The Mathematics of the Wave

The Wave Equation

There are several day-to-day situations in which things happen suddenly, unexpectedly or unexpectedly, but they end up settling in a coming and going rhythm and even start to have repercussions in time and space (propagation of if). It is common to refer to this with the expression: "this is just a wave, it will soon pass". This is due to the fact that such situations are fleeting, examples: pandemics, epidemics, the price of commodities, tsunamis, earthquakes, even news or everyday events. Sometimes such situations present oscillating behavior, such as on the surface of a liquid in a swimming pool, or even in natural water sources, as in rivers, seas and oceans during the passage of a vessel.

It is important to note that there are different characteristics in such situations. Some simply oscillate or vibrate and in others, in addition to oscillating or vibrating, this condition is subsequently transmitted to other places in the environment where such situations occur. In this second case, it is said that there is a propagation of the phenomenon. Such that, oscillation, wave, propagation are correlated terms.

Etymologically, the word wave would have to do with the meaning of something that floats on water. The discoveries and developments in this theme in its beginnings derived from the study of musical sounds. The modern study of waves and acoustics is said to have originated with Galileo Galilei.

Figure 1 presents the evolution, in a historical perspective, of the scientific knowledge of this theme, through the contribution of several researchers.

Figure 1. Diagram of the evolution of knowledge of the wave phenomenon.

It is observed, according to the information in Figure 1, that physics consolidated its knowledge on this subject in a physical model, translated through an equation, differential equation.

Following this perspective, we will present here a development of the one-dimensional differential equation of the wave from its hourly equation described by a generic analytical function “f(x)”, illustrated in figure 2:

Figure 2. A one-dimensional generic function representing a disturbance in a medium.

Let's assume that, keeping the profile of the hypothetical physical disturbance in the form of this generic function, it will start propagating to the right (walking), that is, in the positive direction of the "x" axis and at a speed "v". Whenever such a function is observed, it will, analytically, correspond to a mathematical transformation in the form:

situation depicted in the diagram in Figure 3.

Figure 3. Representation of a wave as a function transformation.

Seeking to represent such a situation in the perspective of time "t", this function must now be presented in two variables:

and in the transformed form, making "a=v×t":

where "v" is the speed with which the function "f(x)" moves (wave velocity).

Defining the expression:

we can in advance calculate the first partial derivatives

Rewriting the time equation, we have

With this, their first partial derivatives would be transformed as follows:

and

Developing, from there, the respective partial second derivatives, we have:

and

Replacing the term in the second member of this last expression by its correspondent already determined in the previous step, results:

Which is the one-dimensional expression of the wave differential equation.