8.11. Energy of Elastic Waves
Any wave transfers the energy in the form of vibration motion from one particle of the medium to another one. The elastic medium, in which mechanical waves propagate, posses the kinetic energy of the vibration motion of particles and the potential energy caused by a deformation. If vi is the velocity of the medium particles, then the volume density of the medium kinetic energy is given by
where is the medium density, dWk is the kinetic energy of all particles in the small medium volume dV chosen in such a manner that within it the velocity v1 is the same.
The volume density of the potential energy of the elastic deformed medium is given by
where dWp is the potential energy of homogeneously deformed small part of medium with the volume dV, v is the phase wave velocity in the medium, is the longitudinal strain.
The volume density of the elastic wave energy is given by:
8.12. Wave Superposition and Interference
It is common to have waves from different sources moving through a medium at the same time.
The principle of superposition:
when two or more wave motions pass the given point at the same instant the displacement is the resultant of the displacements that would be produced by each of the waves acting separately.
If we have n waves in a medium with linear properties then the resultant displacement S, the velocity v and the acceleration a for the particles of a medium at an arbitrary moment t are equal:
where Si, vi and ai are displacements, velocities and accelerations for the i-th wave. This is a statement of a very important law of nature. It is valid for all kinds of waves, provided the amplitudes are not too great. Let us apply the principle of superposition to the particular case in which two coherent waves are traveling in the same direction (or close directions) at the same time.
Two waves are called coherent if the difference of their phases does not depend on time:
Two waves with the same frequencies 1= 2= are always coherent. The waves with different frequencies are coherent only during the time of coherency of the generated vibrations.
The interference is the superposition of waves resulting in mutual stable in time amplification or weakening in certain points of the space in dependence on a correlation of phases.
Point sources S1 and S2 (Fig. 8.23) create two plane waves in the space:
y1 = A1sin1 = A1sin(t – kr1 + 01),
y2 = A2sin2 = A2sin(t – kr2 + 02).
Figure 8.23
The amplitude A and the phase of the resultant harmonic vibration in the point M:
y = y1 + y2 = Asin,
where (for 01 = 02 = 0)
A2 = A12 + A22 + 2A1A2cosk(r1 - r2),
The result of the interference depends on the value = r2 – r1, which is called the difference of traveled ways.
The interference maximums take place when cosk = 1 and
where m = 0,1,2,… is the order of maximum.
The interference minimums take place when cosk = -1 and
