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Representation of Waves Using Exponentials

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It is convenient to represent waves using exponential functions. For example a stationary wave can be described by the function: ψ(x) = Aeikx, where the constant A is a real number. The function ψ(x) can be resolved into its real and imaginary parts using Euler's equation,

ei θ = cosθ + isinθ,

to obtain

ψ(x) = Acos(kx) + iAsin(kx).

A travelling stationary wave can be described by exponential function

ψ(x),(t) = Aei(kx - ωt)

The exponential function has mathematical properties, which makes it more convenient to use than trigonometric functions. For instance the product of an exponential function eA and a second exponential function eB can be evaluated by simply adding the exponents:

eA + eB = eA + B

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