Waves - Transverse and Longitudinal Waves

A wave is a disturbance that travels through a medium, transporting energy and momentum without the actual physical transfer of matter.

In a Transverse Wave, the particles of the medium oscillate perpendicular to the direction of wave propagation. These waves consist of 'crests' (highest points) and 'troughs' (lowest points). Examples include waves on a plucked string or electromagnetic waves.

In a Longitudinal Wave, the particles of the medium oscillate parallel to the direction of wave propagation. These waves consist of 'compressions' (regions of high pressure/density) and 'rarefactions' (regions of low pressure/density). Examples include sound waves in air.

Wavelength ($\lambda$) is the distance between two consecutive points in the same phase, such as two consecutive crests or compressions.

Frequency ($\nu$ or $f$) is the number of oscillations per unit time, while the Time Period ($T$) is the time taken for one complete oscillation, related by $T = 1/\nu$.

The displacement of a sinusoidal wave traveling along the positive $x$-axis is given by the function $y(x, t) = A \sin(kx - \omega t + \phi)$, where $A$ is the amplitude, $k$ is the angular wave number, and $\omega$ is the angular frequency.

Transverse waves can only propagate through solids and on the surface of liquids because they require shear strength. Longitudinal waves can propagate through solids, liquids, and gases.