1. Pg. 298 – Damping
Underdamping
Overdamping
Critical damping – reaches equilibrium in shortest time possible
(See examples at bottom of page)
11-6: Forced Vibrations; Resonance
Objects have a natural frequency (f0) due to properties of that object
We can also exert a force that has its own frequency (f)—this causes a forced vibration.
Amplitude of forced vibration depends on difference between f and f0 ; it is at maximum when f0 = f—this is known as resonance.
f0 is also called resonant frequency.
Examples

2. 11-7: Wave Motion
Mechanical wave – oscillation of particles in a medium
Wave travels across a distance, while particles themselves only oscillate about an equilibrium point.
A wave is not matter—it carries energy through matter.
Pulse – a single “bump” in a wave
The source of any wave is a vibration; if it vibrates in SHM, then the wave itself will be sinusoidal in both space and time

3. Wave Properties
frequency(f) – number cycles/unit time
period(T) – time for one cycle
wave velocity(v) – the velocity at which the wave itself moves

4. 2 Types of Waves:
Transverse – particles vibrate perpendicular to wave’s motion
Longitudinal – particles vibrate along direction of wave’s motion

5. Waves transfer energy by transferring vibrational energy from one particle to the next.
For a wave in SHM, each particle has energy ½kA2.
Intensity (I) = power transported across area perpendicular to flow:
I = power/area (units: W/m2)

6. For 2-D or 3-D waves, we usually look at wave fronts—the set of all points along the crests of the multiple waves.
A line drawn in the direction of the motion of these wave fronts is called a ray.
When waves fronts have travelled far from their source, they have become nearly straight and are called plane waves.

7. Reflection & Refraction
When a wave strikes an obstacle or boundary between media, two things can happen: reflection and refraction.
At least part of the wave will be reflected (ex: echo)
Angle of reflection = Angle of incidence (Law of Reflection)
Refraction – wave goes through boundary but “bends” based on properties of media (Law of Refraction or Snell’s Law)

8. If the wave encounters a heavier medium, less energy is able to be transferred through, so the wave slows down.
However, the particles are still oscillating at the same rate, so frequency stays the same.
 If f stays the same and the wave slows down, what does that mean about v and λ?

9. Interference & Superposition
Interference – what happens when two wave pulses pass each other
Superposition Principle – in the region where they pass each other, the resulting amplitude is the algebraic sum of the individual amplitudes
2 Types of Interference:
Constructive
Greater amplitude than either individually
Happens when crests line up with crests (in phase)
Destructive
Less amplitude than either individually
Can be partially or completely out of phase
Completely by ½ λ, when crests line up with troughs

12. Standing Waves & Resonance
If you shake or pluck a fixed cord, it will cause a wave. That wave will be reflected, and the two will interfere.
If oscillated at just the right frequency, it will produce a standing wave.

13. Standing waves can occur at multiple frequencies.
The lowest frequency that produces a standing wave has one “bump,” twice this same frequency has two “bumps,” etc.
These are the natural or resonant frequencies of the cord.
The lowest resonant frequency is called the fundamental frequency.
Other natural frequencies are called overtones (whole # multiples of fundamentals.
Another name for this set of frequencies is harmonics:
fundamental frequency = 1st harmonic
2 loops = 2nd harmonic
3 loops = 3rd harmonic
Etc.

14. 𝐿= 𝑛 2 λ 𝑛
n = 1, 2, 3, … (# of the harmonic)
𝑓 𝑛 =𝑛 𝑓 1

15. Diffraction – when waves “bend” to pass around an obstacle

16. Chapter 12: Sound
To have “sound,” three things must be present:
1-Source
2-Wave
3-Receiver
Characteristics of Sound
Must have a medium to travel through
Speed different in various media (vair ≈ 343 m/s)
Loudness – intensity (how much energy it carries)
Pitch – high/low (determined by frequency; low pitch  low f)
Audible range ≈ 20 to 20,000 Hz
Above this = ultrasonic
Below this = infrasonic

17. Intensity
Intensity = amplitude
energy transferred per unit area (W/m2)
Due to way humans perceive sound, logarithmic scale is used:
Units: bel
Commonly used: decibel (1 dB = 0.1 bel)

18. Music
Source of any sound is a vibrating object
With instruments, standing waves are produced & the source vibrates at its natural resonant frequency.
Frequency of sound waves emitted are the same as the source, but v and λ could vary.
Same fundamental frequency/harmonics concepts apply.
Stringed Instruments
If you hold down a guitar string, you shorten it…what happens to the frequency?
λ decreases, so f must increase
This means the pitch is higher
Strings are different densities; v of heavier ones will be lower. If L is constant, f decreases and produces lower pitch.
Tension can also be adjusted.
For pianos & harps, strings are various lengths.
Box amplifies the sound (greater surface area in contact with air).

19. Wind Instruments
Sound comes from standing waves in a column of air
At each end, tube can be:
-Open (air can move freely in and out)
-Closed (air not free to move)
For displacement of air in tube open at both ends:

20. For displacement of air in tube closed at one end:

21. For pressure of air:
Tube open at both ends: Tube closed at one end:

22. For multiple sources of sound, interference occurs
Quality- terms like “timbre” or “tone color” sometimes used; what gives an instrument its unique sound
When a note is played, both the fundamental frequency and overtones are usually present; they add together using the superposition principle.
For multiple sources of sound, interference occurs
Person C will hear loud sound (constructive interference)
Person D will here very little sound (destructive interference)

23. Beats
If two sources have close but not identical frequencies, when they interfere, a phenomenon called beats occurs.
Sometimes the waves will be in phase (constructive interference) and sometimes they will be out of phase (destructive interference).
This interference is regularly spaced, and the listener hears “beats.”
Example: Wave A has fA= 50 Hz and Wave B has fB = 60 Hz.
Beat frequency = difference in frequency in two waves

24. Doppler Effect
Occurs when a source of sound is moving with respect to the listener
Source moving toward you  pitch is higher than it is at rest
Source moving away from you  pitch is lower than it is at rest
The same effect occurs when the source is at rest and the listener is moving