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

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