In physics, a sinusoidal plane wave is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the distance from some fixed plane. It is also called a monochromatic plane wave, with constant frequency (as in monochromatic radiation).
Basic representation For any position x → {\displaystyle {\vec {x}}} in space and any time t {\displaystyle t} , the value of such a field can be written as
F ( x → , t ) = A cos ( 2 π ν ( x → ⋅ n ^ − c t ) + φ ) {\displaystyle F({\vec {x}},t)=A\cos \left(2\pi \nu ({\vec {x}}\cdot {\hat {n}}-ct)+\varphi \right)}
where n ^ {\displaystyle {\hat {n}}} is a unit-length vector, the direction of propagation of the wave, and " ⋅ {\displaystyle \cdot } " denotes the dot product of two vectors. The parameter A {\displaystyle A} , which may be a scalar or a vector, is called the amplitude of the wave; the coefficient ν {\displaystyle \nu } , a positive scalar, its spatial frequency; and the adimensional scalar φ {\displaystyle \varphi } , an angle in radians, is its initial phase or phase shift. The scalar quantity d = x → ⋅ n ^ {\displaystyle d={\vec {x}}\cdot {\hat {n}}} gives the (signed) displacement of the point x → {\displaystyle {\vec {x}}} from the plane that is perpendicular to n ^ {\displaystyle {\hat {n}}} and goes through the origin of the coordinate system. This quantity is constant over each plane perpendicular to n ^ {\displaystyle {\hat {n}}} . At time t = 0 {\displaystyle t=0} , the field F {\displaystyle F} varies with the displacement d {\displaystyle d} as a sinusoidal function
F ( x → , 0 ) = A cos ( 2 π ν ( x → ⋅ n ^ ) + φ ) {\displaystyle F({\vec {x}},0)=A\cos \left(2\pi \nu ({\vec {x}}\cdot {\hat {n}})+\varphi \right)}
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