compare the minimum velocities of surface waves at 10 ◦ C for mercury and
water if the surface tensions are 544 and 74 dyne/cm, respectively, and the
specific gravity of mercury is 13.56.
8.39 It is only when a string is perfectly flexible that the phase velocity of a wave
on a string is given by
√
T /μ. The dispersion relations for the real piano wire
can be written as
ω 2
k 2 =
T
μ
+ ak
2
where α is a small positive quantity which depends on the stiffness of the
string. For perfectly flexible string, α = 0. Obtain expressions for phase
velocity (v p ) and group velocity v g and show that v p increases as wavelength
decreases.
8.40 The dispersion relation for water waves of very short wavelength in deep water
is ω 2 =
S
ρ k 3 , where S is the surface tension and ρ is the density.
(a) What is the phase velocity of these waves?
(b) What is the group velocity?
(c) Is the group velocity greater or less than the phase velocity?
8.41 The general dispersion relation for water waves can be written as
ω
2
=
gk +
s
ρ
k
3
tanh kh
where g is acceleration due to gravity, ρ is the density of water, S is the surface
tension and h is the water depth. Use the properties of tanh x function viz. for
x >> 1, tanh x = 1 and for x << 1, tanh x = x.
Show that (a) in shallow water the group velocity and the phase velocity are
both equal to
√
gh if the wavelength is long enough to ensure that Sk 2 /v =
4π 2 S/λ 2 ρ << g. (b) Show that for deep water the phase velocity is given by
v p =
g
k + Sk/ρ and find the group velocity.
8.42 For water ρ = 10 3 kg/m
3 and S = 0.075 N/m. Calculate v p and v g in deep
water for small ripples with λ = 1 cm and for large waves with λ = 1 m.
8.43 The relation for total energy E and momentum p for a relativistic particle is
E 2 = c 2 p 2 + m 2 c 4 , where m is the rest mass and c is the velocity of light.
Using the relations, E = ¯
hω and p = ¯
hk, where ω is the angular frequency
and k is the wave number and ¯
h = h/2π , h being Planck’s constant. Show
that the product of group velocity v g and the phase velocity v p , v p v g = c 2 .
8.44 Taking the surface tension of water as 0.075 N/m its density as 1000 kg/m
3 ,
find the wavelength of surface waves on water with a velocity of 0.3 m/s.
351
8.38 Compare the minimum velocities of surface waves at 10 ◦ C for mercury and
water if the surface tensions are 544 and 74 dyne/cm, respectively, and the
specific gravity of mercury is 13.56.
8.39 It is only when a string is perfectly flexible that the phase velocity of a wave
on a string is given by
√
T /μ. The dispersion relations for the real piano wire
can be written as
ω 2
k 2 =
T
μ
+ ak
2
where α is a small positive quantity which depends on the stiffness of the
string. For perfectly flexible string, α = 0. Obtain expressions for phase
velocity (v p ) and group velocity v g and show that v p increases as wavelength
decreases.
8.40 The dispersion relation for water waves of very short wavelength in deep water
is ω 2 =
S
ρ k 3 , where S is the surface tension and ρ is the density.
(a) What is the phase velocity of these waves?
(b) What is the group velocity?
(c) Is the group velocity greater or less than the phase velocity?
8.41 The general dispersion relation for water waves can be written as
ω
2
=
gk +
s
ρ
k
3
tanh kh
where g is acceleration due to gravity, ρ is the density of water, S is the surface
tension and h is the water depth. Use the properties of tanh x function viz. for
x >> 1, tanh x = 1 and for x << 1, tanh x = x.
Show that (a) in shallow water the group velocity and the phase velocity are
both equal to
√
gh if the wavelength is long enough to ensure that Sk 2 /v =
4π 2 S/λ 2 ρ << g. (b) Show that for deep water the phase velocity is given by
v p =
g
k + Sk/ρ and find the group velocity.
8.42 For water ρ = 10 3 kg/m
3 and S = 0.075 N/m. Calculate v p and v g in deep
water for small ripples with λ = 1 cm and for large waves with λ = 1 m.
8.43 The relation for total energy E and momentum p for a relativistic particle is
E 2 = c 2 p 2 + m 2 c 4 , where m is the rest mass and c is the velocity of light.
Using the relations, E = ¯
hω and p = ¯
hk, where ω is the angular frequency
and k is the wave number and ¯
h = h/2π , h being Planck’s constant. Show
that the product of group velocity v g and the phase velocity v p , v p v g = c 2 .
8.44 Taking the surface tension of water as 0.075 N/m its density as 1000 kg/m
3 ,
find the wavelength of surface waves on water with a velocity of 0.3 m/s.