Complex Addition of Harmonic Motions and the Phenomenon of Beats
Complex Addition of Harmonic Motions and the Phenomenon of Beats
Harmonic motion can be represented by either trigonometric or complex exponential functions. Any vector in the - plane can be written as a complex number: =A(cosθ+isinθ), where is the amplitude (also called the norm) and is the phase (also called the argument). The projection onto the axis of such a rotating vector is the imaginary part of . According to Euler's formula, a complex number can be expressed as =A=A, where denotes the frequency of rotation in the counterclockwise direction and is time. Multiplying by shifts the phase.
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This Demonstrations shows the addition of two independent harmonic motions in terms of complex numbers. The sum of two harmonic motions with nearly equal frequencies exhibits a phenomenon known as beats.