Harmonic waves

Once we have become familiar with simple harmonic motion, we will continue to describe the result of this type of motion: harmonic waves .

Definition: We call the generated wave in a simple harmonic motion harmonic wave. It appears when we tighten a string at one end making the MAS describe in the perpendicular direction at the other end.
If we take the free end as the coordinate origin, the displacement of the equilibrium position will be given by:
where ω is the angular frequency of the harmonic disturbance and φ or the initial phase.

This movement is transmitted to the adjacent portions of the string with speed υ thanks to its elasticity and inertia, so that a particle located at a distance x from the free end undergoes a displacement: Taking into account that x / υ is the time that the disturbance takes time to reach it, we obtain the equation that describes the state of vibration of the points of the medium located at a distance x from the origin at time t: where k = ω / υ is a certain characteristic of the wave called number of waves

. For simplicity, we will henceforth take our initial phase as 0: φo = 0; whereby the equation that describes a harmonic wave that propagates along the + X axis with velocity υ = ω / k is: Observation: To find the equation that describes along the -X axis, we would only have to change the + by the -.

The wave motion represented by the previous equation is doubly periodic, both in time and space:
a) In time:
A determined particle of the medium (fixed x) describes a MAS of angular frequency ω and period T such that: substituting in the wave equation: Therefore ωT = 2Π and every T = 2Π / ω second the particle describes a complete oscillation, or what is the same, in one second it will have made f = 1 / T oscillations . This magnitude, f, is called the wave frequency and is measured in hertz (Hz) in the SI.

  1. b) In space:
    If at a certain instant (fixed t) we observe the medium through which the wave propagates we will find the particles in different states of vibration. The distance between two consecutive points that oscillate in phase is called the wavelength (λ), so: That is: In this case, it leads to: k λ = 2Π or λ = 2Π / kat a given instant, all the points in the middle reached by a wave and distant from each other an integer number of wavelengths will be found oscillating in phase:

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Note: Everything mentioned above can also be used when we substitute the sine function for the cosine function in the wave equation, since the change would simply suppose a correction in the phase of Π / 2 radians.

 

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