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This Concept Map, created with IHMC CmapTools, has information related to: plane waves, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <munderover> <mtext> k </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> ⊥ </mtext> <munderover> <mtext> B </mtext> <none/> <mtext> → </mtext> </munderover> </mrow> </math> which imply "in-plane" (transverse) fields, sine waves (like a wave on a rope) advancing in <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> the </mtext> <munderover> <mtext> k </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> direction </mtext> </mrow> </math>, monochromatic plane waves are far-field limits of spherical waves, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <munderover> <mtext> B </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> = </mtext> <mfrac> <mrow> <munderover> <mtext> k </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> × </mtext> <munderover> <mtext> E </mtext> <none/> <mtext> → </mtext> </munderover> </mrow> <mtext> ω </mtext> </mfrac> </mrow> </math> which requires <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <munderover> <mtext> k </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> ⊥ </mtext> <munderover> <mtext> B </mtext> <none/> <mtext> → </mtext> </munderover> </mrow> </math>, vectors at every point in 3-D space having the same (complex) field directions, sine waves (like a wave on a rope) in turn represent connected tips of vectors corresponding to the field strength along a straight line, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <munderover> <mtext> B </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> = </mtext> <mfrac> <mrow> <munderover> <mtext> k </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> × </mtext> <munderover> <mtext> E </mtext> <none/> <mtext> → </mtext> </munderover> </mrow> <mtext> ω </mtext> </mfrac> </mrow> </math> which requires <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mfenced open="|" close="|"> <mtext> B </mtext> </mfenced> <mtext> = </mtext> <mfrac> <mfenced open="|" close="|"> <mtext> E </mtext> </mfenced> <mtext> ω/k </mtext> </mfrac> </mrow> </math>, a set of parallel lines in turn represent iso-field surfaces, Maxwell's E&M equations in free space only if they obey <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <munderover> <mtext> B </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> = </mtext> <mfrac> <mrow> <munderover> <mtext> k </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> × </mtext> <munderover> <mtext> E </mtext> <none/> <mtext> → </mtext> </munderover> </mrow> <mtext> ω </mtext> </mfrac> </mrow> </math>, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <munderover> <mtext> B </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> = </mtext> <mfrac> <mrow> <munderover> <mtext> k </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> × </mtext> <munderover> <mtext> E </mtext> <none/> <mtext> → </mtext> </munderover> </mrow> <mtext> ω </mtext> </mfrac> </mrow> </math> which requires <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <munderover> <mtext> E </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> ⊥ </mtext> <munderover> <mtext> B </mtext> <none/> <mtext> → </mtext> </munderover> </mrow> </math>, iso-field surfaces advancing unchangingly in <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> the </mtext> <munderover> <mtext> k </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> direction </mtext> </mrow> </math>, phases have <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <munderover> <mtext> k </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> ⋅ </mtext> <munderover> <mtext> r </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> spatial
 variation </mtext> </mrow> </math>, a stack of planes in turn represent iso-field surfaces, vectors at every point in 3-D space having the same amplitudes, a stack of planes orthogonal to <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> the </mtext> <munderover> <mtext> k </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> direction </mtext> </mrow> </math>, no divergence which requires <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <munderover> <mtext> k </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> ⊥ </mtext> <munderover> <mtext> E </mtext> <none/> <mtext> → </mtext> </munderover> </mrow> </math>, monochromatic plane waves are often represented by a set of parallel lines, monochromatic plane waves are often represented by sine waves (like a wave on a rope), a set of parallel lines advancing in <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> the </mtext> <munderover> <mtext> k </mtext> <none/> <mtext> → </mtext> </munderover> <mtext> direction </mtext> </mrow> </math>, monochromatic plane waves are fields that satisfy the 3D wave equation