Wave energy

In today’s article we continue in the case of harmonic waves, that is, the case in which a wave is propagated by a taut string. Furthermore, the results that we will obtain below can be generalized to most mechanical waves.

When a wave propagates, a small element of the string of length ∆x located at a distance x from the origin that describes a MAS of amplitude i, has an energy :

where d∆x is the mass of the string element.

The MAS is transmitted to the adjacent string elements, so that energy travels through the medium with a speed: but ∆E / ∆t is the power P transferred along the string and ∆x / ∆t the velocity of propagation, obtaining: From this equation it is deduced that the power of the wave (speed with which energy travels through the medium) is equal to the energy density by the velocity of propagation. It is worth noting the proportionality between P and the square of the amplitude of the oscillations.

is defined the intensity of a wave at a point as the power that propagates through unit area perpendicular to the propagation direction at that point: I = P / S . In the SI They are measured in W / m ^ 2.
In the case of three-dimensional waves generated by a point emitting focus, the energy spreads by distributing itself on a spherical front with a larger section the further it moves away from the focus, as we can see in this image.
If it is transmitted with a power P, the intensity at point Q1 and at point Q2 respectively are I1 and I2: Therefore, if there are no energy losses, the following will be true:

Then, we can conclude that the intensity of a wave at a point is inversely proportional to the square of the distance to the emitting focus. While the amplitude of the three-dimensional waves at a point is inversely proportional to the distance to the emitting focus. The loss of amplitude, and therefore the intensity when moving away from the focus, is a consequence of the principle of conservation of energy and the widening of the wavefront that causes the energy to be distributed among more and more points in the medium. This phenomenon is known as wave attenuation.

Until now we have assumed that the wave energy is transmitted without loss from some particles in the medium to others. However, in practice, friction between oscillators causes losses that are generally manifested as heat and lead to a decrease in wave intensity as it travels through the medium. The phenomenon is called absorption . The intensity lost, -dI, when traversing a thickness dx of material is proportional to the incident intensity and the distance traversed: -dI = α Idx . Where the negative sign refers to the decrease in intensity and α is a constant characteristic of the medium called the absorption coefficient.

From the previous equation we obtain dI / I = – α dx , which, integrated between the incidence value (Io for x = 0) and that corresponding to a thickness crossed x, leads to: What gives us the intensity of the wave when crossing the thickness x.


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