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Maxwell’s Equations in Free Space

Electromagnetic Waves in Vacuum. Maxwell’s Equations in Free Space. wave equation in 3D. General solution The one dimensional wave equation is unusual for a partial differential equation in that a very simple general solution may be found by defining new variables:.

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Maxwell’s Equations in Free Space

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  1. Electromagnetic Waves in Vacuum Maxwell’s Equations in Free Space wave equation in 3D

  2. General solution The one dimensional wave equation is unusual for a partial differential equation in that a very simple general solution may be found by defining new variables:

  3. Jean le Rond d'Alembert (1717–1783) In other words, solutions of the 1D wave equation are sums of a right traveling function F and a left traveling function G. "Traveling" means that the shape of these individual arbitrary functions with respect to x stays constant, however the functions are translated left and right with time at the speed c.

  4. For an initial value problem, the arbitrary functions F and G can be determined to satisfy initial conditions: d'Alembert's formula

  5. Example

  6. The sinusoidal wave wave number phase shift wavelength angular frequency frequency sinusoidal wave traveling to the right sinusoidal wave traveling to the left

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