## 4. Interference.

The mechanical behavior of point P of a continuous elastic medium where overlap two different waves
Ψ_{1} e Ψ_{2}, emitted by two sources O_{1} e O_{2} from which it has
distances *r*_{1} e *r*_{2}, is the same as if there were only one wave given by the
sum of Ψ_{1} e Ψ_{2}.

In particular, if the sources O_{1} e O_{2} oscillate in phase with equal amplitudes and
angular frequencies, the prosthaphaeresis formulas give

The amplitude of the sum Ψ is

This amplitude has its maximum value *2A* when the absolute value of the cosine in (4.2) is 1, that is when

**The points of the medium such that the difference between their distances from the sources O**_{1} and
O_{2} is equal to an even number of half-wavelengths oscillate with the greatest amplitude.

The resulting amplitude instead is null, that is the point P doesn't oscillate, when the cosine is null, that is when

**The points of the medium such that the difference between their distances from the sources O**_{1} and
O_{2} is equal to an odd number of half-wavelengths do not oscillate.

In conclusion, when there are two sources with equal amplitudes and angular frequencies, the points of a medium
oscillate with different amplitudes, from a minimum value 0 to a maximum value *2A*.
In a three-dimensional medium the sets of contiguous points which do not oscillate are hyperboloidal nodal surfaces
and the sets of contiguous points which oscillate with greatest amplitude are hyperboloidal antinodal surfaces.

In particular, in a one-dimensional medium, these surfaces reduce to points said respectively nodes and antinodes.
In this case, the intermediate point between two sources is an antinode.

You can see some maybe useful animations in Waves.

If instead the sources O_{1} e O_{2} are in antiphase, that is they have phase difference π or,
more simply, they oscillate like the pans of a balance, the wavefunctions are, with an appropriate choice of the
origin of the time,

The situation is the same as the sources were oscillating in phase, but at a distance from each other increased
by an half-wavelength. Now the points of the medium are nodes when the difference between their distances from the
sources O_{1} and O_{2} is equal to an even number of half-wavelengths and vice versa.

In particular, the intermediate point between two sources is a node.