The Principle of Superposition

# The Principle of Superposition - constructive interference...

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The Principle of Superposition One property of the differential wave equation is that it is linear. This means that if you find two solutions ψ 1 and ψ 2 that both satisfy the equation, then ( ψ 1 + ψ 2 ) must also be a solution. This is easily proved. We have: = = Adding these gives: + = + ( ψ 1 + ψ 2 ) = ( ψ 1 + ψ 2 ) This means that when two waves overlap in space, they will simply 'add up;' the resulting disturbance at each point of overlap will be the algebraic sum of the individual waves at that location. Moreover, once the waves pass each other, they will continue on as if neither had ever encountered the other. This is called the principle of superposition. When waves add up to form a greater total amplitude than either of the constituent waves it is called

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Unformatted text preview: constructive interference, and when the amplitudes partially or wholly cancel each other out it is called destructive interference. Identical waves that overlap completely are said to be in-phase and will constructively interfere at all points, with an amplitude double that of either constituent wave. Otherwise identical waves (that is they have the same frequency and amplitude) that differ in phase by exactly 180 o ( Π radians) are said to be out-of-phase, and will destructively interfere at all points. Some examples are illustrated in and . The principle of superposition will come to be of vital importance in the rest of our study of optics. Figure %: Constructive interference. Figure %: Destructive interference....
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