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Introduction to the Physics of Waves by Tim Freegarde

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5

Complex wavefunctions

5.1 Complex harmonic waves

We have seen in Chapter 4 that the property of orthogonality, together with a particular relevance to waves from oscillating and rotating sources, makes sinusoidal waves a convenient set of solutions to many common wave equations. Indeed, the alternation of successive differentiations of a sinusoidal function between sine and cosine means that sinusoidal functions of the wave coordinates xit, x, y, z, … will in general be solutions to partial differential wave equations of the form

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provided that the orders of the derivatives (i.e. the terms (ni + nj + …)) are either all odd or all even. ...

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