Slides for the web

1
todays music Pan Pipes
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)

Pan and nymph
First exam on Thursday, Feb, 21. Study guide and
answers have been handed out. Covers Ch. 1-4, and
homework 1-4
2
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.
3
String stiffness
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.
4
Pipes
(woodwinds, brass, organ pipes)
an open pipe (open at both ends)
at open end, no pressure build-up because air is
free to escape OPEN END is always a
PRESSURE NODE
Longitudinal wave
5
Fundamental Oscillation

fundamental freq
f is (almost) independent of pipe diam!
Examplefind length of flute of frequency C
260 Hz
demo 1.25 m long pipe
6
air flow
half a period later
How change pitch of pipe? f v/2L can ONLY
change L (fingerholes on flute) cant change
speed of sound v! diameter has (almost) no
effect! But can overblow to higher modes!!
7
Higher modes of flute
example f1 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?
8
Closed Pipe
Pipe closed at ONE end
9
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 A1 organ pipe? (Answ
1.56m 5 ft if closed pipe vs. 3.12 m
10 ft if open pipe)
10
Higher modes of closed pipe need pressure NODE
at open end pressure BELLY at closed end
closed end
1st mode (fundamental) f1 v/4L (first harmonic)
2nd mode (first overtone) f 3f1 third harmonic
0
3rd mode (second overtone) f 5f1 fifth
harmonic
0
odd multiples of fundametal
press distribution
11
conical pipes oboe, bassoon
same frequency as cylindrical pipe
why? - not obvious and theory is difficult
math! (text book tries to explain it...)
12
Resonance

• oscillating system has natural frequency f0
  when
• it is oscillating on its 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 Df measures how far you can be
  off
• in frequency before amplitude drops to 1/2 of
  peak

13
RESONANCE Resonance
curves for different amount damping (friction)
less damping - narrower res curve
amplitude of osc.
width of res curve at half max
more friction - wider res curve
driving frequency fD (Hz) frequency of push
strings small damping winds large damping
wider resonance - can pull frequency of
instrument
14
RELATIONSHIP BETWEEN DAMPING
TIME Df AND RESONANCE WIDTH t
amplitude of osc.
Df
Df width of res curve at half max
more friction - wider res curve
driving frequency fD (Hz)
Df t 4/9
width of resonance curve and damping time
inverse relation
15
Examples 1. Sitar (N. India) 7 strings 11
sympathetic strings 2. Marimba
16
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
17
Sound Spectrum (Fourier Spectrum)
Fourier represent complicated periodic
oscillation (period T) as sum of
sinusoidal oscillations of frequencies
f1 (1/T) and harmonics f22f1, f33f1 etc.
easy visualization of harmonic content (timbre)
but contains no information about relative timing
of overtones (phase).
18
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)
19
Tone Quality (Timbre)
In acoustic theory, what exactly is timbre?
Timbre is that attribute that differentiates two
tones of same loudness and same pitch.
HOWEVERThe 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)