Sums of Harmonic Functions

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Section 2.7 Sums of Harmonic Functions

Harmonic (sinusoidal) functions have many remarkable properties. It is almost always easiest to prove these properties by using Euler's formula to turn the harmonic functions into exponentials, see (2.5.1).

In this section, we explore the effects of adding two harmonic functions. The results here are very useful when you want to describe waves with different algebraic representations. Figure 2.7.1 below shows the graph of \(a\cos\theta+b\sin\theta\text{,}\) subject to the constraint that \(a^2+b^2=1\) while Figure 2.7.2 shows the effect of allowing \(a\) and \(b\) to vary independently.

Figure 2.7.1. The graph of \(a\cos\theta+b\sin\theta\text{.}\)

Figure 2.7.2. The graph of \(a\cos\theta+b\sin\theta\text{.}\)

Activity 2.7.1. 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 \(b\text{.}\)

Use Euler's formula (2.5.1)

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