Physics of Periodic Motion and Wave Propagation

Posted on Sep 8, 2025 in Physics

Periodic and Oscillatory Motion

A movement is called periodic when its position, velocity, and acceleration repeat after a certain time interval, known as the period. An example is uniform circular motion.

These movements are also called oscillatory movements. Whenever an object returns to its starting position, it has completed an oscillation (or swing), and the time taken is its period. If these oscillations are very fast, they are called vibrations or vibratory motion.

A typical case of vibratory motion is Simple Harmonic Motion (SHM). This motion considers three situations:

  • Equilibrium (centered) position: The point where the net force is zero.
  • External deformation: Displacement from the equilibrium position.

This recovery or elastic restoring force always acts in the opposite direction to the movement and is proportional to the position of the mass relative to the central or equilibrium position. Mathematically, this is expressed as:

F = -K r = -K x i

where r (or x in one dimension) is the position vector of the mass relative to the central position.

The key features of this oscillatory motion, called Simple Harmonic Motion (SHM), are:

  • Its path is straight (rectilinear).
  • It is produced by a restoring force that is proportional to the displacement vector but acts in the opposite direction.
  • The forces that produce SHM are generally elastic in nature.

Simple Harmonic Motion: Key Concepts

The unique and repetitive nature of Simple Harmonic Motion leads us to define several key magnitudes:

  • Center of Oscillation: The central equilibrium position of the particle’s oscillatory motion.
  • Elongation (x): The distance of the moving particle from the center of oscillation. We consider positive values for positions to the right of the center and negative values for positions to the left.
  • Amplitude (A): The maximum elongation or maximum displacement from the equilibrium position.
  • Period (T): The time taken for one complete oscillation.
  • Frequency (f): The number of oscillations per unit time, measured in Hertz (Hz).
  • Angular Frequency (ω) or Pulsation: The frequency multiplied by 2π. It is measured in radians per second (rad/s).

Dynamics of Simple Harmonic Motion

Given that the acceleration of Simple Harmonic Motion is a = -ω2x, if we apply Newton’s second law (the fundamental principle of dynamics):

F = ma = -mω2x

Recalling that the restoring force producing this type of motion is F = -K r (or F = -Kx for its magnitude in one dimension), we can deduce that:

K = mω2

This equation shows that the elastic constant K of the medium is related to the body’s mass m and the angular frequency ω of the oscillations. Consequently, the period of these oscillations will also be determined by these factors.

Wave Motion

To understand what a wave is, consider what happens when you throw a stone onto the surface of pond water. A disturbance is created that causes the water particles to rise and fall, generating ripples that spread to the edges of the pond. It’s important to note that it is not the water itself that moves towards the walls of the pond, but rather the disturbance that propagates.

The same principle applies if you shake a rope at one end; this disturbance travels to the other end, but the rope itself does not move along its entire length.

In general, wave motion is a way to transmit energy from a disturbance without transporting matter. To produce a wave, two conditions are necessary:

  • An initial disturbance produced at a point, which we call the focus or emitter.
  • A transmission mechanism that allows for a delay in the arrival of this energy at all points.

Types of Waves

Waves can be classified in several ways:

Classification by Medium Requirement:

  • Mechanical Waves: These waves require a material medium with some elasticity to propagate energy. Examples include sound waves and seismic waves.
  • Electromagnetic Waves: These are produced when variable electric and magnetic fields coexist in a region of space. They do not require a material medium for propagation. Examples include light, radio waves, and X-rays.

Classification by Particle Motion Relative to Wave Propagation:

If we consider how particles move in the material environment when the wave reaches them, in relation to the direction of wave propagation, waves are classified into:

  • Transverse Waves: These occur when the particles vibrate perpendicularly (e.g., vertically) to the direction in which the wave propagates (e.g., horizontally).