In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, a plane wave is a special case of
wave or
field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position
in space and any time
, the value of such a field can be written as
:
where
is a
unit-length vector, and
is a function that gives the field's value as dependent on only two
real parameters: the time
, and the scalar-valued
displacement of the point
along the direction
. The displacement is constant over each plane perpendicular to
.
The values of the field
may be scalars, vectors, or any other physical or mathematical quantity. They can be
complex numbers, as in a
complex exponential plane wave.
When the values of
are vectors, the wave is said to be a
longitudinal wave if the vectors are always collinear with the vector
, and a
transverse wave
In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example o ...
if they are always orthogonal (perpendicular) to it.
Special types
Traveling plane wave
Often the term "plane wave" refers specifically to a
traveling plane wave, whose evolution in time can be described as simple translation of the field at a constant wave speed
along the direction perpendicular to the wavefronts. Such a field can be written as
:
where
is now a function of a single real parameter
, that describes the "profile" of the wave, namely the value of the field at time
, for each displacement
. In that case,
is called the direction of propagation. For each displacement
, the moving plane perpendicular to
at distance
from the origin is called a "
wavefront". This plane travels along the direction of propagation
with velocity
; and the value of the field is then the same, and constant in time, at every one of its points.
Sinusoidal plane wave
The term is also used, even more specifically, to mean a "monochromatic" or
sinusoidal plane wave: a travelling plane wave whose profile
is a
sinusoidal function. That is,
:
The parameter
, which may be a scalar or a vector, is called the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
of the wave; the scalar coefficient
is its "spatial frequency"; and the scalar
is its "phase".
A true plane wave cannot physically exist, because it would have to fill all space. Nevertheless, the plane wave model is important and widely used in physics. The waves emitted by any source with finite extent into a large homogeneous region of space can be well approximated by plane waves when viewed over any part of that region that is sufficiently small compared to its distance from the source. That is the case, for example, of the
light waves from a distant star that arrive at a telescope.
Plane standing wave
A
standing wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
is a field whose value can be expressed as the product of two functions, one depending only on position, the other only on time. A
plane standing wave, in particular, can be expressed as
:
where
is a function of one scalar parameter (the displacement
) with scalar or vector values, and
is a scalar function of time.
This representation is not unique, since the same field values are obtained if
and
are scaled by reciprocal factors. If
is bounded in the time interval of interest (which is usually the case in physical contexts),
and
can be scaled so that the maximum value of
is 1. Then
will be the maximum field magnitude seen at the point
.
Properties
A plane wave can be studied by ignoring the directions perpendicular to the direction vector
; that is, by considering the function
as a wave in a one-dimensional medium.
Any
local operator,
linear or not, applied to a plane wave yields a plane wave. Any linear combination of plane waves with the same normal vector
is also a plane wave.
For a scalar plane wave in two or three dimensions, the gradient of the field is always collinear with the direction
; specifically,
, where
is the partial derivative of
with respect to the first argument.
The
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
of a vector-valued plane wave depends only on the projection of the vector
in the direction
. Specifically,
:
In particular, a transverse planar wave satisfies
for all
and
.
See also
*
Plane wave expansion
*
Rectilinear propagation
Rectilinear propagation describes the tendency of electromagnetic waves (light) to travel in a straight line. Light does not deviate when travelling through a homogeneous medium, which has the same refractive index throughout; otherwise, light s ...
*
Wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ...
*
Weyl expansion
References
Sources
*
*{{cite book , last=Jackson , first= John David , author-link= John David Jackson (physicist) , date= 1998 , title=
Classical Electrodynamics
Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fi ...
, location=New York , publisher=
Wiley , isbn= 9780471309321 , edition=3
Wave mechanics