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Section 10.11 Sums of Harmonic Functions

Harmonic (sinusoidal) functions have many remarkable properties. In this section, we explore the effects of adding two harmonic functions. Figure 10.11.1 below shows the graph of \(a\cos\theta+b\sin\theta\text{,}\) subject to the constraint that \(a^2+b^2=1\) while Figure 10.11.2 shows the effect of allowing \(a\) and \(b\) to vary independently.

Figure 10.11.1. The graph of \(a\cos\theta+b\sin\theta\text{.}\)
Figure 10.11.2. The graph of \(a\cos\theta+b\sin\theta\text{.}\)
Activity 10.11.1. The sum of two harmonic functions.
In the animations above, it looks as if the sum of two harmonic functions is another harmonic function. Show algebraically that this is true, i.e. show that \(a\cos{\theta}+b\sin{\theta}=r\cos{(\theta-\delta)}\text{.}\) Furthermore, find expressions for \(r\) and \(\delta\) in terms of \(a\) and \(b\text{.}\)
Hint

Use Euler's formula (2.5.1)

Answer

\(r=\sqrt{a^2+b^2}\) and \(\tan{\delta}=\frac{b}{a}\text{.}\)