§ 4 The interference of waves.

Superposition principle. The concept of a coherent wave

If the environment is distributed several waves simultaneously, the oscillations of the particles among equals sum of the geometric variations that made the particles in the propagation of each wave separately. Consequently, the waves just imposed, without disturbing each other - the principle of superposition (overlay) waves.

Two waves are called coherent if the phase difference is independent of time

- The condition of coherence .

Sources of coherent waves are called coherent sources.

because for coherent sources of difference in the initial phase , the amplitude of Ares at different points depending on the value , called the path difference. If

  • then there is a maximum.


  • minimum is observed.

In imposing waves of coherent sources observed minima and maxima, the resulting amplitude, ie mutual reinforcement in some points of space and weakening in others, depending on the relation between these phases, waves - are interference phenomena.

§ 5 The standing waves

A particular case of interference are standing waves - waves that are generated when two traveling waves propagating in opposite directions with equal amplitudes of waves and frequencies.
To derive the equation of a standing wave will take:

1) the waves propagate in a medium with no attenuation,

2) А1 = А2 =А - have equal amplitude, and

3) ω1 = ω2= ω - equal frequencies, and

4) φ10 = φ20 = 0.

The equation of a traveling wave propagating along the positive direction of the x axis (ie, the equation of the incident wave)



The equation of a traveling wave propagating in the negative direction of the x axis (ie, the equation of the reflected wave):


Adding (1) and (2) get the equation of a standing wave:


Feature of the standing wave is that the amplitude depends on the x coordinate. As you move from one point to another amplitude varies in accordance with:

- The amplitude of the standing wave. Those points of the medium in which the amplitude of the standing wave maximum and equal to 2A, are called antinodes. The coordinates of the antinodes can be found from the condition that

from here

The distance between two antinodes is equal .

The points at which the amplitude of the standing wave is minimal and equal to 0 are called nodes. Coordinate nodes can be found from the condition

from here


The distance between two nodes is .

In contrast to the traveling wave, all points of which fluctuate with the same amplitude but different phases,depending on the x coordinate of the point (),the point of a standing wave between two nodes varies with different amplitudes, but with the same phase (Sst). In going through the node multiplier  

changes sign, so the phase fluctuations on both sides of the node differs by π, ie points that lie on either side of the node vibrate in antiphase

 A standing wave is the result of interference of the incident and reflected waves. Reflection on the nature of impact between two media, from which there is a reflection. If the wave is reflected from a less dense medium II (Fig. a), the phase of the wave at the interface does not change and at the boundary between two media will antinode.

If the wave is reflected from more dense medium III, then its phase is reversed, ie reflection from more dense medium occurs with the loss of half of the wavelength λ/2) (Fig. b). The traveling wave carries energy of the vibrational motion in the direction of propagation. Standing wave energy does not transfer, because incident and reflected waves of equal amplitude are the same energy in opposite directions. The total energy of the resulting standing wave of between nodes is constant. Only within a distance equal to λ/2 is the conversion of kinetic energy into potential energy.


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