384
8 Waves
Using (4) in (2), we get
v =
γ P 0
ρ 0
(Laplace formula)
(5)
where P 0 and ρ 0 refer to equilibrium conditions of pressure and density.
The velocity υ 0 at 0 ◦ C can be found out by substituting γ = 1.4, P 0 =
1.013 × 10 5 N/m 2 and ρ 0 = 1.293 kg/m
3 . We find v 0 = 331.2 m/s, in good
agreement with the experiment.
With the assumption of isothermal changes we would have obtained the
formula v =
√
P 0 /ρ 0 , (Newton’s treatment) which gives a value of 20%
lower.
8.70 (a) v| t=20 = 1403 + 5 × 20 − 0.06 × (20) 2 + 0.0003 × (20) 3 = 1481.4 m/s
(b)
dv
dt
= 5 − 0.12t + 0.0009t 2
dv
dt
| t=20 = 5 − 0.12 × 20 + 0.0009 × (20)
2
= 2.62 m/s/
◦ C
8.3.5 Doppler Effect
8.71 v s = 72 km/h = 72 ×
5
18
m/s = 20 m/s
f 0 =
v f s
v + v s
(direct)
f
0 =
v f s
v − v s
(reflected from the wall of the rock)
f
0
f 0
=
v + v s
v − v s
=
340 + 20
340 − 20
=
9
8
8.72 v 0 = v s = 90 km/h = 90 ×
5
18
= 25 m/s
v = 350 m/s
f
=
f (v + v 0 )
v − v s
=
520(350 + 25)
350 − 25
= 600 Hz.
8.73 f
=
f (v − v 0 )
v + v s
= 520
(330 − 25)
330 + 25
= 446.8 Hz.
8.74 Let the whistle rotate clockwise, Fig. 8.7. At point A the linear velocity of
the whistle will be towards the distant listener and at B away from the listener.