1. Dr. Andrew Tomasch 2405 Randall Lab atomasch@umich.edu
Physics 106 Lesson #24
Resonance
Dr. Andrew Tomasch
2405 Randall Lab

2. Review: Waves Two Defining Features: Two Main Types of Waves:
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)

3. Review: Transverse and Longitudinal Waves
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

4. 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

5. 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

6. Review: Wave Speed on a String
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.
This is a dynamics equation. It tells us how the speed of the wave is related to the physical parameters of the system.

7. 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

8. 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
Ultrasonic: f > 20,000 Hz
Example: bats echo locate
objects with f > 60,000 Hz
Infrasonic: f < 20 Hz
Example: Whales

9. 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

10. 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

11. 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

12. Review: Overtones The length of the pipe is L
Standing sound waves for the first two modes:
n=1
n=2
“Stopped" Pipe (open on one end)
"Open" pipe (open both ends)

13. 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 :

14. 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
Demonstration
f is the natural frequency of a mass-and-spring oscillator.

15. 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
f is the natural frequency of a simple pendulum.

16. 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

17. Underdamped Motion
Amplitude
The restoring force is more important than the damping force
The oscillations decay away slowly
Examples: Mass on a Spring or Pendulum

18. 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

19. 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 .

20. 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

21. Resonance
Demo: Mass and Spring on Finger
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

22. Resonance in Action Tacoma Narrows Bridge (1940) Tacoma WA
Demo: Shattering
the Wine Glass
Resonance in Action
Tacoma Narrows Bridge (1940) Tacoma WA
Wind forces led to a catastrophic failure by driving the bridge at its natural frequency