Waves transfer energy and information through a medium (or through fields) without permanently transporting matter from one place to another.
A convenient way to classify waves is by the direction of the medium’s oscillation compared to the direction the wave travels.
In a transverse wave, the displacement of the medium is perpendicular to the direction of propagation. A classic example is a wave on a
stretched string: the string elements move up and down while the wave travels along the string. In a longitudinal wave, the displacement is
parallel to the direction of propagation. Sound in air is the standard example: air molecules oscillate back and forth along the direction the
pressure disturbance travels.
These directions matter because they determine important physical properties such as polarization and the type of restoring force involved.
Transverse waves can be polarized: the oscillations can occur in different perpendicular directions. For instance, a string can vibrate
vertically or horizontally, and the wave still travels along the string. Electromagnetic (EM) waves are also transverse; their electric and magnetic
fields oscillate perpendicular to the direction of propagation and to each other, allowing linear, circular, or elliptical polarization.
In contrast, longitudinal waves do not support the same kind of polarization, because the motion is constrained along the propagation direction.
(One can still discuss polarization in more advanced elastic solids where both shear and compressional components exist, but in simple fluids,
sound is purely longitudinal.)
Wave speed models. The speed of a mechanical wave depends on the medium’s inertia and its restoring force. For a stretched string, the inertia
is set by the linear mass density \(\mu\) (kg/m) and the restoring force is provided by the tension \(T\) (N). The wave speed is
\[
v_{\text{string}}=\sqrt{\frac{T}{\mu}}.
\]
Increasing the tension makes the wave travel faster; increasing the mass per unit length makes it travel slower. For sound waves in a fluid,
compression and expansion are resisted by the bulk modulus \(B\) (Pa), while the inertia is provided by the fluid density \(\rho\) (kg/m\(^3\)).
The sound speed is
\[
v_{\text{sound}}=\sqrt{\frac{B}{\rho}}.
\]
A stiffer medium (larger \(B\)) transmits sound faster, while a denser medium (larger \(\rho\)) transmits it slower.
Energy transport and intuition. In both cases, energy moves through the medium even though the particles mainly oscillate about equilibrium.
For a transverse string wave, energy is stored alternately as kinetic energy of moving string elements and potential energy associated with the string’s
curvature under tension. For a longitudinal sound wave, energy alternates between kinetic energy of particle motion and potential energy associated with
compression (pressure) and rarefaction. A helpful visualization is compression “packets” in a longitudinal wave: regions of higher density move along
the direction of travel, even though individual particles only shift slightly.
This calculator compares these concepts side-by-side. It computes typical speeds from the formulas above (if you enter the needed parameters), states
the polarization difference, and animates the two wave types: a transverse sinusoidal displacement curve and a longitudinal particle-motion pattern.
At university level, you can extend the comparison to elastic solids (which support both transverse shear waves and longitudinal compressional waves),
seismic S-waves (transverse) and P-waves (longitudinal), and electromagnetic transverse waves in vacuum and materials.
