Wave Superposition
When dealing with wave mechanics the principle of linear superposition specifies that when two or more waves interact they interfere with each other. This interference may be constructive or destructive interference. When wave crests match constructive interference takes place, when their form (phases) do not match destructive interference can occur. With identical waves (i.e., waves of the same wavelength and amplitude) emitted by the same source, the waves will undergo constructive interference if the distance traveled by one wave is equal to, or differs by an integral number of wavelengths from, the distance traveled by the second wave. Whenever path distances differ by exactly one-half a wavelength, the waves are completely out of phase and undergo complete destructive interference.
The general formulation of the superposition principle is that, when a number of influences act on a system, the total influence on that system is the linear addition of the individual influences. In quantum mechanics, the equivalent principle states that the resultant wave function due to two or more individual wave functions is the sum of the individual wave functions. In wave physics, the principle of superposition can be applied to waves whenever two or more waves are traveling through the same medium at the same time and they pass through each other without being disturbed. In this case, the net displacement of the medium at any point is simply the sum of the individual wave displacements. This applies to wave pulses and to continuous sine waves. For example, the net displacement of two gaussian wave pulses traveling in opposite directions in a given non-dispersive medium and passing through each other without being disturbed is the sum of the two individual displacements. Another example is the case of two sine waves with the same amplitude, frequency, and wavelength traveling in the same direction in a given medium. Applying the principle of superposition, the resulting medium displacement may be described by summing the two waves.
The result is a traveling wave whose amplitude depends on the phase angle. When the waves are in-phase, constructive interference occurs with the resulting wave amplitude being twice that of the individual waves. When the waves are out-of-phase, destructive interference is produced and the respective amplitudes cancel each other out. Another example is of two sine waves also with the same amplitude, frequency, and wavelength but that are traveling in opposite directions. Again, applying the principle of superposition, the resulting wave is no longer a traveling wave but a standing wave.
In the case of electromagnetic waves, when two such waves with the same frequency superpose in space, the resultant electric and magnetic field strength of any point of space and time is the sum of their respective fields, except when they are in phase or out-of-phase, which leads to interference. In the first case, constructive interference occurs because the resultant field is twice that of each individual wave and the resultant intensity is four times the intensity of each superposed wave because intensity is proportional to the square of the field strength. In the second case, destructive interference occurs since the respective electric field vectors are in opposite directions, so the result is zero and no wave results.
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