Slides for the web Physics 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: 2 nd harmonic has belly where string is plucked : STRONGEST 4 th harmonic has NODE where string is plucked: ABSENT 8 th 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 pitch effect 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

air press. L Fundamental Oscillation: fundamental freq: Example:find length of flute of frequency C = 260 Hz demo: 1.25 m long pipe f is (almost) independent of pipe diam!

air press. L L half a period later: How change pitch of pipe? f = v/2L can 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: example f 1 first mode (fundamental) 260Hz f 2 = 2f 1 second mode (first overtone) 520Hz f 3 = 3f 1 third mode (second overtone) 780Hz f n = nf 1 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 end pressure antnode air press. L = /4 L

fundamental frequency of closed pipe: note: this is half the frequency of an open pipe of same length (octave below ) open end: pressure NODE (motion antinode) closed end: pressure antinode (motion node) example: how long is a A 1 organ pipe? (Answ: 1.56m = 5 ft if closed pipe vs m = 10 ft if open pipe)

L 1 st mode (fundamental) f 1 = v/4L (first harmonic) 2 nd mode (first overtone) f = 3f 1 third harmonic 3 rd mode (second overtone) f = 5f 1 fifth harmonic closed end press distribution press 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 as cylindrical pipe: why? - not obvious and theory is difficult math! (text book tries to explain it...) L

Resonance: oscillating system has “natural” frequency f 0 when it is oscillating on it’s own push on oscillating system at steady rate - driving frequency f D observe amplitude of oscillation as you vary f D : amplitude peaks at resonance frequency f 0 with of resonance  f measures how far you can be off in frequency before amplitude drops to 1/2 of peak

RESONANCE Resonance 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 f D (Hz) = frequency of push amplitude of osc. wider resonance - can “pull” frequency of instrument strings: small damping winds: large damping

RELATIONSHIP BETWEEN DAMPING TIME  f AND RESONANCE WIDTH   f width of res curve at half max more friction - wider res curve driving frequency f D (Hz) amplitude of osc. width of resonance curve and damping time: inverse relation  f  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 f 1 = (1/T) and harmonics f 2 =2f 1, f 3 =3f 1 etc. amplitude of harmonic freq of harmonic (Hz) easy visualization of harmonic content (timbre) but contains no information about relative timing of overtones (phase).

Fourier Synthesizer….. produces frequencies f 1, 2f 1, 3f 1, 4f 1, 5f 1 6f 1, 7f 1 etc of adjustable amplitude and phase. e.g. f 1 = 440 Hz = A 4 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)