Waves

A wave is a disturbance that moves through a medium, transferring energy from one point to another without permanently displacing the medium. Wave motion refers to the process of transmitting a disturbance through a medium without transferring its particles.

Types of Waves

Waves can be categorized in two ways:

Based on the medium of propagation: Mechanical waves and Electromagnetic waves.
Based on the wave’s direction relative to particle vibration: Transverse waves and Longitudinal waves.

Production and Propagation of Waves

1. Mechanical Waves

Mechanical waves require a medium to propagate and transfer energy. Examples include waves in springs, water waves, and sound waves.

2. Electromagnetic Waves

Electromagnetic waves do not require a medium to propagate. Examples include radio waves, visible light, ultraviolet rays, X-rays, and gamma rays. These waves travel at the speed of light, approximately \( 3.0 \times 10^8 \) m/s.

Types of Waves Based on Motion

1. Progressive Waves

Progressive waves move through a medium, transferring energy continuously. They can be either transverse or longitudinal.

2. Stationary (Standing) Waves

Stationary waves form when two waves moving in opposite directions interfere. This occurs due to the superposition of an incident wave and its reflection, creating variations in amplitude along the wave.

Classification of Waves Based on Particle Motion

1. Transverse Waves

In transverse waves, the vibrations occur perpendicular to the wave's direction of travel.

2. Longitudinal Waves

Longitudinal waves move in the same direction as the vibrations of the medium.

Key Terms in Wave Motion

Phase: Particles that are at the same vertical position and moving in the same direction are said to be in phase.
Cycle: A complete oscillation or to-and-fro movement of a vibrating particle.
Amplitude (A): The maximum displacement of a particle from its rest position, measured in meters (m).
Period (T): The time taken for a particle to complete one full cycle of oscillation.

\[ T = \frac{1}{f} \]

Frequency (f): The number of cycles completed in one second, measured in Hertz (Hz).
Wavelength (λ): The distance a wave travels in one complete oscillation. For transverse waves, it is the distance between successive crests or troughs, while for longitudinal waves, it is the distance between consecutive compressions or rarefactions. It is measured in meters (m).
Wave Velocity (v): The distance traveled by the wave in one second, with the SI unit of meters per second (m/s).

Wave Equations

The relationship between wave velocity, frequency, and wavelength is given by:

\[ v = f\lambda \]

v = Wave velocity (m/s)
f = Frequency (Hz)
\(\lambda\) = Wavelength (m)

Since frequency and period are related by:

\[ f = \frac{1}{T} \]

We can also express wave velocity as:

\[ v = \frac{\lambda}{T} \]

Worked Example

A radio station broadcasts at a frequency of 300 kHz. If the wave speed is \(3 \times 10^8\) m/s, calculate the period and wavelength of the wave.

Solution:

The period \(T\) is:

\[ T = \frac{1}{f} = \frac{1}{300000} = 3.3 \times 10^{-6} \text{ s} \]

The wavelength \(\lambda\) is:

\[ \lambda = \frac{v}{f} = \frac{3 \times 10^8}{3 \times 10^5} = 1000 \text{ m} \]

Mathematical Representation of Wave Motion

Equation for a Progressive Wave

The general equation for a stationary wave is:

\[ y = A \sin \left( \frac{2\pi x}{\lambda} \right) \]

A = Amplitude of the wave
\(\lambda\) = Wavelength of the wave
y = Vertical displacement of the wave
x = Horizontal coordinate of the vibrating particle

Wave Equation with Phase Difference

If two points, O and P, are out of phase by \(\Phi\), we express the wave as:

\[ y = A \sin \left( \frac{2\pi x}{\lambda} - \Phi \right) \]

Where:

\[ \frac{\Phi}{2\pi} = \frac{x}{\lambda} \] \[ \Phi = \frac{2\pi x}{\lambda} \]

Since \(x = vt\), we get:

\[ \Phi = \frac{2\pi v t}{\lambda} \]

Substituting this into the wave equation:

\[ y = A \sin \left( \frac{2\pi x}{\lambda} - \frac{2\pi v t}{\lambda} \right) \] \[ y = A \sin \left( \frac{2\pi}{\lambda} (x - v t) \right) \]

Also, using \( v = f \lambda \), we can rewrite it as:

\[ y = A \sin \left( \frac{2\pi x}{\lambda} - 2\pi f t \right) \]

Since angular frequency is given by \( \omega = 2\pi f \), we obtain the final equation:

\[ y = A \sin \left( \frac{2\pi x}{\lambda} - \omega t \right) \]

Properties of Waves

All waves exhibit the following properties:

Reflection
Refraction
Diffraction
Interference

Additionally, transverse waves exhibit an extra property called polarization.

Reflection

Reflection occurs when a traveling wave strikes a surface and bounces back. The incoming wave is called the incident wave, while the wave that bounces back is the reflected wave.

In the case of water waves in a ripple tank, if waves strike a plane strip perpendicularly, they reflect back along the same path. If they strike at an angle, the angle of incidence equals the angle of reflection, following the laws of reflection.

Laws of Reflection

The incident ray, reflected ray, and normal at the point of incidence all lie in the same plane.
The angle of incidence is equal to the angle of reflection.

Refraction

Refraction is the change in speed and direction of a wave as it moves from one medium to another. When plane waves move from deep to shallow water, their wavelength shortens, and they travel more slowly. This change in speed and wavelength causes a change in the direction of travel.

The refractive index is given by:

\[ \eta = \frac{\sin i}{\sin r} = \frac{v_1}{v_2} \]

Where:

\(\eta\) = Refractive index
i = Angle of incidence
r = Angle of refraction
v₁ = Wave velocity in the first medium
v₂ = Wave velocity in the second medium

Diffraction

Diffraction is the spreading of waves as they pass through a narrow opening or around an obstacle.

If waves pass through a wide gap relative to their wavelength, they remain mostly unchanged.
If waves pass through a narrow gap, they spread out, forming spherical wavefronts.

Diffraction occurs when the wavelength of the wave is comparable to or greater than the width of the opening.

Interference

Interference occurs when two similar waves traveling in the same direction overlap.

Constructive Interference: If two waves are in phase, their amplitudes add up, resulting in a larger wave.
Destructive Interference: If a crest of one wave meets the trough of another, they cancel out, producing no resultant wave.

Polarization

Polarization is a property unique to transverse waves. It restricts wave vibrations to a single plane. A wave vibrating in only one plane is said to be plane-polarized.

Polarized light is produced by passing unpolarized light through a polarizer, such as a Polaroid filter or crystals like calcite, tourmaline, or quartz. The polarizer permits vibrations in one direction while absorbing others.

Applications of Polaroid

Used in sunglasses to reduce glare from reflected light.