MUSICAL ACOUSTICS
RESONANCE
Science of Sound
Chapter 4
Resonance
Resonance occurs when a vibrating system is driven at
its natural frequency.
The amplitude of the vibrator reaches a maximum,
limited only by the damping.
PHASE DIFFERENCE BETWEEN CRANK AND MASS
The Pasco Driven
Harmonic Motion
Analyzer
For extra credit:
Driven harmonic
analyzer (“Texas tower”)
Make graphs of
amplitude vs frequency
and phase angle vs
frequency for 2 amounts
of damping
STANDING WAVES ON A STRING
Partials, harmonics, and overtones
Partials are components of a complex tone or
vibration
The fundamental is the partial on which a
complex vibration or tone is built
Harmonics are partials whose frequencies are
multiples of a fundamental frequency
Overtones are partials above the fundamental
frequency
OPEN AND CLOSED PIPES
OPEN
PIPE
CLOSED
PIPE
STANDING
WAVES IN
CYLINDRICAL
AND
CONICAL
PIPES
ACOUSTIC IMPEDANCE
Acoustic impedance is the ratio of sound
pressure p to volume velocity U
ZA = p/U
APPARATUS FOR
GRAPHING THE
ACOUSTIC
IMPEDANCE OF
WIND
INSTRUMENTS
HELMHOLTZ RESONATOR
HELMHOLTZ RESONATOR
TUNING FORK
D.A. Russell, "On the sound field radiated by a tuning fork," Am. J.
Phys., 68(12), 1139-45 (2000).
CHOIRCHIMES
CHOIRCHIME VIBRATIONS
SINGING RODS AND WINEGLASSES
SINGING RODS
In a bar or rod with free ends the
fundamental mode will have a
node at its center, and the
maximum vibration occurs at the
ends (just as in a pipe open at
both ends).
The next mode has two modes at
¼ L and ¾ L
Stroking an aluminum rod with the
fingers to excite these longitudinal
resonances can create rather loud
sounds
SELF-EXCITATION
A linear force or motion can excite vibratory motion by
a process called self-excitation.
Examples of this are the
stick-slip motion that excites
a wineglass or a violin string
SELF-EXCITATION
A linear force or motion can excite vibratory motion by
a process called self-excitation.
Examples of this are the
stick-slip motion that excites
a wineglass or a violin string
COLLAPSE OF THE TACOMA NARROWS BRIDGE
A DRAMATIC CASE OF SELF-EXCITED OSCILLATION
The Collapse of the Tacoma Narrows
Bridge
A spectacular example of self excitation
Positive feedback (or negative damping) made it
possible for a nearly steady flow of air to excite torsional
oscillations
Vortex shedding played a role, but the feedback
mechanism appears to have been more subtle than
depicted in the Caltech video
Physically it is a little like the self-excitation of a violin
string by the steady motion of a bow.
The Collapse of the Tacoma Narrows
Bridge
Futher reading for extra credit:
Read the following papers, and write a paper on the
physics of the Tacoma Narrows Bridge collapse:
K.Y. Billah and R.H.Scanlon, “Resonance, Tacoma
Narrows bridge failure, and undergraduate physics
textbooks,” American Journal of Physics 59, 118-124
(1991).
D. Green and W.G.Unruh, “The failure of the Tacoma
Bridge: A physical model,” American Journal of Physics
74, 706-716 (2006)
SYMPATHETIC VIBRATION
The vibrating string of a piano or guitar excites the
soundboard to vibrate (and thus to produce much
more sound). This is called sympathetic vibration.
Sympathetic vibration and self-excitation of vibration
should not be confused with resonance.
Assignment for January 20
Read Chapter 5
Exercises 1-8 (p.73)
For extra credit:
Driven harmonic analyzer (“Texas tower”)
Make graphs of amplitude vs frequency and
phase angle vs frequency for 2 amounts of damping
Show the modes of a wineglass that can be excited by
rubbing with a finger; bowing radially with a violin
bow; driving it with a loudspeaker.
Ref: J. Acoust. Soc. Am. 95, 1108-1111 (1994).
Paper on the physics of the Tacoma Bridge collapse.