(edited by Roberto Bigoni)
Wave Mechanics studies how the local deformations in continuous and elastic medium spread around the source.
If we for simplicity consider an elastic, continuous, unidimensional medium under tension (like, e.g., a string of a musical instrument) and we choose a point S on it as the origin, the position of another point P of the medium is determined by its abscissa x.
If the medium is at rest, the deformation ψ of the medium in this point P is null and it remains null during the flowing of the time. We represent this situation by the equation
If we displace the point S from its position of equilibrium, the elastic reaction of the medium push it towards the position of equilibrium with strength proportional to displacement and the point S begins to move with harmonic motion with amplitude A and pulsation ω.
If we start the time when the point S is in its position of equilibrium and is moving away from this position with positive velocity, we can represent the displacement of this point by the equation
But the medium where the point S is located is elastic and continuous, so the deformation is transmitted to every other points P of the medium after a delay depending from the abscissa x of any point and from the nature of the medium itself. These points begin to move with harmonic oscillation of equal amplitude A and equal pulsation ω of the source S, but dephased with respect to that of the source S depending on the time taken by the perturbation to reach their position.
If we assume the velocity v of the perturbation to be constant, the delay is
and so the oscillation of any point P of the medium, at a distance x from the source S, is represented by the equation
This equation, which denotes the state of every point of the medium at the time t, is the (unidimensional) wave equation .
When the displacements of the point P from its position of equilibrium are perpendicular to the velocity, we have transversal waves. When the displacements of the point P from its position of equilibrium are aligned with the velocity, we have longitudinal waves.
If we remember that
we can also write
The product vT, which is dimensionally a length, represents the smallest distance between two not contiguous points of the medium swinging in phase; this distance is the wavelength λ:
If we use the wave number k
we can write the wave equation
We can now generalize these results to bidimensional or tridimensional waves and, by denoting r the distance of a point P from the source S, we can write the wave equation
If a wave is bidimensional or tridimensional, the whole of the contiguous points swinging in phase is called front wave. For a bidimensional wave a front wave is a line, for a tridimensional wave a front wave is a surface.
If the medium is homogeneous and isotropic, the velocity of a wave is the same in all directions so the front wave are circumferences or spheres.
A region of a spheric front wave very distant from the source is indistinguishable from a flat region. Here, we can say that the wave is a plane wave.
In Optics the lines perpendicular to any consecutive the wavefront, representing the direction the wave is moving, are called rays.
A source emitting waves emits energy in the surrounding medium. The ratio between the emitted energy and the duration of the emission is the mean power of the source.
When the duration of the emission is very short (i.e. it tends to zero) the mean power becomes the instantaneous power of the source.
The instantaneous power is directly proportional to the velocity, to the square of the amplitude and to the square of the frequency.
If the medium is not homogeneous, pulsation ω and frequency ν do not change but both the velocity v and the length λ change. The velocity not only changes its intensity but also its direction. This phaenomenon is called refraction. If we can trace the surface separating media with different velocities v1 and v2, then if we trace the normal to the point in which a ray passes trough this surface and we call α and β the angles the ray forms, before and after, with the normal we have the Snell's Low
When the velocity of a wave cannot transmit the whole emitted power there is reflexion.
If a point P is reached by waves emitted by two or more different sources, the state of this point is determined by the the sum of the different waves. This phaenomenon is said interference.
If the sum of the waves is null, the point P is at rest, it does'nt swing: there is destructive interference. When the sum of the waves is greater than any single wave there is constructive interference.
When a wave passes through a slit having width close to the wavelength, we can observe phaenomena of multiple interference: it happens as if there were in the slit many virtual sources emitting in phase (the Huygens-Fresnel principle). This phaenomenon is said diffraction.
If a medium, like e.g. a string of a musical instrument, has limited dimensions, the energy emitted by a source cannot go beyond its ends so the waves are reflected. The reflected waves, going backward, interfere with those going forward, producing a static pattern of points in which there is destructive interference (nodes) and points in which the constructive interference has a maximum (antinodes). This configuration is said standing (or stationary) wave.
If we suppose, as we do in Quantum Mechanics, that the energy propagated by a wave is quantized and has corpuscular properties and, viceversa, that the particles have ondulatory properties, the function which expresses the instantaneous power can provide informations about the probability of finding the abscissa of a particle in the interval [x;x+dx].
