6. Superposition Principle and Complex Representation

Let us say that ψ₁ and ψ₂ are each separate solutions of one-dimensional differential wave equation $\frac{{{\partial }^{2}}\psi }{\partial {{x}^{2}}}=\frac{1}{{{v}^{2}}}\frac{{{\partial }^{2}}\psi }{\partial {{t}^{2}}}$ (see also Equation (3.5)) . The superposition principle says that: the sum of these two wave-functions, i.e. $\left( {{\psi }_{1}}+{{\psi }_{2}} \right)$ , is also a solution of the wave equation. This principle can be proven as follows:

$\frac{{{\partial }^{2}}{{\psi }_{1}}}{\partial {{x}^{2}}}=\frac{1}{{{v}^{2}}}\frac{{{\partial }^{2}}{{\psi }_{1}}}{\partial {{t}^{2}}}$                                                                (6.1)

$\frac{{{\partial }^{2}}{{\psi }_{2}}}{\partial {{x}^{2}}}=\frac{1}{{{v}^{2}}}\frac{{{\partial }^{2}}{{\psi }_{2}}}{\partial {{t}^{2}}}$                                                               (6.2)

Adding Equation (6.1) and (6.2) yields:

$\frac{{{\partial }^{2}}{{\psi }_{1}}}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}{{\psi }_{2}}}{\partial {{x}^{2}}}=\frac{1}{{{v}^{2}}}\left( \frac{{{\partial }^{2}}{{\psi }_{1}}}{\partial {{t}^{2}}}+\frac{{{\partial }^{2}}{{\psi }_{2}}}{\partial {{t}^{2}}} \right)$                                              (6.3)

And then, by simplifying Equation (6.3) we obtain:

$\frac{{{\partial }^{2}}}{\partial {{x}^{2}}}\left( {{\psi }_{1}}+{{\psi }_{2}} \right)=\frac{1}{{{v}^{2}}}\frac{{{\partial }^{2}}}{\partial {{t}^{2}}}\left( {{\psi }_{1}}+{{\psi }_{2}} \right)$                                             (6.4)

 Equation (6.4) establishes that $\left( {{\psi }_{1}}+{{\psi }_{2}} \right)$ is definitely a solution.

This means that when two separate waves arrive at the same place in space wherein they overlap, both waves will add (or subtract from) one another without disrupting either wave. By taking the algebraic sum of the individual constituent waves at a certain location, the disturbance at each point in the region of overlap can be calculated. For instance, when the two constituent waves rise and fall in-step, reinforcing each other, they are said to be in-phase, i.e. their phase-angle difference is zero (see Figure 6.1. below).

Figure 6.1. The superposition of two sinusoidal in-phase waves with amplitudes 1.0 and 0.9.

In contrast to in-phase sinusoidal waves, there are also waves that tend to diminish each other, the so-called the out-of-phase sinusoidal waves when the phase-angle difference equals π (see Figure 6.2. below).

Figure 6.2. The superposition of two sinusoidal out-of-phase waves with amplitudes 1.0 and 0.9.

The complex number representation that describes a wave function offers an alternative description that is mathematically simpler to process. The complex number $\tilde{z}$ has the form:

$\tilde{z}=x+iy$                                                            (6.5)

Where $\sqrt{-1}=i$. Both x and y are real numbers, and they can also be expressed in terms of real and imaginary parts of $\tilde{z}$ respectively, i.e. Re $\left( {\tilde{z}} \right)$ and Im $\left( {\tilde{z}} \right)$.

The complex number $\tilde{z}$ in Equation (6.5) can be expressed graphically in the Argand diagram as given in Figure 6.3.

Figure 6.3. A representation of a complex number in an Argand diagram. In terms of x and y (a). In terms of polar coordinates r and θ  (b). Figure (c) illustrates the complex conjugate $\tilde{z}*$ of $\tilde{z}$. And figure (d) shows when θ is a constantly changing function of time, the arrow rotates at a rate ω.

In terms of polar coordinates (r,θ), the real and imaginary parts in Equation (6.5) can be expressed:

$x=r\cos \theta $                                                                 (6.6)

$y=r\sin \theta $                                                                  (6.7)

$\tilde{z}=x+iy=r(\cos \theta +i\sin \theta )$                                                  (6.8)

Where r denotes the distance from the center of Argand diagram to a point in complex plane (Figure 6.3.(b)), i.e. $r=\left| {\tilde{z}} \right|=\sqrt{{{x}^{2}}+{{y}^{2}}}$ which is also defined as the modulus of complex number $\tilde{z}$. The angle θ is the argument of complex number $\tilde{z}$ which varies between 0 and 2π ($0\le \arg \tilde{z}\le 2\pi $). Therefore complex number $\tilde{z}$ in Equation (6.8) can be written:

$\tilde{z}=\left| {\tilde{z}} \right|[\cos (\arg \tilde{z})+i\sin (\arg \tilde{z})]$                (6.9)

Euler formula defines that a complex number (in term of polar coordinate) can be expressed in term of complex exponential function:

${{e}^{i\theta }}=\cos \theta +i\sin \theta $                                                     (6.10)

Equivalently, for an exponential function with negative power:

${{e}^{-i\theta }}=\cos \theta -i\sin \theta $                                                    (6.11)

From Euler formula, it can be shown that:

$\cos \theta =\frac{{{e}^{i\theta }}+{{e}^{-i\theta }}}{2}$                                                       (6.12)

And

$\sin \theta =\frac{{{e}^{i\theta }}-{{e}^{-i\theta }}}{2i}$                                                      (6.13)

Euler formula also allows us to rewrite the complex number $\tilde{z}$  in terms of polar coordinates (r,θ) as follows:

$\tilde{z}=r{{e}^{i\theta }}=r\cos \theta +ir\sin \theta $                                                (6.14)

For the complex conjugate $\tilde{z}*$ (see Figure 6.3.(c)) applies:

$\tilde{z}*=x-iy$                                                            (6.15)

$\tilde{z}*=r(\cos \theta -i\sin \theta )$                                                     (6.16)

And

$\tilde{z}*=r{{e}^{-i\theta }}$                                                           (6.17)

Notice that, in term of a complex number and its corresponding complex conjugate, the modulus of a complex number can be expressed as follows: $r=\left| {\tilde{z}} \right|\equiv \sqrt{\tilde{z}\tilde{z}*}$.

Addition and substraction of two complex numbers ${{\tilde{z}}_{1}}$ and ${{\tilde{z}}_{2}}$ are straight forward:

${{\tilde{z}}_{1}}\pm {{\tilde{z}}_{2}}=({{x}_{1}}+i{{y}_{1}})\pm ({{x}_{2}}+i{{y}_{2}})=({{x}_{1}}\pm {{x}_{2}})+i({{y}_{1}}\pm {{y}_{2}})$                       (6.18)

For multiplication and division, the following rules of thumb can be applied:

${{\tilde{z}}_{1}}{{\tilde{z}}_{2}}={{r}_{1}}{{r}_{2}}{{e}^{i({{\theta }_{1}}+{{\theta }_{2}})}}$                                                      (6.19)

$\frac{{{{\tilde{z}}}_{1}}}{{{{\tilde{z}}}_{2}}}=\frac{{{r}_{1}}}{{{r}_{2}}}{{e}^{i({{\theta }_{1}}-{{\theta }_{2}})}}$                                                        (6.20)

Another important equation is:

${{e}^{z}}={{e}^{x+iy}}={{e}^{x}}{{e}^{iy}}={{e}^{x}}(\cos y+i\sin y)$                                         (6.21)

Therefore

$\left| {{e}^{z}} \right|={{e}^{x}}$                                                            (6.22)

And for $n\in \mathbb{Z}$ 

$\arg ({{e}^{z}})=y+2\pi n$                                                        (6.23)

For all ${{\tilde{z}}_{1}},{{\tilde{z}}_{2}}\in \mathbb{Z}$  the following equation is also valid:

${{e}^{{{{\tilde{z}}}_{1}}}}{{e}^{{{{\tilde{z}}}_{2}}}}={{e}^{{{{\tilde{z}}}_{1}}+{{{\tilde{z}}}_{2}}}}$ .                                                              (6.24)