Plane wave

 ( Wave Mechanics ) 

In physics, a plane wave is a special case of a wave or field: a physical quantity whose value, at any given moment, is constant through any plane that is perpendicular to a fixed direction in space.

For any position $\backslash vec\; x$ in space and any time $t$, the value of such a field can be written as $$F(\backslash vec\; x,t)\; =\; G(\backslash vec\; x\; \backslash cdot\; \backslash vec\; n,\; t),$$ where $\backslash vec\; n$ is a unit vector, and $G(d,t)$ is a function that gives the field's value as dependent on only two real number parameters: the time $t$, and the scalar-valued displacement $d\; =\; \backslash vec\; x\; \backslash cdot\; \backslash vec\; n$ of the point $\backslash vec\; x$ along the direction $\backslash vec\; n$. The displacement is constant over each plane perpendicular to $\backslash vec\; n$.

The values of the field $F$ 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 $F$ are vectors, the wave is said to be a longitudinal wave if the vectors are always collinear with the vector $\backslash vec\; n$, and a transverse wave 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 Phase velocity $c$ along the direction perpendicular to the wavefronts. Such a field can be written as $$F(\backslash vec\; x,\; t)\; =\; G\backslash left(\backslash vec\; x\; \backslash cdot\; \backslash vec\; n\; -\; c\; t\backslash right)\backslash ,$$ where $G(u)$ is now a function of a single real parameter $u\; =\; d\; -\; c\; t$, that describes the "profile" of the wave, namely the value of the field at time $t\; =\; 0$, for each displacement $d\; =\; \backslash vec\; x\; \backslash cdot\; \backslash vec\; n$. In that case, $\backslash vec\; n$ is called the direction of propagation. For each displacement $d$, the moving plane perpendicular to $\backslash vec\; n$ at distance $d\; +\; c\; t$ from the origin is called a "wavefront". These planes travel along the direction of propagation $\backslash vec\; n$ with velocity $c$; and the value of the field is then the same, and constant in time, at every one of their points.

John David Jackson (1998). 9780471309321, Wiley. ISBN 9780471309321

---

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 $G(u)$ is a sinusoidal function. That is, $$F(\backslash vec\; x,\; t)\; =\; A\; \backslash sin\backslash left(2\backslash pi\; f\; (\backslash vec\; x\; \backslash cdot\; \backslash vec\; n\; -\; c\; t)\; +\; \backslash varphi\backslash right)$$ The parameter $A$, which may be a scalar or a vector, is called the amplitude of the wave; the scalar coefficient $f$ is its "spatial frequency"; and the scalar $\backslash varphi$ is its "phase shift".

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 from a distant star that arrive at a telescope.

---

Plane standing wave A standing wave 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 $$F(\backslash vec\; x,\; t)\; =\; G(\backslash vec\; x\; \backslash cdot\; \backslash vec\; n)\; \backslash ,\; S(t)$$ where $G$ is a function of one scalar parameter (the displacement $d\; =\; \backslash vec\; x\; \backslash cdot\; \backslash vec\; n$) with scalar or vector values, and $S$ is a scalar function of time.

This representation is not unique, since the same field values are obtained if $S$ and $G$ are scaled by reciprocal factors. If $\backslash left|S(t)\backslash right|$ is bounded in the time interval of interest (which is usually the case in physical contexts), $S$ and $G$ can be scaled so that the maximum value of $\backslash left|S(t)\backslash right|$ is 1. Then $\backslash left|G(\backslash vec\; x\; \backslash cdot\; \backslash vec\; n)\backslash right|$ will be the maximum field magnitude seen at the point $\backslash vec\; x$.

---

Properties A plane wave can be studied by ignoring the directions perpendicular to the direction vector $\backslash vec\; n$; that is, by considering the function $G(z,t)\; =\; F(z\; \backslash vec\; n,\; t)$ as a wave in a one-dimensional medium.

Any local operator, linear operator or not, applied to a plane wave yields a plane wave. Any linear combination of plane waves with the same normal vector $\backslash vec\; n$ 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 $\backslash vec\; n$; specifically, $\backslash nabla\; F(\backslash vec\; x,t)\; =\; \backslash vec\; n\backslash partial\_1\; G(\backslash vec\; x\; \backslash cdot\; \backslash vec\; n,\; t)$, where $\backslash partial\_1\; G$ is the partial derivative of $G$ with respect to the first argument.

The divergence of a vector-valued plane wave depends only on the projection of the vector $G(d,t)$ in the direction $\backslash vec\; n$. Specifically, $$\backslash nabla\; \backslash cdot\; \backslash vec\; F(\backslash vec\; x,\; t)\; \backslash ;=\backslash ;\; \backslash vec\; n\; \backslash cdot\; \backslash partial\_1\; G(\backslash vec\; x\; \backslash cdot\; \backslash vec\; n,\; t)$$ In particular, a transverse planar wave satisfies $\backslash nabla\; \backslash cdot\; \backslash vec\; F\; =\; 0$ for all $\backslash vec\; x$ and $t$.

---

See also

• Plane-wave expansion
• Rectilinear propagation
• Wave equation
• Weyl expansion

Post Comment