0In complex form, a wave is represented using an exponential expression that contains a phase term. This phase changes as we move through space.
Wavelength is defined as the distance in space over which the phase of the complex wave increases by one complete cycle (that is, a full rotation in the complex plane). After this distance, the wave repeats its pattern because the exponential term returns to the same value.
In this representation, the quantity that multiplies position inside the phase determines how rapidly the phase changes with distance. Wavelength is inversely related to how fast this phase changes. If the phase changes rapidly with position, the wavelength is small. If it changes slowly, the wavelength is large.
For IIT-JEE level understanding:
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Wavelength is the spatial periodicity of the complex wave.
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It is the distance required for the complex exponential to complete one full phase cycle.
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Even though the wave is written in complex form, the wavelength still represents a real physical distance in space.
If the wave number becomes complex (as in damping situations), the real part determines the wavelength, while the imaginary part represents decay of amplitude.
