Introduction to Waves

  • Waves are disturbances that transfer energy through space and matter.

Classes of Waves

Mechanical Waves

  • Characteristic: Require matter to travel through a medium.
  • Examples:
    • Sound
    • Water waves
    • Earthquake waves

Electromagnetic Waves

  • Characteristic: Do not require matter; can travel through a vacuum.
  • Examples:
    • Visible light
    • X-rays
    • Microwaves
    • Gravitational waves

Differences Between Wave Types

  • Mechanical Waves:

    • Need a medium (material substance) to propagate.
    • Two types:
    • Longitudinal Waves: Particles move parallel to the wave direction.
    • Transverse Waves: Particles move perpendicular to the wave direction.

  • Electromagnetic Waves:

    • Do not require a medium for propagation.

Wave Part Vocabulary

  • Compression:

    • Definition: A part of a longitudinal wave where there is a high-pressure region.

  • Rarefaction:

    • Definition: A part of a longitudinal wave where there is a low-pressure region.

  • Wavelength (\lambda):

    • Definition: The distance over which the wave's shape repeats; one complete wave cycle.

  • Crest:

    • Definition: The highest point of a transverse wave.

  • Trough:

    • Definition: The lowest point of a transverse wave, pronounced as "troff."

  • Node:

    • Definition: The point where the wave crosses the equilibrium position.

  • Amplitude (A):

    • Definition: The maximum extent of a wave's motion from the equilibrium.
    • Measurement: In a transverse wave, measured from the equilibrium to the crest.

How Waves Move

  • Energy: Waves carry energy as they propagate through medium.
  • Movement: Various wave types exhibit distinct particle movement patterns.

Transverse Wave Movement

  • Particles move perpendicular to the direction of wave travel.

Longitudinal Wave Movement

  • Particles move parallel to the direction of wave travel.

Longitudinal Waves

Characteristics:

  • Definition: Waves that propagate by moving particles of the medium parallel to the wave direction (e.g., sound waves).
  • Compression: A high-pressure area in the wave.
  • Rarefaction: A low-pressure area in the wave.
  • Wavelength: Length of one complete cycle (distance from one compression to the next).

Elements of Longitudinal Waves

  1. Compression (high pressure):
    • Areas where particles are closer together.
  2. Rarefaction (low pressure):
    • Areas where particles are spread apart.

Amplitude in Longitudinal Waves

  • Definition: Minimum and maximum values of a wave’s pressure at a given time.

Examples of Longitudinal Waves

  • Sound waves and some earthquake waves (P-waves).

Illustration of Wave Propagation

  • A tuning fork creates compressions and rarefactions which can be visually represented.

Example with Spring:

  • Compression Region: Part of the spring is compressed.
  • Expansion Region: Part of the spring is extended.

Visualization of Sound Waves

  • Rubens Tube: Demonstrates how sound can be visualized through the arrangement of flames responding to pressure waves.

Transverse Waves

Definition:

  • Transverse waves are characterized by energy moving perpendicular to the direction of wave propagation.

Key Features:

  • Crest: The peak or topmost point of the wave.
  • Trough: The lowest point of a transverse wave.
  • Node: The position where the wave crosses the equilibrium line.

Earthquake Waves

P-Waves:

  • Definition: Also known as primary waves; these are the fastest seismic waves and arrive first during an earthquake.
  • Characteristic: Can travel through the entire Earth and cause ground shaking on the opposite side during large earthquakes.

S-Waves:

  • Definition: Secondary waves that travel slower than P-waves and arrive later.

Historical Seismometer

  • Invented by Chinese astronomer Zhang Heng in 132 AD.
  • Function: Detects ground movement during earthquakes.
  • Capability: Indicate direction of seismic activity, even from hundreds of miles away.

Calculating Wave Properties

Frequency

  • Definition: The number of wavelengths that pass a point in one second.
  • Unit: Hertz (Hz).
  • Formula:
    • f = \frac{\text{# of cycles}}{\text{total time}}
    • or f = \frac{1}{T}

Period

  • Definition: The duration for one complete wavelength to pass.
  • Unit: Seconds (s).
  • Formula:
    • T = \frac{\text{total time in s}}{\text{# of cycles}}
    • T = \frac{1}{f}
    • T = \frac{\lambda}{v}

Velocity

  • Definition: Speed at which a wave travels through a medium.
  • Formula:
    • V = \text{(frequency in Hertz)} \times \text{(} \lambda \text{ in meters)}
    • V = f \lambda

Example Problem on Frequency and Period

  • Given: \lambda = 20 cm and it took 4 seconds for one wavelength to complete.

  • Required: Calculate frequency and period.

  • Frequency Calculation:

    • f = \frac{1\text{ wave}}{4\text{ seconds}} = 0.25 Hz

  • Period Calculation:

    • T = 4 s (as it takes 4 seconds for one wavelength).

Standing Waves

  • Description: Waves that appear to be stationary, formed by the interference of two waves traveling in opposite directions.
  • Stable Motion: Requires driven ends to maintain consistent wave energy.
  • Nodes: Points of zero amplitude.
  • Anti-nodes: Points of maximum amplitude.

Example with Jump Rope:

  • Visualization of standing waves can be compared to a jump rope, where nodes and anti-nodes are distinctly observable.

  • Fundamental Wave: The largest stationary wave produced within a certain distance.

  • Formula for Wavelength with Anti-nodes: Each anti-node represents \frac{1}{2}\lambda.

Harmonics

  • Definition: Waves that are whole number multiples of the fundamental frequency.
  • Properties: Harmonics have defined nodes at the boundaries, sound louder, retain energy longer, and require less energy to produce.

Example Frequencies of Harmonics:

  • First 5 Harmonics of a Vibrating String:
    • Harmonic Frequencies:
    • f_1: 4 Hz
    • f_2: 8 Hz
    • f_3: 12 Hz
    • f_4: 16 Hz
    • f_5: 20 Hz

Non-Harmonic Waves

  • Can also be forced into boundaries but tend to decay quickly, sound quieter, and require more energy to produce as compared to harmonic waves.

Solved Examples in Harmonics

  • To find harmonics, use:
    • f= \text{fundamental frequency} \times \text{harmonic number}
  • Third Harmonic Example:
    • f_{H3} = (4 Hz) \times 3 = 12 Hz
  • Fifth Harmonic Example:
    • Given f_{H5} = 55 Hz to find the fundamental frequency:
    • f{1} = \frac{f{H5}}{5} = \frac{55 Hz}{5} = 11 Hz