SOUND
a) There is NO sound in vacuum; you need matter to propagate a sound wave.
b) SOUND: Any mechanical wave whose frequency lies between 20Hz and 20,000Hz, that
is, 20 Hz ≤ f ≤ 20kHz (It is called sound because you can hear it!).
c) We will work with sound in Gases only-then sound is a purely Longitudinal wave.
d) Sound is a longitudinal displacement wave or a longitudinal pressure wave.
e) Periodic Sound wave properties
DISPLACEMENT
Sine wave, ∅ = 0
S = Sm Sin (κx − ωt )
amplitude Sm || x$
ω = vk
→
PRESSURE
To write corresponding pressure wave we have to
realize that the variation occurs so rapidly that there is
no possibility for exchange of heat (DQ) to ensure
equilibrium with surroundings, so DQ=0, sound is an
adiabatic process: Pressure and Volume satisfy:
γ
PV
0 0 = constant.
P0 = ambient pressure
γ =
Cp
Cv
, Cp = sp ht at const P
Cv = sp ht at const V
γ monoatomic =
5
3
7
5
−π
→ ∅ =
2
P = P0 − γ P0 Smκ Cos(κ x − ω t )
γ diatomic =
Displacement oscillates about zero.
Pressure oscillates about P0
Amp of pressure wave
Pm = γ P0 Smκ
Pm || x$
→
π
out of phase with displacement where S
2
is max. (P-P0)=0!
Pressure is
f) Speed of sound in a gas
γP0
v=
ρ0 = ambient density
since PV
0 0 = N k BT
[k
ρ0
B
= 1383
. x10−23 J / K
]
If gas particles have mass m, we can write
P0 =
Nm k B
P k T
T , or 0 = B
V0 m
ρ0 m
v=
v
γ k BT
= r rms
m
3
[T in Kelvin Scale]
g) Intensity of sound wave: Imagine that the wave is traveling with velocity v through a
tube of cross-section A.
Since S = Sm Sin (κ x − ω t ) the
particle velocity is
Vp = Smω cos(κ x − ω t ) and kinetic energy
1
ρ S 2ω 2 .
2 0 m
Volume of wave traveling past every
cross-section will be Av in one second.
Energy transport per second through area
1
A = A ρ 0 Sm2ω 2v
2
Intensity I=energy transport per second per
m2
γP ⎤
1
⎡
= ρ 0 Sm2ω 2v
ρ 0 = 20 ⎥
⎢
2
v ⎦
⎣
ω2
1
= γP0 Sm2
v
2
per unit volume is K ⋅ E =
1 2 ω2
⋅F
Please compare this with energy transport per second on wire P = A
v
2
h) Smallest discernable intensity is I 0 = 10−12 Watt / m2
So we define decibels db = 10 log
That is 90db sound has 9 = log
I
[bel comes from Alexander Graham Bell]
I0
I
, I = 10−3Watt / m2
10−12
. , P0 = 105 N / m2 ) ,
i) Amplitude of displacement wave for I 0 ( w = 103 rad / s, γ = 14
1
S 2 x106
x14
. x105 x m
340
2
− 10
Sm ≅ 10 m
10−12 =
Roughly equal to diameter of hydrogen atom. REMARKABLE!!!
That is your ear is sensitive to motion of air molecules where displacement is equal to the
diameter of a hydrogen atom. Congratulations!!