The propagation of waves is described using the wave function. Its simplest form can be derived as follows:
We begin by considering an oscillator in a medium that performs a harmonic oscillation around its equilibrium position. A wave arises when such oscillators are coupled, allowing the oscillation to be transmitted to neighboring oscillators. This transfer occurs with a time delay, since the wave propagates at a finite speed $c$.
To describe this propagation correctly, we must extend the harmonic oscillation function to include a term inside the trigonometric function that accounts for the spatial dependence. The instantaneous displacement at a specific location $x$ and time $t$ is therefore given by:
$$
y(x,t) = A \sin\left(\omega \left(t - \frac{x}{c} \right)\right)
$$
This general wave function, which applies to all harmonic waves, depends on two variables: position $x$ and time $t$. Therefore, the instantaneous displacement $y(x,t)$ must be represented in two independent plots.
The graph on the left of Figure\~\ref{fig\:wellenfunktion} shows the time evolution of the oscillation at a fixed position $x$. As usual, the period $T$ denotes the time interval between two successive oscillation cycles.
Next, by holding the time $t$ constant, we can plot the displacement as a function of position $x$, as shown on the right of the figure. The result is again a periodic function, typically sinusoidal. The spatial distance between two adjacent wave peaks is called the wavelength, denoted $\lambda$.
In time $T$, the wave travels a distance equal to one wavelength $\lambda$. Substituting $\lambda$ for distance and $T$ for time in the velocity equation, we obtain the wave speed $c$ as the product of wavelength and frequency:
$$
\boxed{c = \frac{\lambda}{T} = f \lambda}
$$
Substituting this into the wave function gives the most common form found in literature:
$$
\boxed{y(x,t) = A \sin(\omega t - kx)}
$$
The constant $k$ is the wavenumber, defined analogously to angular frequency as:
$$
\boxed{k = \frac{2\pi}{\lambda}}
$$
Thus, the wave speed can also be written as:
$$
\boxed{c = \frac{\omega}{k}}
$$
Taking the second partial derivatives of the wave function with respect to both time and space yields:
$$
\frac{\partial^2 y}{\partial t^2} = -\omega^2 A \sin(\omega t - kx), \quad \frac{\partial^2 y}{\partial x^2} = -k^2 A \sin(\omega t - kx)
$$
Dividing both equations and applying $\omega/k = c$, we obtain the classic wave equation:
$$
\boxed{\frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2}}
$$
This fundamental equation describes the propagation of all kinds of waves.