Sinusoidal plane wave
is the number of full cycles per unit of length along the direction
A sinusoidal plane wave could be a suitable model for a sound wave within a volume of air that is small compared to the distance of the source (provided that there are no echos from nearly objects).
, the field will also vary sinusoidally with time; it will be a scalar multiple of the amplitude
The formula above gives a purely "kinematic" description of the wave, without reference to whatever physical process may be causing its motion.
In a mechanical or electromagnetic wave that is propagating through an isotropic medium, the vector
of the apparent propagation of the wave is also the direction in which energy or momentum is actually flowing.
above can also be expressed in terms of sine instead of cosine using the elementary identity
Thus the value and meaning of the phase shift depends on whether the wave is defined in terms of sine or co-sine.
has the effect of reversing the direction of propagation, with a suitable adjustment of the initial phase.
The formula of a sinusoidal plane wave can be written in several other ways: A plane sinusoidal wave may also be expressed in terms of the complex exponential function
The original wave expression is now simply the real part,
Observe that the first term equals the real form of the plane wave just discussed.
The introduced complex form of the plane wave can be simplified by using a complex-valued amplitude
While the complex form has an imaginary component, after the necessary calculations are performed in the complex plane, its real value (which corresponds to the wave one would actually physically observe or measure) can be extracted giving a real valued equation representing an actual plane wave.
The main reason one would choose to work with complex exponential form of plane waves is that complex exponentials are often algebraically easier to handle than the trigonometric sines and cosines.
Specifically, the angle-addition rules are extremely simple for exponentials.
In quantum mechanics the solutions of the Schrödinger wave equation are by their very nature complex-valued and in the simplest instance take a form identical to the complex plane wave representation above.
The imaginary component in that instance however has not been introduced for the purpose of mathematical expediency but is in fact an inherent part of the “wave”.
The equations describing electromagnetic radiation in a homogeneous dielectric medium admit as special solutions that are sinusoidal plane waves.
The equations that describe vibrations in a homogeneous elastic solid also admit solutions that are sinusoidal plane waves, both transverse and longitudinal.
These two types have different propagation speeds, that depend on the density and the Lamé parameters of the medium.
The fact that the medium imposes a propagation speed means that the parameters
The form of the planewave solution is actually a general consequence of translational symmetry.
More generally, for periodic structures having discrete translational symmetry, the solutions take the form of Bloch waves, most famously in crystalline atomic materials but also in photonic crystals and other periodic wave equations.
Because this is a plane wave, each blue vector, indicating the perpendicular displacement from a point on the axis out to the sine wave, represents the magnitude and direction of the electric field for an entire plane that is perpendicular to the axis.
Represented in the second illustration is a circularly polarized, electromagnetic plane wave.
Each blue vector indicating the perpendicular displacement from a point on the axis out to the helix, also represents the magnitude and direction of the electric field for an entire plane perpendicular to the axis.
In the case of the linearly polarized light, the field strength from plane to plane varies from a maximum in one direction, down to zero, and then back up to a maximum in the opposite direction.
In the case of the circularly polarized light, the field strength remains constant from plane to plane but its direction steadily changes in a rotary type manner.
The ratio of the amplitudes of the electric and magnetic field components of a plane wave in free space is known as the free-space wave-impedance, equal to 376.730313 ohms.
