Transverse wave
In contrast, a longitudinal wave travels in the direction of its oscillations.
[1][2] Electromagnetic waves are transverse without requiring a medium.
Light is another example of a transverse wave, where the oscillations are the electric and magnetic fields, which point at right angles to the ideal light rays that describe the direction of propagation.
Transverse waves commonly occur in elastic solids due to the shear stress generated; the oscillations in this case are the displacement of the solid particles away from their relaxed position, in directions perpendicular to the propagation of the wave.
These displacements correspond to a local shear deformation of the material.
Water waves involve both longitudinal and transverse motions.
[6] Mathematically, the simplest kind of transverse wave is a plane linearly polarized sinusoidal one.
"Plane" here means that the direction of propagation is unchanging and the same over the whole medium; "linearly polarized" means that the direction of displacement too is unchanging and the same over the whole medium; and the magnitude of the displacement is a sinusoidal function only of time and of position along the direction of propagation.
be the direction of propagation (a vector with unit length), and
be the direction of the oscillations (another unit-length vector perpendicular to d).
where A is the wave's amplitude or strength, T is its period, v is the speed of propagation, and
will see the particle there move in a simple harmonic (sinusoidal) motion with period T seconds, with maximum particle displacement A in each sense; that is, with a frequency of f = 1/T full oscillation cycles every second.
, with the displacements in successive planes forming a sinusoidal pattern, with each full cycle extending along
with speed V. The same equation describes a plane linearly polarized sinusoidal light wave, except that the "displacement" S(
and time t. (The magnetic field will be described by the same equation, but with a "displacement" direction that is perpendicular to both
In a homogeneous linear medium, complex oscillations (vibrations in a material or light flows) can be described as the superposition of many simple sinusoidal waves, either transverse or longitudinal.
The vibrations of a violin string create standing waves,[7] for example, which can be analyzed as the sum of many transverse waves of different frequencies moving in opposite directions to each other, that displace the string either up or down or left to right.
, we can choose two mutually perpendicular directions of polarization, and express any wave linearly polarized in any other direction as a linear combination (mixing) of those two waves.
By combining two waves with same frequency, velocity, and direction of travel, but with different phases and independent displacement directions, one obtains a circularly or elliptically polarized wave.
In such a wave the particles describe circular or elliptical trajectories, instead of moving back and forth.
It may help understanding to revisit the thought experiment with a taut string mentioned above.
Notice that you can also launch waves on the string by moving your hand to the right and left instead of up and down.
There are two independent (orthogonal) directions that the waves can move.
(This is true for any two directions at right angles, up and down and right and left are chosen for clarity.)
Your motion will launch a spiral wave on the string.
The maxima of the side to side motion occur a quarter wavelength (or a quarter of a way around the circle, that is 90 degrees or π/2 radians) from the maxima of the up and down motion.
To the extent your circle is imperfect, a regular motion will describe an ellipse, and produce elliptically polarized waves.
(Let the linear mass density of the string be μ.)
The kinetic energy of a mass element in a transverse wave is given by:
Using Hooke's law the potential energy in mass element
