Sums of Harmonic Functions

Section 9.2 Sums of Harmonic Functions

Harmonic (sinusoidal) functions have many remarkable properties. In this section, we explore the effects of adding two harmonic functions. Figure 9.2.1 below shows the graph of \(a\cos\theta+b\sin\theta\text{,}\) subject to the constraint that \(a^2+b^2=1\) while Figure 9.2.2 shows the effect of allowing \(a\) and \(b\) to vary independently.

Figure 9.2.1. The graph of \(a\cos\theta+b\sin\theta\text{.}\)

Figure 9.2.2. The graph of \(a\cos\theta+b\sin\theta\text{.}\)

Activity 9.2.1. 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 (1.5.1)

Answer

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

Figure 9.2.3 below shows the \(m\)th order Fourier expansion of a given function.

Figure 9.2.3. The \(m\)th order Fourier expansion of a given function.

Figure 9.2.4 below shows the \(m\)th order Fourier expansion of a given function.

Figure 9.2.4. The \(m\)th order Fourier expansion of a given function.