380
8 Waves
8.54 P = 2.4 sin π(x − 330t) = 2.4 sin 2π
1
2
x − 165t
P = P max sin 2π
x
λ
− f t
(standard expression)
On comparing the two expressions we find
(a) 2.4 N/m 2 , (b) 165 Hz, (c) 2.0 m, (d) v = f λ = 165 × 2 = 330 m/s
8.55 I = 2π
2
ρ 0 A
2 f
2
v
∴ A =
1
π f
I
2ρ 0 v
=
1
1200π
2 × 10 −6
2 × 1.293 × 330
= 1.28 × 10
−4 m
8.56 P e = 1 microbar = 10
−6 bar = 0.1 N/m
2
I =
P 2
e
ρ 0 v
=
(0.1) 2
1.293 × 330
= 2.34 × 10
−5 W/m
2
IL = 10 log
I
I 0
= 10 log
2.34 × 10 −5
10 −12
= 10(7 + log 2.34) = 73.7 dB 74 dB
8.57 IL = 10 log
I
I 0
70 = 10 log
I
10 −12 = 10 [log I + 12]
log I = −5 I = 10
−5 W/m
2
Energy density
E =
I
v
=
10 −5
331
= 3 × 10
−8 J/m
3
Effective pressure
P e =
Iρ 0 v =
10 −5 × 1.293 × 331 = 0.0654 N/m
2
8.58 P max =
2Iρ 0 v =
√
2 × 1 × 1.293 × 331 = 29.26 N/m
2
8.59 v =
γ RT
M
=
1.4 × 8.317 × 273
2.016 × 10 −3 = 1256 m/s
8 Waves
8.54 P = 2.4 sin π(x − 330t) = 2.4 sin 2π
1
2
x − 165t
P = P max sin 2π
x
λ
− f t
(standard expression)
On comparing the two expressions we find
(a) 2.4 N/m 2 , (b) 165 Hz, (c) 2.0 m, (d) v = f λ = 165 × 2 = 330 m/s
8.55 I = 2π
2
ρ 0 A
2 f
2
v
∴ A =
1
π f
I
2ρ 0 v
=
1
1200π
2 × 10 −6
2 × 1.293 × 330
= 1.28 × 10
−4 m
8.56 P e = 1 microbar = 10
−6 bar = 0.1 N/m
2
I =
P 2
e
ρ 0 v
=
(0.1) 2
1.293 × 330
= 2.34 × 10
−5 W/m
2
IL = 10 log
I
I 0
= 10 log
2.34 × 10 −5
10 −12
= 10(7 + log 2.34) = 73.7 dB 74 dB
8.57 IL = 10 log
I
I 0
70 = 10 log
I
10 −12 = 10 [log I + 12]
log I = −5 I = 10
−5 W/m
2
Energy density
E =
I
v
=
10 −5
331
= 3 × 10
−8 J/m
3
Effective pressure
P e =
Iρ 0 v =
10 −5 × 1.293 × 331 = 0.0654 N/m
2
8.58 P max =
2Iρ 0 v =
√
2 × 1 × 1.293 × 331 = 29.26 N/m
2
8.59 v =
γ RT
M
=
1.4 × 8.317 × 273
2.016 × 10 −3 = 1256 m/s