Lecture 8Standing sound waves.

Resonance.

Interference with sound

Superposition works exactly as it did for transversal waves.

Additional complication: 3D waves! (next lecture)

Reflection of sound waves against a surface

Consider a sound pulse (air moves to the right and back to initial position) traveling along a pipe toward a closed end:

A closed end is a “fixed end”

v

s

x

s v

Incoming pulse

s

v

x

s

v

Reflected pulse

No displacement: s

= 0

Wave must be inverted (s becomes –

s)

Reflection of sound at an open end

v

s

…and causes a wave to propagate back in (a reflection!)

v

s

v

(and another wave is transmitted outside)

Pulse travels out into open air

v

…and increases pressure…

Oscillation back from a larger slice moves more air into pipe…

Beyond the open end of the pipe, variations in the pressure must be much smaller than pressure variations (gauge pressure) in pipe.

Just beyond the open end, 0p

Boundary conditions for sound

Closed end• air displacement = 0• maximum (absolute) gauge pressure

Open end• gauge pressure = 0• maximum (absolute) air displacement

Standing sound waves in pipe open at both ends

A harmonic wave and its reflection on an open end:

1 max( , ) sin( )s x t s kx t 2 max( , ) sin( )s x t s kx t

all max( , ) sin( ) sin( )s x t s kx t kx t

sin sin 2cos sin2 2

a b a ba b

Standing wave within pipe: does not travel, bounces back and forth.Amplitude will decrease as energy is transported out of the pipe

all max( , ) 2 sin coss x t s kx t

At the openings:p ~ 0Large displacements

ACT: Pipe open at both ends

This is the air displacement for a standing wave inside this tube.

Sketch the gauge pressure vs position for this wave.

Compare with your neighbor and discuss.

p = 0 at open endsMaximum/minimum p at node

p

x

Higher harmonics

Each harmonic is a standing wave.

1,2,3...2

L n n

2 1,2,3...n

Ln

n

gets shorter, frequency increases

Visualize them: http://www.walter-fendt.de/ph11e/stlwaves.htm

DEMO: Organ pipes

ACT: Pipe closed at one end

First harmonic or fundamental frequency:

Closed end:s = 0, max p

Open end:Max s, p = 0

s

What is the standing wave for the next harmonic?

A

B

C

s = 0 at an open end? (No!)

And s max at a closed end? (No!)

2

4

3L

1 4 L

odd odd

I n general,

1,3...4

L n n

In-class example

A. 4f B. 2f C. f D. f/4 E. None of the above

A tube with both ends open has a fundamental frequency f. What is the fundamental frequency of the same tube if one end is closed?

Close end = node

Open end = antinode

4 2 L

2 L

2 2v v f

f

A little music

When you blow air into a pipe, all the harmonics are present.

DEMO: Xylophone

Example: Blow into a tube of length 19.2 cm open at one end

1 1 5

343 m/ s2 890 Hz Approx. A (La)

2 2 0.192 m

vL f

L

Resonance

To produce a wave, we need to apply an external force (driving force). This driving force can be periodic with frequency fD.

The amplitude of the perturbation is maximum when the frequency of the driving force is equal to one of the natural (or harmonic, or normal) frequencies of the system.

Examples:

Pendulum: resonance occurs when

(A pendulum has only one normal frequency)

D 2 ( length of string)g

f LL

String fixed at both ends: when D f or 1,2,... (see lecture 6)2n

F nff n

L

Pipe closed at one end: when oddD sound odd f or 1,3,...

4n

nff v n

L

DEMO: Resonant

slabs