How does the formula C-6 come about in the following image?

[weezy]

 

[Simon Bridge]

Via the principle of superposition ... where did it lose you?
C-6 is just summing up all the independent solutions already found...

 

[dextercioby]

Do you know anything about Fourier transformation and how you can use it for PDEs?

 

[weezy]

Via the principle of superposition ... where did it lose you?
C-6 is just summing up all the independent solutions already found...

I meant why does the superposition look like this? i.e. a Fourier transform?

 

[weezy]

Do you know anything about Fourier transformation and how you can use it for PDEs?

I do know about Fourier transforms but am not familiar with their relation to PDEs

 

[jtbell]

I meant why does the superposition look like this?

A superposition of two plane waves looks like this: $$\psi(\vec r, t) = g_1 e^{i(\vec k_1 \cdot \vec r - \omega_1 t)} + g_2 e^{i(\vec k_2 \cdot \vec r - \omega_2 t)}$$
A superposition of infinitely many (but still "countable" 1,2,3...) plane waves looks like this: $$\psi(\vec r, t) = \sum_{j=0}^\infty g_j e^{i(\vec k_j \cdot \vec r - \omega_j t)}$$
Suppose the $\omega_j$ and the $\vec k_j$ can be related by ("fitted to") a continuous function $\omega(\vec k)$. Likewise for the $g_j$ and the $\vec k_j$, with another function $g(\vec k)$. Then we can rewrite this slightly as $$\psi(\vec r, t) = \sum_{j=0}^\infty g(\vec k_j) e^{i(\vec k_j \cdot \vec r - \omega(\vec k_j) t)}$$
Finally, suppose we have a superposition of an "uncountably" infinite number of waves with a continuous (not discrete) distribution of values of $\vec k$. In this case the sum becomes an integral: $$\psi(\vec r, t) = \int g(\vec k) e^{i(\vec k \cdot \vec r - \omega(\vec k) t)} d^3 k$$
This is a sort of "volume integral" in 3-dimensional $\vec k$-space. The extra constant in front of your equation C-6 is basically a sort of normalization factor.