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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 ≡ 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 f 1/f T
• For a String – restoring force tension in the string– inertia parameter mass per unit length
• 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.
Tv
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.
Review: Wave Speed on a String
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
• 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
Review: Sound Waves Propagate in Air
Ultrasonic: f > 20,000 HzExample: bats echo locateobjects with f > 60,000 Hz
Infrasonic: f < 20 HzExample: Whales
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 /2 apart
/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
piccolo Demonstration
• The length of the pipe is L• Standing sound waves for the first two modes:
“Stopped" Pipe (open on one end)
"Open" pipe (open both ends)
n=1
n=2
L2
L
L4
L4
3
Review: Overtones
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
– T and f are reciprocals :T
f1
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)
1
2
kf
m
k
m
fT 2
1
f is the natural frequency of a mass-and-spring oscillator.
Hooke’s Law: Fspring= -kx
Demonstration
•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
The Simple Pendulum: Frequency and Period
1 1 2
2
g Lf T
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
• The restoring force is more important than the damping force
• The oscillations decay away slowly
• Examples: Mass on a Spring or Pendulum
UnderdampedMotion A
mp
litu
de
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 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 .
Damping
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
f natural
restoring force parameter
inertia parameter
driving naturalf f
• Natural Frequency: a frequency determined by the physical properties of a vibrating object
• Driving an object at its resonant frequency (≡natural frequency) produces high-amplitude oscillations
• Objects driven at their natural frequency can be damaged or destroyed
Resonance
driving naturalf f
Demo: Mass and Spring on Finger
Resonance in Action• 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
Demo: Shatteringthe Wine Glass
