INDEX FOR RESONANCES, WAVES AND FIELDS

A-B    C-D    E-L    M-Q    R-S    T-V    W-Z

A

• acoustical wave, attenuation,equation.
• acoustical wave equations,equation,written in both real and complex forms.
  

Thomas Edison never did accept AC power, largely because of the lack of mathematical methods

• AC power,oscillating nature of,animation.
• adding two oscillating quantities with phasors,discussion with equations.
• addition of phasors, bungy jumpers animation,animation.
• addition of wavefunctions,explanation of the difficulty with only using sines and cosines.
• algebra of complex numbers,discussion with equations,addition, subtraction, multiplication and division.
• amplitude,simple explanation of.
• amplitude of a wave in the cosine representation,explanation.
• angular frequency,simple explanation of.
• angular frequency of a wave,explanation.
• animation of a complex phasor showing the relationship between the phasor and the oscillating quantity it represents,animation.
• animation of a general traveling wave, with offset, amplitude, etc,animation.
• animation of a traveling cosine wave,animation,illustrates use of shifter equation to make an equation for a traveling wave.
  

The complex phasor for a wave can be considered a helical wave (one axis is imaginary)

• animation of complex helical wave, its phasor, and real part,animation,shows the helical nature of a complex phasor for a wave.
• animation of complex phasor for waves,animation,graphical demonstration of the relationship between a wave and its phasor representation.
• animation of complex phasors for standing waves on a string,animation,this animation has a number of options to show traveling wave components, phasors, and complex helical wave components.
  

The red and blue vectors represent the electric and magnetic fields

• animation of electromagnetic radiation,animation.
• animation of gaussian wave,animations, equation and application of,explanation of a gaussian wave.
  

The red areas are in compression while the blue areas are in rarefraction

• animation of longitudinal pressure waves,animation,compression waves going through balls.
• animation of Marconi's spark-gap wireless telegraph transmitter,animation.
  

Animation of rotating vector and its projection

• animation of rotating vector and its projection, the core of phasor diagrams,animation.
• animation of the complex rotor,animation,animation of exp(iwt) which is the central concept of complex phasors.
• animation of the free surface of a water wave showing a traveling wave with grid pattern, see wave animation with frog.
• animation of the multiplication of a complex phasor by a complex constant,animation.
  

Example of one dimensional waves

• animation of traveling waves on a string,animation.
• animation on the oscillating nature of AC power,animation.
  

The wave-like sine function can be understood from the idea of the sine of an angle

• animation showing derivation of the sine function from the definition of the sine of an angle,animation.
• animation showing traveling waves combining to make a standing wave,animation.
  

Transverse waves propagating on a two dimensional mesh

• animations of 2D waves on a mesh,animation.
  

One of the ways to make waves in flash

• animation, ways to make waves,posting.
• anti-nodes of a standing wave,explanation of.
• attenuation of sound waves,equation.
• author, P. Ceperley,brief bio.
• beating of sounds, audio demonstration of beating,audio.
  

Animation showing beating of two oscillating objects

• beating of two oscillating objects, animation with bungy jumpers,animation.
• Bessel functions,mention of.
• bungy jumper demonstrating a simple phasor and its relationship to an oscillating quantity,animation.
• bungy jumpers animation showing phasor addition,animation.
• bungy jumpers demonstrating beating of two oscillating objects,animation.

C

• Ceperley, the author of this blog,brief bio.
• circular motions in water waves,discussion.
• complex addition, as applied to complex phasors,discussion and math.
• complex constant multiplying a complex phasor,discussion.
• complex helical wave, its phasor, and real part,animation,shows the helical nature of a complex phasor for a wave.
• complex multiplication and division in polar notation,explanation.
• complex numbers, algebra of,discussion with equations,addition, subtraction, multiplication and division.
• complex numbers and Euler's formula,posting and introduction to.
• complex numbers, example of use in complex phasors to represent oscillations and vibrations,posting with discussion and equations.
  

Animation of a complex phasor and the oscillation it represents

• complex phasor, animation showing the relationship between the phasor and the oscillating quantity it represents,animation.
• complex phasor for waves,equation,introduction to and discussion of the complex representation of waves and complex phasor for waves.
  

Animation showing a wave and its complex phasor as a function of time and distance

• complex phasor for waves,animation,graphical demonstration of the relationship between a wave and its phasor representation.
• complex phasor multiplied by a complex constant,discussion.
• complex phasor multiplied by a complex constant, animation of,animation.
• complex phasor multiplied by a complex constant, list of applications,applications.
• complex phasors, advantages,discussion.
• complex phasors for standing waves on a string,animation,this animation has an number of options to show traveling wave components, phasors, and complex helical wave components.
  

Fun with waves

• complex phasors, limitations of,discussion.
  

Art of using complex phasors

• complex phasors, the algebra of,posting,introduction to algebraic manipulations of complex phasors.
• complex phasors to mathematically represent oscillations and vibrations,posting with discussion and equations.
• complex phasors, use in waves,posting,an introduction to the use of complex phasors for analysis of waves as opposed to analysis of oscillatory phenomena, builds on mathematics using cosines.
• complex phasors, use of in math of superposition,derivation,math of standing waves.
• complex plane, see graphical method for complex numbers.
• complex representation of a standing wave,derivation.
• complex representation of waves,equation,introduction to and discussion of the complex representation of waves and complex phasor for waves.
  

The complex rotor is the real genius behind complex phasors

• complex rotor, an animation of,animation,animation of exp(iwt) which is the central concept of complex phasors.
• compression wave,animation,compression waves going through balls.
  

Understanding the right triangle is the key to understanding trigonometric functions

• cosine,introduction to.
• cosine function,explanation of and graphs of.
• cosine function,discussion of the cosine function and its Taylor series.
• cosines, difficulty in adding two,explanation of the difficulty with only using sines and cosines.
• cusps,mention,cusps in water waves.
  

Animation using of a wheel to generate a cycloid shape

• cycloid,description,general description of a cycloid and its generation by a rolling wheel.
• cycloid animation,animation,generation of a cycloid and curtate cycloid by a rolling wheel.
• d'Alembert, Jean le Rond, mathematician,image and discussion,introduction to his "shifter" solution of the wave equation.
• d'Alembert's solutions to the wave equation,discussion.
• differential equations,example of use,as applied to water waves.
• differential equations,example of use in solving for water wave flow,posting.
• dispersion, see dispersive water waves.
• dispersion relation as applied to water waves, example,dispersion relationship, dispersion of water waves.
• dispersive,definition, example of for water waves.
• dispersive water waves,discussion.
• displacement equation,mathematical equation,derivation of the equation for displacement from equilibrium of water parcels in a water wave.
• divergence, see incompressible flow.
• dot product, as used in equation for waves,equation.
• download, how to download animations,.
• dynamics of the free surface of water,discussions and derivation,surface dynamics of water waves,surface dynamics.

E

• earthquakes, see tsunamis.
• Edison, Thomas,mentioned with regard to the rise of AC power.
• Edison, Thomas,animation on oscillating nature of AC power.
• Electromagnetic radiation, an example of transverse waves,discussion.
• electromagnetic radiation, equations for the electric and magnetic fields in,equations.
• elliptical and circular motions in water waves,discussion.
• EM radiation, see Electromagnetic radiation, an example of transverse waves.
• envelope of a wave,definition,as applied to water waves.
• Euler, Leonhard,mention in regard to Euler's formula.
  

Euler's formula is the basis of the complex number system

• Euler's formula,introduction to and rationale for.
• Euler's formula and complex numbers,posting and introduction to.
• Euler's formula and polar notation,explanation.
• exp function,mention of and graph of with regard to Euler's formula.
• exp function,discussion of the exp function and its Taylor series.
• flash programming, ways to make waves,posting and animations.
  

Audio clips of beating of two sound sources

• flute players, audio demonstration of beating,audio.
  

Waves live very near the surface

• free surface,definition,free surface of water.
• free surface of a water water, dynamics of,discussion and derivations.
• frequency, angular,simple explanation of.
  

d'Alembert's shifting method applied to a gaussian function produces a gaussian (or solitary) wave

• gaussian function,animations,equation and application of,explanation of a gaussian wave.
• graphical addition and subtraction of complex numbers,discussion.
• graphical method for complex numbers,explanation of.
• graphical multiplication and division of complex numbers,explanation.
• graphical plotting of exp(i phi),explanation.
• gravity waves,discussion.
• height, mean or average height of a wave,explanation.
• helical waves on a string, real physical helical waves,discussion.
• i, imaginary number base,explanation with regard to Euler's formula.
• imaginary numbers, rationale for creating,explanation with regard to Euler's formula.
• incompressible flow,mathematical definition,incompressible flow with divergence of the velocity vector field.
• infinite series,posting, discussion with equations and figures.
• infinite series, of sine and cosine functions,mention in regard to Euler's formula.
• interference,use of concept,interference of two traveling waves to make a standing wave.
• internal cloud waves,illustration and discussion.
  

Internal waves can be generated at the the interface between water of different temperature or salinity

• internal waves in water,animation.
• internal waves in water,discussion with links.
• irrotational flow,definition,irrotational flow in water waves.
• Laplace's Equation,mathematical definition,example of use for irrotational, incompressible flow in water waves.
• light,animation of electromagnetic radiation,animation.
• logarithm, natural,mention of and graph of with regard to the exp function and Euler's formula.
• longitudinal pressure waves,animation,compression waves going through balls.
• longitudinal waves,discussion.
• LRC circuit, example of in Marconi's wireless spark-gap transmitter,posting.

M

• Marconi, spark-gap transmitter,discussion.
• Marconi, spark gap transmitter,mention of.
• Marconi's spark-gap wireless telegraph transmitter,animation.
• maxima of a standing wave pattern, see anti-nodes of a standing wave.
  

Standing waves can be made by reflection

• mermaid animation,animation,demonstrating the relationship between traveling and standing waves on a string.
• motion of water parcels in water waves,animation,shows the wave motion of water for various depths of water.
• multiplication of a complex phasor by a complex constant,discussion.
  

Animation of the multiplication of a complex phasor by a complex constant

• multiplication of a complex phasor by a complex constant, animation of,animation.
• multiplication of a complex phasor by a complex constant, list of applications,applications.
• muscle man presents a simple phasor, the core of phasor diagrams,animation.
• natural phasor for water waves,animation,shows how the motion of water parcels can be used as a natural phasor for water waves.
• natural phasor for water waves,discussion.
• Newton's second law,example of usage,Newton's second law applied to water waves.
• nodes of a standing wave,explanation of.
• oscillating nature of AC power,animation.
• oscillations, difficulty in adding two,explanation of the difficulty with only using sines and cosines.
• oscilloscope,mention of.
• permitted use of material on this blog,.
• phase, detailed discussion including phase offset, phase difference and phase shift,discussion.
• phase offset,explanation.
• phase shift,simple explanation of.
• phase velocity as applied to water waves, math derivation,derivation of equation of phase velocity of water waves.
• phasor addition,discussion with equations.
• phasor addition, beating of two oscillating objects, animation with bungy jumpers,animation.
  

Animation showing phasor addition

• phasor addition with bungy jumpers in an animation,animation.
• phasor, complex phasor representation of a standing wave,posting and discussion of,extending geometric phasors of standing waves to using complex phasors.
• phasor diagrams, why do they work?,discussion.
• phasor for water waves,animation,shows phasors for water waves, compression waves, and waves on a string.
• phasor for water waves,description.
• phasors, an introduction to geometric or simple phasors,discussion with equations and animations.
• phasors, complex representation of standing waves on a string,animation,this animation has an number of options to show traveling wave components, phasors, and complex helical wave components.
  

Animiation of a simple phasor and the oscillating object it represents

• phasor, simple and its relationship to the graph of an oscillating object,animation.
• phasor, simple, rotating vector and its projection, the core of phasor diagrams,animation.
• phasors in standing waves,discussion.
• phi, the greek letter,discussion of styles of.
• polar notation and Euler's formula,explanation.
• polar notation in complex multiplication and division,explanation.
• product solution,mathematical definition,example of use for water waves.
• programming in flash, ways to make waves,posting and animations.

R

• radial waves in 2D, equation for,equation,approximate equation.
• radians,explanation of.
• real part of, complex operation,explanation.
• real representation of a standing wave,derivation.
• reflected wave,derivation.
• reflection coefficient,discussion using.
• reflection coefficient, complex,mathematical definition.
• reflection, 100%, 180 degrees phase shift,discussion.
• reflection, 100%, no phase shift,discussion.
• reflection, variable,discussion.
• resonator, LRC circuit, example of in Marconi's wireless spark-gap transmitter,posting.
• rotating vector and its projection, the core of phasor diagrams,animation.
• separation constant,mathematical definition,example of use for water waves solution.
• separation of variables,definition,example of use for water waves.
• shifter solution to the wave equation, see d'Alembert, Jean le Rond, mathematician.
• sine,introduction to.
  

Term by term graphing of Taylor series for the sine function

• sine function,discussion of the sine function and its Taylor series.
• sine function, derivation of,animation.
• small amplitude approximation for water waves,discussion.
• smith chart, an application of reflection concepts,discussion and illustration.
• solitary wave, gaussian form of,discussion,discussion of gaussian wave function.
• solutions to differential equations,example of use,for solution of small amplitude water waves.
• solution to the wave equation, d'Alembert's solution,discussion.
• sound, three dimensional wave equation,equation.
• sound wave, attenuation,equation.
• sound wave equations,equation,written in both real and complex forms.
  

Marconi's spark-gap transmitter revolutionize communication

• spark-gap transmitter,animation,animation.
• spark-gap transmitter, details of operation,discussion.
• spark-gap transmitter, Marconi,discussion.
• speed of water waves,equation.
• speed of water waves, deep water,equation.
• speed of water waves, shallow water,equation.
• standing wave, complex phasor representation of a standing wave,posting and discussion of,extending geometric phasors of standing waves to using complex phasors.
• standing wave envelope,derivations,two derivations of the envelope equation using complex math.
• standing wave formation from two traveling waves,shows waves on a string, compression waves, and their phasors.
• standing wave, mathematical derivation with complex phasors,derivation.
• standing wave, mathematical derivation with cosines,derivation.
• standing wave, nodes and anti-nodes,explanation of.
• standing wave, position of minima,mathematical derivation.
• standing wave, position of the maxima,mathematical derivation.
• standing wave ratio,definition.
• standing wave ratio,mathematical definition and derivation.
• standing wave ratio, common use of,discussion.
• standing waves and superposition,posting on,discussion of the superposition process and the math involved, using both real and then complex methods.
  

Animation of the complex phasors for standing waves on a string

• standing waves on a string, complex representation,animation,this animation has an number of options to show traveling wave components, phasors, and complex helical wave components.
• Steinmetz, Charles,mentioned with regard to the rise of AC power.
• streamlines in a water wave,animation.
• superposition and standing waves,posting on,discussion of the superposition process and the math involved, using both real and then complex methods.
• superposition, mathematical derivation with complex phasors,derivation,math of standing waves.
• superposition, mathematical derivation with cosines,derivation,math of standing waves.
• superposition of traveling waves on a string to form standing waves,mathematical derivation and explanation.
• superposition of traveling waves on a string to form standing waves, also illustrates the complex phasors for these,animation,this animation has an number of options to show traveling wave components, phasors, and complex helical wave components.
  

Animation showing the superposition process

• superposition of traveling waves to make a standing wave,animation,shows waves on a string, compression waves, and their phasors,both the two traveling wave components and the combined standing wave.
• surface tension, role in water waves,discussion.

T

• technology of oscillations and waves demand new mathematical methods,posting,discussion of the need for new mathematical methods for the development of AC power and electronic communication.
• total derivative of water velocity, Dv/Dt,math definition,short definition.
• total reflection,derivation,the math of superimposing the incident and reflected waves to form a standing wave.
• total wave field,derivation,math of superimposing the incident and reflected wave fields, done in complex math.
• Transverse waves, in general,discussion.
• traveling wave, animation of using cosines,animation,illustrates use of shifter equation to make an equation for a traveling wave.
  

Various constants can be used to modify the basic equation for a traveling wave

• traveling wave, general with offset, amplitude, etc,animation and detailed explanation of constants.
• traveling wave, introduction to using cosines,introduction and equations,explains the step by step evolution of a cosine function into the equation for a traveling wave.
• traveling waves,posting,introduction and discussion at a simple level of the mathematics of "true", i.e. traveling, waves.
• traveling waves on a string,animation.
• trigonometric functions, use of in superposition of traveling waves,derivation,math of standing waves.
• tsunamis,mention,one of the causes of water waves.
• tsunamis speed,discussion.
• types of waves,posting discussing various types of waves with animations.
• use, permitted use of material on this blog,.
• velocity field for water waves,mathematical solution,equation for small amplitude waves.
  

Water waves with velocity potential and streamlines

• velocity potential,definition,velocity potential use in water waves,example of gradient usage.
• velocity potential in water waves,animation.
• velocity vectors in a water wave,animation.

W

• water parcels in water waves,animation,irrotational flow,incompressible flow.
  

Water waves are both longitudinal and transverse

• water wave animation,animation,animation comparing compression and transvers waves with water waves.
• water wave dynamics of free surface, see free surface of a water water, dynamics of.
  

Varying the reflection coefficients changes the wave pattern

• water wave reflection,animation,demonstrates the process of creating standing waves by reflection, also the reflection coefficient and the standing wave ratio.
• water waves,general description.
• water waves, derivation of irrotational and incompressible flow equations,posting.
• water wave, velocity potential in,animation.
• wave animation with frog,animation,traveling wave animation emphasizing surface film,grid pattern shows motion of water parcels.
• wave as a complex helical phasor,animation,shows the helical nature of a complex phasor for a wave.
• wave, complex phasor for,equation,introduction to and discussion of the complex representation of waves and complex phasor for waves.
• wave, dynamics of the free surface of water waves, see free surface of a water water, dynamics of.
• wave equation in two dimensions,equation.
• wave equation, one dimensional,equation.
• wave equation, three dimensional, acoustical,equation.
• wavefunction,simple explanation of.
• wave, general traveling wave with offset, amplitude, etc,animation and detailed explanation of constants.
• wave, longitudinal,discussion.
• wavenumber,explanation.
• wavenumber, vector-nature,equation and very brief discussion.
  

Wave off Kanagawa, Hokusai, Wikipedia

• Wave off Kanagawa by Japanese artist Hokusai,image and mention of,a very famous image of a wave.
• waves, animation of complex phasor for,animation,graphical demonstration of the relationship between a wave and its phasor representation.
• waves, 2D,discussion.
• waves in 1D (one dimension),discussion.
• wave, standing, complex phasor representation of a standing wave,posting and discussion of,extending geometric phasors of standing waves to using complex phasors.
• wave, standing, nodes and anti-nodes, see standing wave, nodes and anti-nodes.
• waves, three dimensional,discussion and equations.
• waves, true (traveling waves),posting,introduction and discussion at a simple level of the mathematics of "true", i.e. traveling, waves.
• waves, types of waves,posting discussing various types of waves with animations.
• waves using complex phasors,posting,an introduction to the use of complex phasors in waves as opposed to oscillatory phenomena, builds on mathematics using cosines.
• waves, ways to make in flash animations,posting and animations.
  

Animation of a basic traveling wave equation built on the shifter equation

• wave, traveling, animation of using cosines,animation,illustrates use of shifter equation to make an equation for a traveling wave.
• wave, traveling, introduction to using cosines,introduction and equations,explains the step by step evolution of a cosine function into the equation for a traveling wave.
• Westinghouse, George,mentioned with regard to the rise of AC power.
• why do phasor diagrams work?,discussion.
• wind, primary cause of water waves,discussion.
• wireless, Marconi's spark-gap transmitter,animation.
• wireless, spark-gap transmitter, Marconi,discussion.
• zeroes of a standing wave pattern, see nodes of a standing wave.