Disturbances in an Elastic Medium

Introduction to Waves

Waves are disturbances that transfer energy through a medium without transferring matter. They are fundamental to understanding phenomena in physics such as sound, light, and water waves.

• Elastic Medium: A material that can return to its original shape after being disturbed, allowing wave propagation.

• Wave Propagation: The movement of a wave through a medium, carrying energy from one location to another.

• Wave Speed: The rate at which a wave travels through a medium.

• Transverse Speed: The speed at which particles in the medium move perpendicular to the direction of wave propagation.

• Superposition Principle: When two or more waves overlap, the resulting disturbance is the sum of the individual disturbances.

Types of Waves

Transverse and Longitudinal Waves

Waves can be classified based on the direction of particle motion relative to the direction of wave propagation.

• Transverse Waves: Particles move perpendicular to the direction of wave travel. Example: waves on a string, electromagnetic waves.

• Longitudinal Waves: Particles move parallel to the direction of wave travel. Example: sound waves in air, compression waves in a spring.

• Pulses: Localized disturbances that travel through a medium (e.g., a single push on a slinky).

• Periodic Waves: Repetitive patterns of disturbance (e.g., continuous oscillation of a string).

Examples and Demonstrations

• Slinky Demo: Demonstrates both transverse and longitudinal waves using a spring.

• Water Ripples: Visual example of wave propagation in two dimensions.

Mathematical Description of Waves

Wave Equations

Waves can be described mathematically using functions that represent displacement as a function of position and time.

• General Wave Equation: where:

  • = amplitude (maximum displacement)

  • = wave number (), units: rad/m

  • = angular frequency (), units: rad/s

  • = position

  • = time

• Wave Speed: where:

  • = wave speed

  • = frequency

  • = wavelength

• Direction of Propagation: (positive speed) (negative speed)

Graphical Representation

• Plots of and show the displacement of the wave as a function of position for waves traveling in opposite directions.

Wave Properties

Key Properties

• Amplitude (): Maximum displacement from equilibrium.

• Wavelength (): Distance between successive crests or compressions.

• Frequency (): Number of oscillations per second.

• Period (): Time for one complete oscillation ().

• Wave Speed (): How fast the wave propagates through the medium.

Superposition Principle

Interference of Waves

When two or more waves meet, their displacements add together according to the superposition principle.

• Constructive Interference: When waves add to produce a larger amplitude.

• Destructive Interference: When waves add to produce a smaller (or zero) amplitude.

• Standing Waves: Formed by the superposition of two waves traveling in opposite directions, resulting in nodes (no motion) and antinodes (maximum motion).

Energy in Waves

Wave Energy Transfer

• Waves transfer energy through the medium, but the medium itself does not travel with the wave.

• Energy is proportional to the square of the amplitude.

Summary Table: Types of Waves

Example Problem

Calculating Wave Speed

• Given: Frequency Hz, wavelength m

• Find: Wave speed

• Solution: m/s

Additional info:

• These notes cover material relevant to Chapter 14: Wave Motion, and also touch on concepts from Chapter 13: Oscillatory Motion and Chapter 15: Fluid Motion.