Oscillations & WavesDamping, Forcing, Resonance

Damping

• So far we have assumed that wave propagation is lossless

• In reality, friction and otherresistive forces exist

• The energy of oscillationis eventually dissipatedas heat

• 3 categories of damping

Under-damping

• Small resistive forces cause a gradual, exponential drop in amplitude

• The period of oscillation increases with the degree of damping

Critical Damping

• Resistance returns the system to equilibrium as quickly as possible without oscillation

• Critical damping is a particular case of damping

Over-damping

• The resistance is so great that no oscillations occur (as in critical damping)

• The return to equilibrium is slower than in the case of critical damping

Damping Summary

Forcing

• To counteract resistive forces, one can force an oscillation

• Forcing usually involves the application of a periodic force

• The oscillator eventually adopts the forced frequency

A different kind of forcing!!

Natural Frequency, f0

• If allowed to move freely, oscillations tend to occur at a natural frequency, denoted f0

Mismatched Forcing

• What happens if the driving frequency fD is different from the natural frequency?

• Oscillations will occur at the driving frequency, but with limited amplitude

Resonance

• What happens if the driving frequency fD matches the natural frequency?

• The driving force is synchronized with the oscillation

• Amplitude is magnified with each wave cycle

Putting it All Together

• How are damping, forcing, and resonance related?

Examples

Earthquake Preparedness

• An earthquake may excite a building’s resonant frequency, with catastrophic results

Wheel Balancing

Microwave Cooking

• Multi-atomic molecules can be visualized as mass-spring systems

• Each molecule has natural frequencies• f0 for H2O is 2.45 GHz

Timekeeping

• Clocks contain small quartz crystal tuning forks which oscillate at known frequencies

• In most watches, the crystal oscillates at 32.768 kHz

Summary

• Damping– Under-damping, critical damping, over-damping

• Forcing• Natural Frequency• Mismatched Forcing• Resonance• Examples

Homework

• In Tsokos:– Ch 4.1 - #3, 6, 8, 9, 12, 15, 17, 25, 29, 31, 37– Ch 4.2 - #1, 3, 6, 7, 12, 15, 17– Ch 4.3 - #1, 4, 5, 7, 10– Ch 4.4 - #1, 4, 7