Resonance_ppt_RevW10
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Physics 106 Lesson #24
Resonance
Dr. Andrew Tomasch
2405 Randall Lab
[email protected]
Review: Waves
• Two Defining Features:
– A wave is a traveling disturbance
– A wave transports energy one place to
another
• Two Main Types of Waves:
– Transverse: Electromagnetic Waves (radio,
visible light, microwaves and X-Rays)
– Longitudinal (sound waves)
Review: Transverse
and Longitudinal Waves
• Transverse
– Definition: a disturbance
perpendicular to the
direction of travel
– Example: transverse
pulses on a slinky
• Longitudinal
– Definition: a disturbance
parallel to the direction of
travel
– Example: compression
waves in a slinky
Review:
Periodic Waves
The pattern of the
disturbance is repeated
in time over and over
again (periodically) by
the source of the wave.
The red curve is a
“snapshot” of the wave
at t = 0
The blue curve is a
“snapshot” later in time
A is a crest of the wave
B is a trough of the wave
Review: Periodic Waves
l ≡ wavelength (distance between two successive
corresponding points on the wave e.g. peak-to-peak)
T ≡ period the time it takes for one wave cycle
f ≡ frequency the number of wave cycles per second
A ≡ amplitude (largest displacement from equilibrium)
v ≡ the speed of the disturbance → the magnitude of the
wave velocity
v fl
f 1/ T
Review: Wave
Speed on a String
• For a String
– restoring force tension in the string
– inertia parameter mass per unit length
v
T
m L
This is a dynamics equation. It
tells us how the speed of the
wave is related to the physical
parameters of the system.
• The speed of a wave on a string is greater for
strings with a large tension and lower for
thicker (heavier) strings compared to thinner
strings placed under the same tension.
Review: Sound Waves
• Condensation ≡ region of
increased pressure
• Rarefaction ≡ region of
decreased pressure
• A pure tone is an harmonic
(sine or cosine) sound wave
with a single frequency
Review: Sound Waves
Propagate in Air
• The energy of a sound wave
propagates as an elastic disturbance
through the air
• Individual air molecules do not travel
with the wave
• A given molecule vibrates back and
forth about a fixed location
• When we speak of sound, we mean
frequencies within the range of
human hearing:
20 Hz < f < 20,000 Hz
Infrasonic: f < 20 Hz
Example: Whales
Ultrasonic: f > 20,000 Hz
Example: bats echo locate
objects with f > 60,000 Hz
Review: Interference
• Principle of Linear
Superposition:
– When adding one wave to
another the resulting wave
is the sum of the two
original waves
• This leads to the
phenomenon of
interference:
– Constructive interference:
waves add to a larger
amplitude
– Destructive interference:
waves add to smaller or
zero amplitude.
Constructive
Destructive
Review: Standing Waves
• Occur when a returning reflected wave
interferes with the outgoing wave
• Special points:
– nodes = places that do not vibrate at all
– antinodes = maximum vibration at a fixed point
• Adjacent nodes are spaced a distance l/2
apart
l/2
Review: Standing Waves in Pipes
• Standing sound waves (longitudinal standing
waves) can be set up in a pipe or tube
• Wind instruments (trumpet, flute, clarinet,
pipe organ, etc.) depend on longitudinal
standing waves to produce sounds at
specific frequencies (notes)
• Two kinds
– Open Pipe: tube open at both ends
– Stopped Pipe: tube open at only one end
Demonstration
piccolo
Review: Overtones
• The length of the pipe is L
• Standing sound waves for the first two modes:
l
L
4
n=1
l
L
2
3l
L
4
n=2
lL
“Stopped" Pipe
(open on one end)
"Open" pipe
(open both ends)
Oscillations
• Oscillating Systems:
– Back-and-forth motion in a regular, periodic
way about a stable equilibrium point
– Three main attributes: amplitude (A), period
(T) and frequency ( f )
– Periodic waves are generated by
oscillations and the medium through which
the wave propagates (air for sound or a
string) oscillates about its equilibrium
position as the wave passes
1
f
– T and f are reciprocals :
T
Simple Harmonic Motion (SHM)
• For a mass-and-spring
oscillator, the oscillation
frequency f is governed by
inertia (mass) and the
restoring force (spring
constant)
– A bigger mass produces a
smaller f (slower oscillation
→ longer period)
– A bigger restoring force
(bigger spring constant)
produces a bigger f (faster
oscillation→shorter period)
Hooke’s Law: Fspring= -kx
1
f
2
k
m
1
m
T 2
f
k
Demonstration
f is the natural frequency of a mass-and-spring oscillator.
The Simple Pendulum:
Frequency and Period
•The frequency and period are
independent of the mass of the
pendulum bob
• Assuming g is fixed, the only way to
change the period is to change the
length of the pendulum
•A pendulum with a fixed length can
be used to measure g → different f, T
on a planet with a different g or at
different places on Earth
1
f
2
g
1
L
T 2
L
f
g
f is the natural
frequency of a
simple pendulum.
Damped Harmonic Motion
• An object undergoing ideal SHM oscillates
forever since no non-conservative forces
dissipate any mechanical energy by doing
negative work
• Real oscillations eventually stop because
a dissipative non-conservative force acts
(examples: friction, air drag)
• This is called damped oscillation
• Two possibilities: underdamped and
overdamped
Amplitude
Underdamped
Motion
• The restoring force is more important
than the damping force
• The oscillations decay away slowly
• Examples: Mass on a Spring or Pendulum
Overdamped Motion
• The damping force is more important than
the restoring force
• The system does not oscillate at all. It just
relaxes back to the equilibrium position
Damping
• Damping forces are often put into a system
deliberately to prevent it from oscillating
indefinitely
• Example: the shock absorbers in your car
– Too little damping too “bouncy” a ride
– Too much damping too “stiff” a ride
• Q: How can you tell if a system is
underdamped or overdamped?
A: Disturb it. If it oscillates for a while it’s
underdamped. If it returns to the
equilibrium position without oscillating it’s
overdamped .
Natural Frequency
• All objects are made up of atoms. The
electric forces between atoms act like
springs and the atoms are masses
• Objects will oscillate if you disturb
them, but they don’t just oscillate at
any random frequency
• They oscillate at a natural frequency
fnatural
restoring force parameter
inertia parameter
Resonance
Demo: Mass and
Spring on Finger
• Natural Frequency: a
frequency determined by
the physical properties
of a vibrating object
f driving f natural
• Driving an object at its
resonant frequency
(≡natural frequency)
produces high-amplitude
oscillations
f driving f natural
• Objects driven at their
natural frequency can be
damaged or destroyed
Resonance in Action
Demo: Shattering
the Wine Glass
• Tacoma Narrows Bridge
(1940) Tacoma WA
• Wind forces led to a
catastrophic failure by
driving the bridge at its
natural frequency
http://www.ketchum.org/tacomacollapse.html