Complex waves - Complex waves From Physics Notes Complex...

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Complex waves From Physics Notes Complex numbers are extremely useful for describing the propagation of waves through space and time. Waves are ubiquitous physical phenomena, including electromagnetic waves (radio waves, visible light, X-rays, etc.), sound waves, quantum mechanical wavefunctions, and more. It is therefore very important to have a good mathematical understanding of them. Contents 1 The wave equation 1.1 Real solutions 1.2 Complex representation 1.3 3D generalization 2 Waves in non-uniform media 3 Harmonic waves 4 Exercises The wave equation Wave propagation can be described using a partial differential equation known as the time-dependent wave equation. For simplicity, we will restrict our attentions to a single spatial coordinate, denoted . The time coordinate is denoted . A wave is represented by a real function , called the wavefunction . The wavefunction specifies the value of some measurable physical quantity at each point in space ( ) and time ( ). For example, for a sound wave, stands for the local mechanical pressure. The time-independent wave equation is the following partial differential equation: The constant is called the "wave speed", for a reason that will shortly become clear. More precisely, the above differential equation is the (1+1) dimensional wave equation for a uniform medium. "(1+1) dimensional", or " (1+1)D", refers to the fact that there is one space variable and one time variable . "Uniform medium" means that the wave is propagating in a featureless space, in which every point has the same physical properties as every other point; we'll introduce the non-uniform case in a moment. For compactness, we can also re-write the wave equation by putting everything on one side: Real solutions The wave equation has solutions of the form
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Such a solution describes a sinusoidal wave propagating to the right (+) or left (-) with speed . To see, this,
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