Lecture 7Sound waves. Speed of

sound.Intensity.

ACT: Dust in front of loudspeaker

Consider a small dust particle, suspended in air (due to buoyancy)

dust particlespeaker

When you turn on the speaker, the dust particle

A. oscillates back and forth horizontally, and moves slowly to the right

B. steadily moves to the rightC. oscillates back and forth horizontally

DEMO: Candle

Pressure/density oscillations

Gas in equilibrium: pressure and density are uniform.

Sound wave: periodic longitudinal oscillations of particles in the gas

Consider one slice of air:1. Oscillation to the right causes pressure to increase2. Increase in force causes neighboring air to be displaced sound wave propagates3. Slice of air oscillates back to region of low pressure

The small volumes of air do not propagate with wave, they only oscillate around their equilibrium position.

DEMO: Sound in vacuum.

Harmonic longitudinal waves

Consider a gas in long, thin, horizontal tube. Each particle of gas oscillates horizontally in a harmonic way:

max( , ) cos( )s x t s kx t

air normally at x = 0, displaced to right by 10 m

air normally at x = 50 cm, displaced to left by 10 m

Pressure, density oscillations

Zero displacement ↔ Maximum density and pressure

Air from both sides momentarily accumulates in middle

Here air is to the right of where it should be

Here air is to the left of where it should be

Pressure and density oscillations

maxDisplacement ( , ) cos( )s x t s kx t

It all boils down to a phase difference:

maxPressure ( , ) sin( )p x t p kx t

Note that p is the gauge pressure. The pressure of air in equilibrium is patm. The oscillations give a total pressure atmtotal( , ) ( , )p x t p p x t

maxDensity ( , ) sin( )x t kx t

Density oscillations are also about the regular air density. Total density is total 0( , ) ( , )x t x t

Relation between displacement and pressure

Consider a pipe of cross-sectional area A filled with air, and a small element at x with thickness Δx.

In equilibrium:x

Δx

p0p0

Due to a wave, element moves and changes its size

x + s

Δx + Δs

p0 + p2 p0 + p1

Pressure and displacement are related through the bulk modulus of the air!

V A x

V A s

p

BV

V

(gauge pressure)

0 x

V s sV x x

( , )( , )

s x tp x t B

x

max ( , ) sin( )p x t Bks kx t

maxpmax( , ) cos( )s x t s kx t

The harmonic case:

x + s

Δ x + Δs

p0 + p2 p0 + p1

2

1 2 2

21 2

2

( )

( )

sp p A A x

tp p s

x t

m A x Mass of the element:

x

Δ x

p0p0

Sound wave speed

1 2( )F p p A Net force on the element:

Acceleration of the element:

2

2

sa

t

2

2

p sx t

0x

2

2

p sx t

sp B

x

2 2

2 2

s sB

t x

2 2

2 20

s sBx t

Wave equation with

Bv

2

2

p sB

x x

DEMO: Organ pipe with different gases.

Video.

video

In-class example: Sonar

A sound wave in water has a frequency of 1000 Hz. What is its wavelength? (B water = 2.0 GPa, ρ water = 1000 kg/m3)A. 1.4 mmB. 0.14 mC. 14 mD. 1400 mE. None of the above

9

3 3

2.0 10 Pa1414 m/ s

10 kg/ mB

v

1414 m/ s1.4 m

1000 Hzvf

Wave speed, in general

Fv

String

:

Bv

Sound in a fluid:

restoring f orce property

inertial propertyv In general:

Yv

Sound in a solid:

ACT: A sixth sense?

A large ammunition factory and a town are separated by a rocky hill, at a horizontal distance of about 5 km. An accident produces a huge explosion in the middle of the night. What do the town inhabitants experience?

A. First the room shakes, and then they hear an explosion.

B. First they hear an explosion, and then the room shakes.

C. They hear an explosion and the room shakes at the same time.

5 km

Time for sound wave to reach the town:

granite

granite

6000 m/ sY

v

Through hill (granite):

5000 m0.8 s

6000 m/ sx

tv

This happened in California during WWII. Most people woke up (distressed…) to the light quake and then heard the explosion. Many attributed this to a “sixth sense” that had warned them of the imminent disaster. The “sixth sense” was just the laws of wave propagation…

Through air:

5000 m14.6 s

343 m/ sx

tv

14 seconds later!

Intensity

Example: A siren emits a sound of power 2W at 100 m from you. How much power reaches your ear (eardrum area = 0.7 cm2)Intensity at distance r from source:

5 2at source

2 2

2 W1.6 10 W/ m

4 4 100 mR

PI

r

area

PI

Average power (over time) in wave

Area of the surface where this power is distributed

5 2 4 2eardrum area of eardrum 1.6 10 W/ m 0.7 10 m 1.1 nWRP I

Power absorbed by eardrum:

r

2

Sphere of

area 4 r

Distance and amplitude

At distance r from the source, the power isr rP I2

1

r

2AmplitudeP We also know that

1Amplitude decreases as

r

Intensity for harmonic waves

PI

A

F v

A

x xF v

A

dsp

dt

max sin( )ds

s kx tdt

max( , ) cos( )s x t s kx t

max( , ) sin( )p x t Bks kx t

2 2max sin ( )I Bk s kx t 2

max

12

Bk s

max maxOr, in terms of p Bks2maxp

IB

2 2max

12

I B s Bv

Useful to include frequency effects

Sound intensity level

12 20

0

10log with 10 W/ mI

II

Units: decibels

Threshold of human hearing: 10-12 W/m2

0

Normal conversation: 10-6 W/m2

65 decibels

Threshold of pain: 1 W/m2

120 decibels

Twice the decibels does NOT feel twice as loud!