Wave: Definition and Mathematical Representation

Introduction

In physics, a wave is a disturbance or oscillation that travels through space and matter, transferring energy from one point to another without the permanent displacement of the medium. Waves are classified into different types based on the direction of particle motion and the medium through which they propagate.

Types of Waves

1. Mechanical Waves

Require a medium to propagate.

Examples: sound waves, water waves, seismic waves.
Subtypes:
- Transverse wave – particles move perpendicular to wave direction.
- Longitudinal wave – particles move parallel to wave direction.

2. Electromagnetic Waves

Do not require a medium; can travel through a vacuum.

Examples: light, radio waves, X-rays.

3. Matter Waves

Arise from the quantum mechanical behavior of particles.

Example: de Broglie waves associated with electrons.

Wave Parameters

A typical wave is described by the following properties:

**Wavelength** $λ$ : Distance between two consecutive crests or troughs.
**Frequency** $f$ : Number of oscillations per unit time.
**Period** $T$ : Time taken for one complete oscillation.

$T = \frac{1}{f}$

**Amplitude** $A$ : Maximum displacement from equilibrium.
**Wave speed** $v$ : Speed at which the wave propagates through the medium.

$v = f λ$

General Wave Equation

A traveling wave (in one dimension) is represented as:

$y (x, t) = A \sin (k x - ω t + ϕ)$

Where:

$y (x, t)$ is displacement at position $x$ and time $t$ ,
$A$ is amplitude,
$k = \frac{2 π}{λ}$ is the wave number,
$ω = 2 π f$ is the angular frequency,
$ϕ$ is the phase constant.

Wave Equation (Differential Form)

The wave equation in one dimension:

$\frac{\partial^{2} y}{\partial x^{2}} = \frac{1}{v^{2}} \frac{\partial^{2} y}{\partial t^{2}}$

This is a second-order partial differential equation describing wave propagation.

Superposition Principle

When two or more waves overlap, the resulting displacement is the algebraic sum of individual displacements:

$y_{total} = y_{1} + y_{2} + \dots$

This leads to phenomena such as:

Reflection and Refraction

**Reflection**: Wave bounces back on hitting a boundary.
**Refraction**: Wave changes direction and speed when entering a new medium.

Applications of Waves

Sound and music
Communication (radio, TV, mobile phones)
Optics and light behavior
Quantum mechanics (wave-particle duality)
Medical imaging (ultrasound, MRI)

Wave

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