Figure 8.16
At the point with the coordinate r vibrations do not appear immediately but with the time lag = r/v.
Hence,
The last formula is the equation of the running wave, which determines the oscillating value at any moment of time and at any point of the space. If to define the wave number by the relation k = /v then in general case:
where 0 is the initial phase.
Characteristics of a harmonic wave
1) A is the wave amplitude; (A(r) const).
2) = t – kr + 0 is the wave phase.
3) The wave periods: a wave has two periods – the time period and the space period.
T
he
time period T = 2/
is the time necessary for one complete vibration at the certain
point of the space (Fig. 8.17).
Figure 8.17
T
he
space period – the
wavelength
is the distance in the propagation direction between two successive
points that are in the same phase of vibration (Fig. 8.18).
Figure 8.18
hence = v·T.
The wavelength equals to the distance that the wave pass during one time period.
The wave surface is the geometric place of points in the wave for which oscillations take place with the same phase. From the point S the wave propagates in all directions with the same velocity. In isotropic medium the wave surface is a sphere (Fig. 8.19).
The wave front is the wave surface for which oscillations of all points take place with the phase =0 (Fig, 8.19).
Figure 8.19
Depending on the shape of the wave front (or the wave surface) waves are differentiated between plane, spherical, elliptic waves.
The wave front propagates in the space with the phase velocity.
The phase velocity is the velocity of the displacement in the space of the wave surface points with fixed harmonic wave phase:
=
const = t – kr + 0
,
The last equation is valid only for mediums with linear properties. In non-linear mediums the phase velocity can be not equal to v.
8.9. Differential Wave Equation
The propagation of waves in homogeneous isotropic medium that does not absorb the wave energy is described by the differential equation in partial derivatives, namely the differential wave equation:
where S is physical value, which describes the perturbation,
is operator that is called Laplacian.
It is easy to demonstrate that the plane wave y=Asin(t-kr+0) satisfies (is the solution of) the following differential wave equation:
It is also possible to show that the spherical wave is the solution of the wave equation.
8.10. Mechanical Waves in Elastic Medium.
Longitudinal and Transverse Waves
When studying objective laws of mechanical wave propagation in gases, liquids and solid bodies they are considered as continuous medium, namely without considering the molecular structure of these bodies. The elementary volume dV with sizes much more than intermolecular distance is considered as the particle of the medium performing enforced vibrations.
D
emonstrations:
The elastic wave is the elastic deformation that is propagated in a medium. The velocity of the elastic wave depends on the medium density and elastic modulus of the medium.
It is convenient to classify waves in terms of how the motion of the individual particles of the medium is related to the movement of the wave itself.
If a series of equal masses are joined together by spring (Fig. 8.20) and
Figure 8.20
the particle A is moved toward the particle B, a wave is propagated down the system. The motion of A toward B compresses the spring between them. B is accelerated toward C, which in turn moves toward D. A compressional wave is produced. Similarly, when mass A is pulled away from B, the spring between them is stretched and B is accelerated toward A. If A is moved back and forward with simple harmonic motion, a wave motion is propagated along the system with each particle describing simple harmonic motion along the axis of the system. The various masses do not all vibrate at phase.
A wave motion in which the individual particles vibrate back and forth along the direction in which the wave propagates is called a compressional or a longitudinal wave (Fig. 8.21).
Figure 8.21 Figure 8.22
The velocity of a longitudinal wave is given by the formula:
where E is Young’s modulus of stretching, is a density of a medium.
A transfer wave is the one in which the vibrations take place perpendicular to the direction in which the wave propagates (Fig. 8.22). Its velocity is given by formula
where G is Young’s modulus of shift.
Usually G E and hence v v. This fact is the basis for the seismography.