# Velocity and wavelength

If we frequently do simple harmonic motion at one end of a fixed string stretched horizontally, we will produce a sequence of waves with equal repetition periods, that is, we will produce a periodic wave, also known as a wave train, which propagates with constant speed. .

In the figure above we have the example, that is, the basic configuration of a periodic wave at an instant t , right after the periodic movements have started.

According to the figure below, if we know the configuration of the periodic wave, we have the possibility to identify some of these characteristics. The highest point of the string in the figure below, for example, is called the crest of the wave ; the lowest point of the string is called the wave valley . Therefore, the distance between two consecutive crests or valleys is defined as the wavelength .

We represent the length of any wave using the following symbol: ( λ ).

Still in relation to the figure below, we can say that the distance between a consecutive crest and valley of a periodic wave is equal to half a wavelength, that is, (λ/2); and the distance from a crest or trough to the equilibrium position is equal to a quarter of the wavelength (λ/4).

As we know that each point on the string performs an MHS, it is also possible to determine the wavelength as the shortest distance between two points in phase agreement . We say that two points are in phase agreement if they perform the same MHS, that is, if they have the same acceleration, velocity and elongation.

Thus, we can say that always two crests and two valleys are in phase agreement. As for a crest and a valley, we say that they are in phase opposition. Therefore, we can define that:

The wavelength (λ) corresponds to the shortest distance between two points on the wave in phase agreement.

By the definition of the average speed, we have:

Since v m = v, Δs = λ and Δt = T, we have that:

Knowing that the period is the inverse of the frequency, T = 1/f, comes the Fundamental Equation of Waves :

v=λ .f

The above equation, which gives the propagation speed of a wave, is valid for both mechanical and electromagnetic waves.

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