home | index | units | counting | geometry | algebra | trigonometry & functions | calculus analysis | sets & logic | number theory | recreational | misc | nomenclature & history | physics

# Waves

If you don't make waves, nobody
will ever know that you are alive.
Theodore Isaac Rubin  (b. 1923)

### Related Links (Outside this Site)

Huygens' Principle  by  Kevin Brown  (mathpages.com).
Thomas Young's Experiment  by Walter Scheider.
Longitudinal and Transverse Wave Motion:  Animations by  Dr. Dan Russell.

Wikipedia:   Fresnel zone plates

Video:  MIT OpenCourseWare   Vibrations & Waves  (8.03)  by  Walter Lewin.

## Waves

(2005-05-21)   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.

The principle was given a more precise form be  Augustin Fresnel  in 1816 to apply it to the computation of  diffracrion patterns.

Huygens-Fresnel principle   |   Christiaan Huygens  (1629-1695)

(2007-09-09)   Diffraction   (Grimaldi, c. 1650)
What occurs when a wave originates from a bounded source.

In 1816, Augustin Fresnel (1788-1827; X1804) had the idea to combine this with  Huygens' principle  to compute the diffraction patterns formed by light when it encounters obstacles...

(2005-05-21)   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 (1773-1829)   Nov. 24, 1803

(2005-09-29)   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.

(2007-09-09)   Standing Waves
Nodes and Antinodes.

Standing wave   |   Node

(2008-01-12)   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 so-called  index of refraction  n.

 n1 sin q1   =   n2 sin q2 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).

### Small angle approximation:  Lens-maker's formulas.

1/f   =   (n-1) ( 1/R1 + 1/R2 )

Fermat's Principle of Least Time  (c. 1655)

(2015-09-14)   Total Internal Reflection  (TIR)
At incidences exceeding the  critical angle, 100% of the light is reflected.

Fermat's Principle of Least Time  (c. 1655)

(2008-01-14)   Birefringence betrays polarization   (Bartholinus, 1669)
Birefringence of  Iceland spar  (optical calcite, CaCO3 ).

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).

(2008-01-12)   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  qB  was discovered by Etienne-Louis Malus (1775-1812) in 1808.

In 1815,  Sir David Brewster (1781-1868)  found that angle to be a simple function of the ratio of the two refractive indices involved:

• n1 = n i   pertains to the medium where the incident beam travels.
• n2 = n t   pertains to the medium of the transmitted (refracted) beam.

 q B =   arctan  ( n2 / n1 )

This expression can be derived from Fresnel equations by imposing  Rp = 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.

(2008-01-12)   Fresnel Equations   (1821)
Intensity of polarized light reflected or transmitted by a  planar diopter.

Unlike Snell's Law, the  Fresnel Equations  apply specifically to light and involve the different  polarizations of light  which Augustin Fresnel (1788-1827) firmly established himself in 1821.

Fresnel determined that light consists entirely of transversal vibrations  without any longitudinal component whatsoever.  This went against the opinion of Thomas Young (1773-1829) 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).

 Incident beam polarizedin the plane of incidence Incident beam withorthogonal polarization Coeff.  ofReflection Coeff.  ofTransmission Fresnel'sEquations rp = tan ( q1 - q2 ) rs = - sin ( q1 - q2 ) tan ( q1 + q2 ) sin ( q1 + q2 ) tp = 2 cos q1 sin q2 ts = 2 cos q1 sin q2 cos ( q1 - q2 )  sin ( q1 + q2 ) sin ( q1 + q2 )

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:

Rs  =  | rs |2     Ts  =  | ts |2               Rp  =  | rp |2     Tp  =  | tp |2

### Derivation from Maxwell's Equations :

The Discovery of Polarization  by  J. Alcoz.
Fresnel Equations  (8.03 Vibrations and Waves)  by  Walter Lewin, MIT.
Fresnel Relations  (531 Optics)  by  Cass Sackett, University of Virginia.
"Fresnel Equations" by Bob Eagle  (DrPhysicsA) :   Boundary Conditions  |  Deriving the Equations

Wikipedia   |   Weisstein   |   HyperPhysics (calculator)  by  Rod Nave

(2009-03-12)   Stokes Parameters   (1852)
A standard description of the  state of polarization.

Wikipedia

(2010-12-27)   Transverse Wave on a Rope
(celerity) 2   =   (tension) / (linear mass density)

On a rope of linear density  m  stretched along the x-axis with constant tension  F,  let's consider the behavior of  small  transverse perturbations of amplitude  h(x,t)  in the direction of the  y-axis.

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 x-axis 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.569-475 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.

Mersenne's laws   |   Harmonie universelle (1636)  by  Marin Mersenne  (1588-1648)

Video :   Deriving the wave equation for a rope  by  Walter Lewin   (MIT, Fall 2004)

(2018-02-01)   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).

Vibrations of a circular membrane

Standing oscillations on a surface feature lines of nodes.

Ernst Chladni (1756-1827) 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  clad-knee.)

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 (1776-1831)  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 (1781-1840; 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.

Chladni Patterns for Violin Plates  by  Joe Wolfe  (UNSW, Sydney, Australia)

Videos :   Chladni Patterns on a Square Plate   |   Holding a Chladni Plate  (and ruining a bow)
Fantastic DIY Speakers for less than \$30  in New England  (Tech Ingredients, 2018-01-29).

(2017-06-23)   Wave Inertia   (G.H. Ryan, 1890)
Hemispherical-Resonator Gyroscope  (1996).

How to point a Space Telescope (7:53)  by  Meghan Gray  (Sixty Symbols by Brady Haran, 2017-06-21) The Hemispherical Resonator Gyro  by  David M. Rozelle   (Northrop Grumman Co, 2011).

Wikipedia :   Hemispherical resonator gyroscope (HRG)   |   George Hartley Bryan (1864-1928; FRS 1895)