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## General description of waves

The illustration shows an oscillation with its usual names.

The quantity u is the elongation (deflection) and is generally dependent on both the Ortx and the time t. The maximum value of the elongation is called AmplitudeA.

Harmonic oscillations are those that occur in systems to which Hooke's law applies:

- Note
- The restoring force is proportional to the elongation (deflection from the rest position):
- $\text{F.}=\text{const}\cdot \text{u}$

Vibrations in elastic systems propagate spatially (v = speed of propagation). The elongation u is therefore a function of place and time.

- $$\begin{array}{ll}\text{one-dimensional case}& :\text{u}=\text{u}\left(\text{x, t}\right)\\ \text{two-dimensional case}& :\text{u}=\text{u}\left(\text{x, y, t}\right)\\ \text{three-dimensional case}& :\text{u}=\text{u}\left(\text{x, y, z, t}\right)\end{array}$$

Harmonic waves can be described mathematically using the oscillation equation. In the one-dimensional case:

- $$\left(\frac{{\partial}^{2}\text{u}}{\partial {\text{x}}^{\text{2}}}\right)=\frac{1}{{\text{v}}^{\text{2}}}\left(\frac{{\partial}^{2}\text{u}}{\partial {\text{t}}^{\text{2}}}\right)$$

Solutions to this differential equation are all functions u (x, t) that satisfy the differential equation. For example is

- $$\text{u}=\text{A.}\cdot {\text{e}}^{\text{i}\overline{\omega}\left(\text{t-}\frac{\text{x}}{\text{v}}\right)}$$

such a solution. It represents the equation of a harmonic wave with the amplitude A, the angular frequency ω and the propagation velocity v.

For vibrations of two-dimensional systems (surface waves) the following applies accordingly for the two position coordinates x, y;

- $${\left(\frac{{\partial}^{2}\text{u}}{\partial {\text{x}}^{\text{2}}}\right)}_{y}+{\left(\frac{{\partial}^{2}\text{u}}{\partial {\text{y}}^{\text{2}}}\right)}_{x}=\frac{1}{{\text{v}}^{\text{2}}}\left(\frac{{\partial}^{2}\text{u}}{\partial {t}^{\text{2}}}\right)$$

and for the three-dimensional systems (sky waves, spherical waves):

- $${\left(\frac{{\partial}^{2}\text{u}}{\partial {\text{x}}^{\text{2}}}\right)}_{y,z}+{\left(\frac{{\partial}^{2}\text{u}}{\partial {\text{y}}^{\text{2}}}\right)}_{x,z}+{\left(\frac{{\partial}^{2}\text{u}}{\partial {z}^{\text{2}}}\right)}_{x,y}=\frac{1}{{\text{v}}^{\text{2}}}\left(\frac{{\partial}^{2}\text{u}}{\partial {t}^{\text{2}}}\right)$$