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

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)

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

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

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.

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 Ultrasonic: f > 20,000 Hz Example: bats echo locate objects with f > 60,000 Hz Infrasonic: f < 20 Hz Example: 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 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: n=1 n=2 “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 T and f are reciprocals :

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.

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.

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

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

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

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

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 http://www.ketchum.org/tacomacollapse.html