(20050521) Wave Propagation and Huygens' Principle (1678)
A helpful fiction to describe wave propagation.
Originally, the principle was formulated strictly from a geometrical standoint
by noting that a wavefront can be seen as the
envelope of circular
(or spherical) wavelets centered on a previous wavefront.
Christiaan Huygens himself noticed that this principle was sufficient
to derive the optical laws of reflexion and refraction in the wave theory of light
which he was pioneering.
(20070909) Diffraction
(Grimaldi, c. 1650)
What occurs when a wave originates from a bounded source.
In 1816, Augustin Fresnel (17881827; X1804)
had the idea to combine this with Huygens' principle
to compute the diffraction patterns formed by light when it encounters obstacles...
(20050521) Thomas Young's "Double Slit" Experiment
Demonstrating the undulatory nature of light.
[ This ] may be repeated with great ease, whenever the sun shines,
and without any other apparatus than is at hand to everyone. Thomas Young
(17731829) ^{ } Nov. 24, 1803
(20050929) Celerity
The celerity of a wave is the product of its frequency by its wavelength.
The celerity u of a wave is always equal to the product
l n of its wavelength by its frequency.
When this celerity is constant, the medium is said to be
nondispersive. In a nondispersive medium, a planar wave
would retain its shape as it propagates.
This is not generally true in a dispersive medium.
(20080112) Snell's Law (Harriot, July 1601. Snell, 1621.)
Direction of the refracted wave.
By definition, a diopter is the surface (usually a plane or
a sphere) which separates two regions where specific waves (light, sound, etc.)
travel at different celerities (celerity = phase velocity).
Incidentally, the name "diopter" also denotes a
unit of curvature equal to the reciprocal of
a meter (m) which is used to rate an optical element by specifying the
reciprocal of its focal length.
Snell's Law applies not only to waves but also to other objects
at a boundary between two domains where the travelling speeds are proportional
to different values of a socalled index of refraction n.
n_{1} sin q_{1}
=
n_{2} sin q_{2}
This law of refraction was discovered by Thomas
Harriot in July 1601 and independently by
Willebrord Snell
(1621) and
René Descartes
(1637) who was the first to publish it. The Dutchman
Christiaan Huygens
was instrumental in attributing the
Law to Snell (1678).
Some transparent minerals like Iceland spar exhibit a strange optical
property known as birefringence.
Surprisingly enough, this was the phenomenon which prompted
Huygens to fomulate the
principle bearing his name (1678).
(Fresnel would only apply the principle to diffraction
much later, in 1816).
(20080112) Brewster's Angle (Malus 1808, Brewster 1815)
One angle of incidence makes the reflected beam 100% polarized.
The full polarization of light by optical reflection at a particular angle of incidence
q_{B}
was discovered by EtienneLouis
Malus (17751812) in 1808.
In 1815, Sir David Brewster
(17811868) found that angle to be a simple function of the ratio
of the two refractive indices involved:
n_{1} = n_{ i} pertains to the medium where the incident beam travels.
n_{2} = n_{ t} pertains to the medium of the transmitted (refracted) beam.
q_{ B} =
arctan ( n_{2} / n_{1} )
This expression can be derived from Fresnel equations by
imposing R_{p} = 0.
Snell's Law makes
this equivalent to the observation that Brewster's angle of incidence is
such that the reflected and the refracted beam are perpendicular.
Apparently, Malus did not come up with this relation experimentally because
he focused on just the two cases of water and glass. It turns out that the
type of glass available at that time could have surface properties
unrelated to the index of refraction of its bulk.
Brewster was faced with the same difficulty but he could establish the above
general law by considering a variety of transparent minerals and (ultimately) disgarding
the peculiarities of glass.
(20080112) Fresnel Equations (1821)
Intensity of polarized light reflected or transmitted by a planar diopter.
Fresnel determined that light consists entirely of transversal
vibrations without any longitudinal component whatsoever.
This went against the opinion of Thomas Young
(17731829) who held that light was mostly a
longitudinal phenomenon with only small transversal components.
The Fresnel equations
describe how the strength of an incident
light beam is split between a reflected (r)
and a refracted (t) beam.
Coefficients of Reflection and Transmission :
These coefficients are the ratios of the amplitudes of the emergent
beams (either reflected or refracted) to the amplitude of the
incident one.
The coefficients depend on the polarization of the incident beam.
Traditionally, the linear polarization of an electromagnetic wave
(light) where the electric field is parallel to the plane of incidence
is denoted by the subscript "p" whereas the perpendicular polarization
is denoted by the subscript "s" (the word
senkrecht means "perpendicular" in German).
Fresnel's Equations
Incident beam polarized in the plane of incidence
Incident beam with orthogonal polarization
Coeff. of Reflection
r_{p}
=
tan ( q_{1 } q_{2 })
r_{s}
=

sin ( q_{1 } q_{2 })
tan ( q_{1 }+ q_{2 })
sin ( q_{1 }+ q_{2 })
Coeff. of Transmission
t_{p}
=
2 cos q_{1} sin q_{2}
t_{s}
=
2 cos q_{1} sin q_{2}
cos ( q_{1 } q_{2 })
sin ( q_{1 }+ q_{2 })
sin ( q_{1 }+ q_{2 })
A coherent incident beam whose polarization is neither "p" nor "s"
can be viewed as a superposition of two such beams.
The above formulas give the amplitudes of the emergent beams corresponding
to both components.
Each emergent beam (transmitted or reflected)
has two components of orthogonal polarization (p or s)
corresponding to a superposition obtained by adding those emerging amplitudes.
Reflectance and Transmitance :
These are the ratios of the intensities of the emergent
beams to the intensity of the incident one.
They are respectively equal to the squares of the above coefficients
of reflection and transmission, for both polarizations:
(20101227) Transverse Wave on a Rope
(celerity)^{ 2} = (tension) / (linear mass density)
On a rope of linear density
m stretched along the xaxis
with constant tension F,
let's consider the behavior of
small transverse perturbations
of amplitude h(x,t) in the direction of the yaxis.
If the rope has no rigidity, the forces exerted on each other by adjacent
pieces of the rope are strictly tangential to it.
Thus, they have the same slope j with respect
to the xaxis as the rope itself.
The usual "small angle" approximation holds:
j =
sin j = tg j
= ¶h / ¶x
Mersenne's Three Laws (1636) :
The resonnant frequencies of a string stretched between two fixed points are:
Inversely proportional to the length of the string (law of Pythagoras).
Proportional to the square root of the stretching tension.
Inversely proportional to the square root of the mass per unit length.
The first law is attributed to
Phythagoras (c.569475 BC)
and was the earliest result in
acoustics.
The other laws are direct consequences of the above expression for
the celerity of a transverse wave in a stretched string.
(20180201) Transverse Waves on a Membrane
Standing waves and resonnant frequencies of a circular diaphragm.
Tranverse waves in an horizontal membrane under tension correspond to
small vertical displacements. There's only one tranverse polarization
(there are two in the case of a rope).
(20080413) Chladni Plates & Chladni Patterns
Standing oscillations on a surface feature lines of nodes.
Ernst Chladni (17561827)
was a German physicist whose family hailed from the medieval mining town of
Kremnica
(Kingdom of Hungary, now in central Slovakia).
Chladni has been called the father of acoustics.
He obtained the speed of sound for several
gases and experimented
with vibrating plates peppered with sand to visualize node lines
(the sand accumulates wherever the motion of the plate is minimal).
Similar experiments now go by the name of Chladni plate
experiments and the intriguing patterns so obtained are dubbed
Chladni patterns.
Ernst Chladni was also an avid meteorite collector and he successfully argued
in favor of the celestial origin of meteorites.
(In English and in French, at least, his name is usually
pronounced like cladknee.)
In 1808, Chladni visited Paris and caused quite a stir with the demonstration
of his patterns: The Institut de France set up
a prize competition (including a 1 kg solid gold medal )
with the following challenge, to be met within two years (deadline in 1811).
Formulate a mathematical theory of elastic surfaces and indicate just how it agrees with empirical evidence.
Lagrange himself went on record to state
that all available mathematical methods were inadequate to solve that problem.
In 1811, the only entrant was Sophie Germain (17761831)
who could not justify her hypothesis from physical principles because, at the time, she lacked the
proper knowledge of the calculus of variations
(a brainchild of Lagrange's, by the way).
She did not get the prize. Instead, Lagrange (who was one of the judges)
suggested a new approach and the contest was extended for another two years.
In 1813, Lagrange died.
Germain, still the only entrant, showed that the approach of Lagrange accurately described
Chladni's patterns in several special cases. She didn't provide a physical justification for it.
She just received an honorable mention and the contest was extended, again, for another two years.
In 1815, the third attempt of Sophie Germain was deemed worthy of the prize (in spite
of a few deficiencies in the rigor of her mathematics).
However, she didn't show up at the award ceremony...
It's been suggested that she was protesting the lack of appreciation of her work by
some of the judges, including her younger rival on the subject of
elasticity (that she had arguably founded
in the process)
Siméon Poisson
(17811840; X1798) who ignored her in public...
In 1825, Sophie Germain sent an extension of her research to a commission
whose members included Poisson, Laplace
and de Prony.
The paper was ignored until it was retrieved from de Prony's
personal archives (long after the death of everyone involved)
and belatedly published, in 1880.