Slides for the webPhysics 371 February 12, 2002 and February 14, 2002

• Strings: effect of stiffness• Pipes open pipes - harmonics

closed pipes

• Resonance width of resonance curve dependence on damping demos• Sound Spectrum (Fourier)

First exam on Thursday, Feb, 21. Study guide and answers have been handed out. Covers Ch. 1-4, and homework 1-4

Pan and nymph

today’s music: Pan Pipes

Plucked (or Bowed) String

example: pluck string at 1/4 point from end. which harmonics will be strong? which harmonics will be absent?

Answer:

2nd harmonic has belly where string is plucked: STRONGEST

4th harmonic has NODE where string is plucked: ABSENT

8th harmonic …. ABSENT

other harmonics: more or less present, depending how much amplitude they have at pt. where plucked.

effects from stiffness of string:higher modes are sharp (more bending required at string end)larger stiffness: more inharmonicity more damping of higher modes commercial strings: steel, gut, or "synthetic" e.g.gut core, nylon overwrap and outer wrap of silver or aluminium

finger on finger board changes tension -> change in pitcheffect is largest for metal strings.

String stiffness

Pipes

(woodwinds, brass, organ pipes)

an “open” pipe(open at both ends)

L

at open end, no pressure build-up because air

is free to escape:

OPEN END is always a PRESSURE NODE

Longitudinal wave

L =λ2

air press.

L

Fundamental Oscillation:

f1 =v

2L

fundamental freq:

Example:find length of flute of frequency C = 260 Hz

L =v

2f1=

344msec

2×2601sec

=0.66m

demo: 1.25 m long pipe

f is (almost) independent of pipe diam!

air press.

L

air press.

L

half a period later:

How change pitch of pipe? f = v/2Lcan ONLY change L (fingerholes on flute)can’t change speed of sound v!diameter has (almost) no effect!But can “overblow” to higher modes!!

air flow

Higher modes of flute: examplef1 first mode (fundamental) 260Hz

f2 = 2f1 second mode (first overtone) 520Hz

f3 = 3f1 third mode (second overtone) 780Hz

fn = nf1 MODES ARE HARMONICS

demo: modes of pipe - plastic tube

graphs of pressure and air velocity on blackboard

at pressure node air speed has antinode

at pressure antinode air speed had node why?

Closed Pipe

Pipe closed at ONE end: closed endpressureantnode

air press.

L = /4

L

fundamental frequency of closed pipe:

f1 =v4L

note: this is half the frequency of anopen pipe of same length (octave below)

open end: pressure NODE (motion antinode)closed end: pressure antinode (motion node)

example: how long is a A1 organ pipe? (Answ: 1.56m = 5 ft if closed pipe vs. 3.12 m = 10 ft if open pipe)

L

1st mode (fundamental)f1 = v/4L (first harmonic)

2nd mode (first overtone) f = 3f1 third harmonic

3rd mode (second overtone) f = 5f1 fifth harmonic

closed end

press distribution

press.

0

0

0

Higher modes of closed pipe: need pressure NODE at open end pressure BELLY at closed end

odd multiples of fundametal

conical pipes: oboe, bassoon

same frequency ascylindrical pipe:

why? - not obvious and theory is difficult math! (text book tries to explain it...)

L

f1 =v

2L

Resonance:

• oscillating system has “natural” frequency f0 when it is oscillating on it’s own• push on oscillating system at steady rate - driving frequency fD

• observe amplitude of oscillation as you vary fD: amplitude peaks at resonance frequency f0

• with of resonance f measures how far you can be off in frequency before amplitude drops to 1/2 of peak

RESONANCEResonance curves for different amount damping (friction)

width of res curve at half max

more friction - wider res curve

less damping - narrower res curve

driving frequency fD (Hz)= frequency of push

amplitudeof osc.

wider resonance - can “pull” frequency of instrument

strings: small dampingwinds: large damping

RELATIONSHIP BETWEENDAMPING TIME f AND RESONANCE WIDTH

f width of res curve at half max

more friction - wider res curve

driving frequency fD (Hz)

amplitudeof osc.

width of resonance curve and damping time: inverse relation

f

f

Examples:

1. Sitar (N. India) 7 strings + 11 sympathetic strings

2. Marimba

More Examples:

3. Soundboards of instruments avoid resonances

4. loudspeaker: flexible cardboard speaker cone supported on springy rim.

It is supposed to respond almost uniformly over a wide frequency range

Thus: wide resonance curve and short damping time Thus: large friction -> inefficient (100W amp for few Wsound) tweeter + midrange + woofer to even out freq. response.

5. tone dialing: resonance circuits at phone center

Sound Spectrum (Fourier Spectrum)

Fourier: represent complicated periodic oscillation (period T) as sum of sinusoidal oscillations of frequencies f1= (1/T) and harmonics f2=2f1, f3=3f1 etc.

amplitudeof harmonic

freq of harmonic (Hz)0 200 400 600

easy visualization of harmonic content (timbre) but contains no information about relative timing of overtones (phase).

Fourier Synthesizer….. produces frequencies

f1 , 2f1, 3f1, 4f1, 5f1 6f1, 7f1 etc

of adjustable amplitude and phase .

e.g. f1 = 440 Hz = A4 can synthesize any

440 Hz wave shape.

Fourier Analyzer….. shows graph of Fourier spectrum

(amplitude and frequency of sine wave components)

of periodic wave (voice or instrument)

Tone Quality (Timbre)

In acoustic theory, what exactly is “timbre”? Timbre is that attribute that differentiates two tones of same loudness and same pitch.

HOWEVER:The Fourier Spectrum (frequencies and intensities of overtones) is only one aspect of timbre……..

Other aspect of tone quality: rise and decay An example of two tonal presentations which show importance of the tone envelope (attack and decay)