Mechanical Waves

Introduction to Waves

• Waves are seen in everyday life.
• A wave is a vibratory disturbance transmitted through a material or space.
• A wave transfers energy but not mass.
• Examples: seismic, ocean, light, sound, microwaves, x-rays, UV, gamma, radio, heat/infrared.

Pulses vs. Periodic Waves

• Pulse: One vibration sent down the wave.
  • Think of "the wave" at a sports game.
• Periodic Wave: A series of evenly timed pulses.
• Demonstrations of pulses & periodic waves: https://www.acs.psu.edu/drussell/Demos/waves-intro/waves-intro.html

Wave Types

• Wave type is determined by the direction of particle movement about an equilibrium position relative to the direction of wave propagation (wave movement).
• Two types of waves:
  • Electromagnetic
  • Mechanical

Electromagnetic Waves

• Do not require a medium to travel through.
• Can travel through space/a vacuum or a medium.
• Travels at the speed of light: c = 3 \times 10^8 m/s
• Examples: x-rays, gamma, micro, UV, infrared, TV, radio, visible light.
• Reference Table information: see page 2 of RT

Mechanical Waves

• Must be transmitted through a material (medium).
  • Examples of mediums: air, water, glass, rope, etc.
  • Transverse
    • Particles in a medium vibrate perpendicular to propagation.
    • Examples: "the wave", all EM waves, slinky pushed side to side.
    • T \perp
  • Longitudinal
    • Particles in a medium vibrate parallel to propagation.
    • Examples: sound waves, slinky pushed forward and back.
  • Torsion
    • Vibrations in a twisting motion.
    • Torsion is NEVER the answer on the REGENTS!
    • Parallel Two L’s in = ∥
      parallel to remind us of Longitudinal = Long & Parallel

Transverse Wave

• The direction of wave is perpendicular to the direction of the vibration.

Longitudinal Wave

• The direction of wave is parallel to the direction of the vibration.

Wave Characteristics: Transverse Waves

• Crest: Top
• Trough: Bottom
• Equilibrium: Middle → rest position between crest & trough

Wave Characteristics: Longitudinal Waves

• Compression: Points A & C = particles/lines are close together
• Rarefaction: Points B & D = particles/lines are far apart & spaced out

More Wave Characteristics: Wavelength (λ)

• Units: meters [m]
• Length of a wave between 2 consecutive points that are in the same position on a wave (in phase)
• Transverse:
  • 1 wave = 4 quarters
  • crest to crest
  • trough to trough
  • equilibrium to equilibrium
• Longitudinal:
  • compression to compression
  • rarefaction to rarefaction

Determining Wavelength Example

• There are 3 full waves in a 6 m picture.
• 3 waves = 6 m
• Proportion: \frac{3 \text{ waves}}{6 \text{ m}} = \frac{1 \text{ wave}}{x}
• Solve for x: x = 2 \text{ meters}
• Therefore, the wavelength (λ) is 2m.

More Wave Characteristics: Amplitude (A)

• Units: meters [m]
• Related to the amount of energy a wave carries.
• How big/strong/bright/loud the wave is
• Basically, amplitude is the maximum distance between equilibrium and crest OR maximum distance between equilibrium and trough
• Bigger A → brighter, louder, or stronger

More Wave Characteristics: Frequency (f)

• Units: Hertz [Hz] or s^{-1}
• The number of waves or cycles that occur in one second.
• Determined by the source of the wave. NOT affected by the medium.
• frequency = \frac{waves}{second}
• Example: If a woodpecker knocks on a tree 4 times in a second, what is the frequency of the knock? The frequency of bad physics jokes Hertz!!!

More Wave Characteristics: Period (T)

• Units: seconds [s]
• The time it takes for a wave to complete 1 wave.
• period = \frac{seconds}{waves}
• Example: If a woodpecker knocks on a tree 4 times in a second, what is the period of the knock?
• Frequency and period are inversely proportional
• Equation: T = \frac{1}{f}

The Electromagnetic Spectrum - RT page 2

• Includes:
  • Gamma Rays
  • X-rays
  • Ultraviolet
  • Visible Light
  • Infrared
  • Microwaves
  • TV, FM, AM, Long Radio Waves
• Frequency is displayed in Hertz (Hz)
• Wavelength in a vacuum (m)

Waves & Color

• Color of a wave is determined by Frequency (source)
• Red light: 3.84 \times 10^{14} Hz - 4.82 \times 10^{14} Hz
• Violet light: 6.59 \times 10^{14} Hz - 7.69 \times 10^{14} Hz
• Blue light: 6.10 \times 10^{14} Hz - 6.59 \times 10^{14} Hz
• Green light: 5.2 \times 10^{14} Hz - 6.10 \times 10^{14} Hz
• 5.0 \times 10^{14}Hz is Orange
• Color cannot be determined by wavelength.

The Wave Equation

• Velocity (v) - the speed of the wave
• Determined by the type of medium → air, glass, water, metal, rope, etc.
• Equation: v = f \lambda
  • v = velocity [m/s]
  • f = frequency [Hz]
  • λ = wavelength [m]
• If the medium is uniform the wave will travel at a constant speed.
• Frequency depends on the source.
• Velocity and wavelength depend on the medium.

The Wave Equation Examples

• Example 1: Determine the wavelength of light with a frequency of 5.10 \times 10^{14}Hz.
  • f = 5.10 \times 10^{14} Hz
  • v = 3 \times 10^8 m/s
  • λ = ?
  • v = f \lambda
  • (3 \times 10^8 m/s) = (5.10 \times 10^{14} Hz) (\lambda)
  • \lambda = 5.88 \times 10^{-7} m
• Example 2: Determine the color of light that has a wavelength of 5.10 \times 10^{-7}m.
  • \lambda = 5.10 \times 10^{-7} m
  • v = 3 \times 10^8 m/s
  • color = ? → f = ?
  • v = f \lambda
  • (3 \times 10^8 m/s) = (f) (5.10 \times 10^{-7} m)
  • f = 5.88 \times 10^{14} Hz
  • Green!

Phases

• Two points at equal displacements from their rest positions and are moving in same direction.
• These two points are separated by 0° or 360°.
• This equals 1 full wavelength, or 2 full wavelengths, or 3 full wavelengths, etc.
• Examples of two points that are IN PHASE with each other:
  • A & E
  • E & H
  • B & F
  • D & G
  • A & H (2 λ)
  • C & J (2 λ)

180° Out of Phase

• Two points that are at equal displacements but moving in opposite directions.
• These two points are ½λ apart.
• Examples of two points that are 180° OUT OF PHASE with each other:
  • A & C
  • H & J
  • G & I
  • C & E
  • D & I (1.5 λ)
  • A & J (2.5 λ)

Phases

• 2 points that are 90° out of phase: ¼ λ (ex - D & E)
• 2 points that are 180° out of phase: ½ λ
• 2 points that are 270° out of phase: ¾ λ (ex - A & D)
• 2 points that are in phase (0° or 360° between them): 1 λ, 2 λ, …

Which direction will a point go next?

• In a transverse wave, particles move PERPENDICULAR to the wave movement.
• The wave below moves to the RIGHT.
• So the particles/points can ONLY move UP & DOWN.
• Hint: Look behind to see what is coming next!
  • Wave movement
  • Direction of each point as the wave moves right
    • A. down
    • B. up
    • C. up
    • D. down
    • E. down
    • F. up
    • G. down
    • H. down
    • I. up
    • J. up

Boundaries

• When a wave travels from one medium to another (or comes in contact with a boundary), the wave can be:
  • Reflected - waves bounce back off boundary
  • Absorbed - wave enters new material and some energy stays inside the molecules
  • Transmitted - wave enters and continues through the new material

Boundaries

• At a fixed boundary, the wave reflects back on opposite side it came in on.
• At an open or loose boundary, the wave reflects back on the same side it came in on.

Drawing Waves

• 1. Draw 1 complete wave in 1 second with an amplitude of 5 boxes. Graph → a dot every 5 boxes
• 1 wave in 1 second → So 1 wave is 20 boxes wide
• So equilibrium is half way through 20 boxes at 10 boxes
• The crest is halfway between equilibrium and the start; after 5 boxes

Drawing Waves Steps for

• 1: Determine the frequency (# of waves per 1 second) - keep in mind that the entire length of the x-axis is 1 second, which is 20 boxes.
  • ➢ In this example, the frequency is 1 Hz or 1 wave per 1 second.
• 2: Draw a dot at the start (0 boxes) and end (20 boxes).
• 3: Draw a dot at the end of wave 1, wave 2, etc.
  • ➢ In this example, the end of wave 1 is after 20 boxes.
• 4: Draw a dot half way through each wave at equilibrium.
  • ➢ In this example, halfway through the wave is after 10 boxes.
• 5: Draw a dot halfway between the start and half way at the maximum amplitude to represent the crest.
  • ➢ In this example, the crest is after 5 boxes and it is 5 boxes high.
• 6: Draw a dot halfway between the start and half way at the maximum amplitude to represent the trough.
  • ➢ In this example, the trough is after 15 boxes and it is 5 boxes high.

Drawing Waves

• 2. Draw a wave that is in phase with wave 1 but with a greater energy. Graph → a dot every 5 boxes
• In phase with wave 1 means it lines up with wave 1
• More energy means more amplitude (crest and trough = 6m)
• More energy = More amplitude
• Crest lines up with crest / trough lines up with trough

Drawing Waves Steps for #2

• : Determine the frequency (# of waves per 1 second) - keep in mind that the entire length of the x-axis is 1 second, which is 20 boxes.
  • ❏ In this example, the frequency is 1 Hz or 1 wave per 1 second.
• 2: Draw a dot at the start (0 boxes) and end (20 boxes).
• 3: Draw a dot at the end of wave 1, wave 2, etc.
  • ❏ In this example, the end of wave 1 is after 20 boxes.
• 4: Draw a dot half way through each wave at equilibrium.
  • ❏ In this example, halfway through the wave is after 10 boxes.
• 5: Draw a dot halfway between the start and half way at the maximum amplitude to represent the crest.
  • ❏ In this example, the crest is after 5 boxes and it is 6 boxes high.
• 6: Draw a dot halfway between the start and half way at the maximum amplitude to represent the trough.
  • ❏ In this example, the trough is after 15 boxes and it is 6 boxes high.

Drawing Waves

• 3. Draw 2 complete waves in 1 second with an amplitude of 5 boxes. Graph → a dot every 2.5 boxes
• The wavelength is 10 boxes
• The crest is after 2.5 boxes, trough after 7.5 boxes,
• Next crest is after 12.5 boxes, and next trough after 17.5 boxes
• 2 waves in 1 second = 2 waves in 20 boxes = 1 wave in 10 boxes

Drawing Waves Steps for #3

• : Determine the frequency (# of waves per 1 second) - keep in mind that the entire length of the x-axis is 1 second, which is 20 boxes.
  • ➢ In this example, the frequency is 2 Hz or 2 waves per 1 second.
• 2: Draw a dot at the start (0 boxes) and end (20 boxes).
• 3: Draw a dot at the end of wave 1, wave 2, etc.
  • ➢ In this example, the end of wave 1 is after 10 boxes.
• 4: Draw a dot half way through each wave at equilibrium.
  • ➢ In this example, halfway through the wave is after 5 boxes.
• 5: Draw a dot halfway between the start and half way at the maximum amplitude to represent the crest.
  • ➢ In this example, the first crest is after 2.5 boxes and it is 5 boxes high.
• 6: Draw a dot halfway between the start and half way at the maximum amplitude to represent the trough.
  • ➢ In this example, the first trough is after 7.5 boxes and it is 5 boxes high.

Drawing Waves

• 4. Draw a wave that is 180° out of phase with wave 3 but has less energy. Graph → flipped from #3’s graph with smaller A
• The wavelength is 10 boxes
• The trough is after 2.5 boxes, crest after 7.5 boxes,
• Next trough is after 12.5 boxes, and next crest after 17.5 boxes
• This means when there was a crest in wave 3, there must be a trough in wave 4.
• When there is a trough in wave 3, there is a crest in wave 4.
• Less energy = Less amplitude

Drawing Waves Steps for #4

• : Determine the frequency (# of waves per 1 second) - keep in mind that the entire length of the x-axis is 1 second, which is 20 boxes.
  • ➢ In this example, the frequency is 2 Hz or 2 waves per 1 second.
• 2: Draw a dot at the start (0 boxes) and end (20 boxes).
• 3: Draw a dot at the end of wave 1, wave 2, etc.
  • ➢ In this example, the end of wave 1 is after 10 boxes.
• 4: Draw a dot half way through each wave at equilibrium.
  • ➢ In this example, halfway through the wave is after 5 boxes.
• 5: Draw a dot halfway between the start and half way at the maximum amplitude to represent the crest.
  • ➢ In this example, the first crest is after 7.5 boxes and it is 4 boxes high.
• 6: Draw a dot halfway between the start and half way at the maximum amplitude to represent the trough.
  • ➢ In this example, the first trough is after 2.5 boxes and it is 4 boxes high.

Drawing Waves

• 5. Draw 4 complete waves in 1 second with an amplitude of 4 boxes. Graph → a dot every 1.25 boxes
• The wavelength (length of 1 wave) is 5 boxes.

Drawing Waves Steps for #5

• : Determine the frequency (# of waves per 1 second) - keep in mind that the entire length of the x-axis is 1 second, which is 20 boxes.
  • ➢ In this example, the frequency is 2 Hz or 2 waves per 1 second.
• 2: Draw a dot at the start (0 boxes) and end (20 boxes).
• 3: Draw a dot at the end of wave 1, wave 2, etc.
  • ➢ In this example, the end of wave 1 is after 5 boxes.
• 4: Draw a dot half way through each wave at equilibrium.
  • ➢ In this example, halfway through the wave is after 2.5 boxes.
• 5: Draw a dot halfway between the start and half way at the maximum amplitude to represent the crest.
  • ➢ In this example, the first crest is after 1.25 boxes and it is 4 boxes high.
• 6: Draw a dot halfway between the start and half way at the maximum amplitude to represent the trough.
  • ➢ In this example, the first trough is after 1.25 boxes and it is 4 boxes high.

Drawing Waves

• 6. Draw a wave that is double the frequency of wave 5 with an amplitude of 2. Graph → a dot every 0.625 boxes
• The wavelength (length of 1 wave) is 2.5 boxes.

Drawing Waves Steps for #6

• Total length of 20 boxes divided by the frequency of 8 Hz = 2.5 boxes per wave
• 2. 5 waves divided by 4 (because there are 4 quarters per wave) = 0.625 boxes per dot