This page is under construction, remarks are welcome.
2. Geometrical theory of waves. Fermat principle. Rays, wave fronts. Singularities. Caustics. Huyghens principle.
3. Underlying theories. Wave equation. Approach of geometrical optics.
"The general term applied to the description of a disturbance which
propagates from one point in a medum to other points without giving
the medum as a whole any permanent displacement"
(McGraw Hill Encyclopedia of Science and Technology, 1997)
"Waves are the solution of the wave equation" (What
are Signals
and what are
Waves?) http://www22.pair.com/csdc/ed3/ed3fre24.htm)
"Waves: travel in a material medium, such as water waves, or waves on
a string. Waves...have as their source a vibration. ...the medium
through which a wave travels itself vibrates." (Physics for Scientists
and Engineers, Douglass C. Giancoli)
"The rudimentary notion of a wave is an oscillatory disturbance that moves away from some source and transports no discernible amount of matter over large distances of propagation." (Allan D. Pierce: The Wave Theory of Sound, Excerpts from Chapter 1 of Acoustics: An Introduction to Its Physical Principles and Applications)
"The definition of a wave is a means of transferring energy from one place to another without bulk motion. (" Waves Group: What we do, Department of Applied Mathematics and Theoretical Physics, Cambridge)
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Primarily a wave is a disturbance in a medium. This medium can
be
any arbitrary material, even a vacuum for electromagnetic waves.
Water is a fine medium for the conduction of water waves. Air
works well for sound. The crust of the earth, which is comprised
of a host
of minerals, conducts earthquake waves. So the media in which
waves may propagate are varied and diverse.
Within a given medium the source of a mechanical wave is a
vibration or pulse which disturbs the particles within that medium.
This disturbance then is carried outward from its source as adjacent
particles within the medium interact. In the case of periodic
waves, a temporary disturbance of the medium causes its particles to oscillate
about their equilibrium positions. However, in general, the
motion of particles has nothing to do with the motion of wave fronts (for
transverse elastic waves, the motion of particles disturbed by wave
motion is orthogonal to the direction of wave propagation).
There are several important properties of waves. In a two dimentional
transverse wave the highest point a particle reaches in a
local area of highs and lows is known as the “crest.” The lowest
point is the “trough.”
More generally, wave propagation is always associated with
spatio-temporal variation of one or more quantities described by
wave functions (Y).
The maximum value of the wave function is the amplitude (A). For
periodic waves the wave function is periodic with respect to time and
with respect to space, the time period is denoted by T, while the
spatial period is l (wavelength). The inverse
of T is the frequency n,
while the invese of l is the wave number
k*.
For harmonic planar waves the wave function is sinusoidal:
Y=Acos(wt-kx+jo)
j = wt-kx+jo
phase
jo phase constant,
initial phase
w = 2p/T circular frequency
k= 2p/l circular wave number.
The harmonic wave has no beginning or end. In practice, any wave
has a source, propagates, and decays. For most of the waves
studied in physics wave propagation is attached to energy transfer.
Wave: A traveling wave is a spatiotemporal function of the
form:
Y(r, t) = A(r) * S(t-g(r)) (1.1)
A: amplitude
S: phase
g: eikonal
Wave front: for a given t the surfaces (S = constant) are called
wave fronts.
From eq(1.1) it follows that the wave fronts (equiphase surfaces) are
identical to the equieikonal surfaces (g=constant).
Propagation of the waves is a continuous transformation of the
stationary family of equieikonal surfaces into itself.
A specified wave front which belongs to g1 at t1
will be transformed
at t2 into the surface g2 = g1 + t2
– t1.
If movement is parallel to the gradient of g then:
1/v = |grad g| (1.2)
This equation is called the eikonal equation.
(Derivation: S= const, consequently t-g(r)=const,
from which (grad
g).v=1, v is the phase velocity).
The wave is propagating perpendicular to the wave front with the
velocity v.
Rays: orthoginal trajectories of wave fronts. The wave
propagates
along the rays with velocity v.
We remark that the wave equation has solutions which differ from
the form (1.1). E.g. the linear functions are solutions. On the other
hand, not only the wave equation has the solutions of the form (1.1),
but several other partial differential equations have.
The most important special case for traveling waves is the harmonic
wave, when S = cos(wt-f(r)). In case
of planar waves, f=k.r.
Wave fronts were defined as the level surfaces of the phase S. If
these surfaces differ from the level surfaces of the amplitude,
the
wave is said to be inhomogeneous (Max Born & Emil Wolf:
Principles of Optics - Electromagnetic Theory of Propagation
Interference and Diffraction of Light)
In addition to traveling waves eq. (1.1) also includes standing waves
(g(r) = const):
Y(r,t) = A(r)S(t)
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Greek letters:
q w e r t y u i o p a s d f g h j
k l J w V z x c v b n m
Q W Y P Õ S D F G J L
X S D F G L
É - _ É áú Í ' "
Ö Ü Ó ÛÚÉ