Phys 1240: Sound and Music
LAST: strings and harmonics
TODAY: harmonics, perception &
ears
NEXT: continuing with harmonics
and spectra
Reading: Hall 8.2
STANDING WAVES
Resonances on strings
Wavelengths “fit” in L
Only certain frequencies:
v
fn = n "
2L
“Harmonics”
!
Waves on a string vs waves in air
v
fn = n "
2L
Speed of waves
on string!
Wiggle air => frequency of sound
wave same as freq of string!
speed = " # f
λ of wave on string is NOT the
same as λ of sound wave!
CT 10.1.8a
A string vibrates in the
fundamental, producing an “A”
(440 Hz) sound. Suppose the
speed of sound in the air could be
suddenly doubled (but the string is
left unchanged)
What would you HEAR?
A) Same pitch (440 Hz)
B) Lower pitch
C) Higher pitch, but not double
D) Double pitch = one octave
higher
E) ??
Waves on a string
v
fn = n "
2L
Speed of waves
on string!
What does the speed depend on?
Tension
v=
mass/length
The string’s
tension
The string’s
mass/length
With your neighbor:
what is the mass of 10 m of rope
with mass/length = 2 kg/m ?
<ct.10.1.8b>
T
v=
mass/length
If the tension is increased by a
factor of 9 (nine times the
Tension force!) what happens to
! the speed of waves on a string?
A. Goes up by a factor of 3
B. Goes up by a factor of 4.5
C. Goes up by a factor of 9
D. Goes up by a factor of 81
E. None of these / I don’t know
What happens to the frequency
of the fundamental?
v
fn = n "
2L
<ct.10.1.8b>
If you want to lower the pitch of
a guitar string by two octaves,
what must be done to its
tension?
A. Raise it by a factor of 4
B. Lower it by a factor of 4
C. Lower it by a factor of 2
D. Lower it by a factor of 16
E. None of these / I don’t know
To think about: is this a realistic
way to lower the pitch by this much?
(Why?)
CT 10.1.9a
The clothesline is being driven
(frequency “f”) and is in the 2nd
harmonic (one node in the
middle)
Now I ADD some weights on the
end, increasing T a little.
What happens?
A) Looks same, wiggles faster
B) A second node appears
C) Goes into the fundamental
D) Lose the resonance (just flops
a little, not “pretty”)
E) ??
CT 10.1.9c
If you increase Tension by a factor
of 4 (so the speed doubles)…
A) The frequency of the fundamental
doubles, all other harmonics stay the
same as they were
B) The frequency of EVERY harmonic
doubles
C) None of the frequencies change, the
wavelengths double
D) f1 goes up by 2, f2 by 4, (etc…)
E) Something else/???
ct.10.1.10a
A string on an instrument plays an
A (440 Hz) when plucked. If you
put your finger down one half of
the way along the string, and then
pluck, you are mostly likely to
hear…
A: Still 440 Hz
B: 220 Hz
C: 880 Hz
D: Something entirely different
L
ct.10.1.10b
A string on an instrument plays an
A (440 Hz) when plucked. If you
put your finger down one third of
the way along the string, and then
pluck the longer side, you are
mostly likely to hear…
A: 3*440Hz
B: (1/3)*440 Hz
C: 3/2 * 440 Hz
D: 2/3 * 440 Hz
E: Something entirely different
L
3 * 440Hz
3 * 440Hz
!
L
2L/3
1.5 * 440Hz