In-Phase and Out-Phase

[Antonio Lao]

Given a traveling wave W=Asin(\omega t + \phi), where A is the amplitude, \omega is the angular frequnecy, t is the time variable, and \phi is the phase angle.

For two waves of the same properties and traveling in the same direction, the waves vanish if the phase angle is 180 degrees. The amplitudes are doubled if the phase angle is zero or 360 degrees.

For two waves of the same properties and traveling in opposite directions, the waves formed standing waves if the phase angle is 180 degrees. What happens when the phase angle is zero or 360 degrees?

[Russell E. Rierson]

http://www.gmi.edu/~drussell/Demos/superposition/superposition.html


A travelling wave moves from one place to another, whereas a standing wave appears to stand still, vibrating in place. Two waves (with the same amplitude, frequency, and wavelength) are travelling in opposite directions on a string. Using the principle of superposition, the resulting string displacement may be written as:

y(x,t) = y_m sin(kx - wt) + y_m sin(kx + wt)

= 2y_m sin(kx) cos(wt)



This wave is no longer a travelling wave because the position and time dependence have been separated. The displacement of the string as a function of position has an amplitude of 2y_m sin(kx). This amplitude does not travel along the string, but stands still and oscillates up and down according to cos(wt). Characteristic of standing waves are locations with maximum displacement (antinodes) and locations with zero displacement (nodes).

[Antonio Lao]

Russell,

Thanks. But I still can't see where the phase angle fit into the overall picture of the wave whether travelling or standing.

[Russell E. Rierson]

Wave Tutorials:


http://www.physicsclassroom.com/Class/waves/wavestoc.html

http://www.physicsclassroom.com/Class/waves/U10L4a.html

http://www.learningincontext.com/Chapt08.htm




Standing Waves:

http://www.oreilly.cx/phi/combining_waves/standing_waves.html

http://www.glafreniere.com/sa_spherical.htm

http://www.upscale.utoronto.ca/IYearLab/Intros/StandingWaves/StandingWaves.html

http://id.mind.net/~zona/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.html

http://www.upscale.utoronto.ca/IYearLab/Intros/StandingWaves/StandingWaves.html



Resonance:

http://hyperphysics.phy-astr.gsu.edu/hbase/sound/reson.html#resdef

http://www.colorado.edu/physics/2000/microwaves/standing_wave2.html

http://www.pha.jhu.edu/~broholm/l29/node4.html


Damped Harmonic Oscillator:

http://hyperphysics.phy-astr.gsu.edu/hbase/oscda.html#c1

[Russell E. Rierson]

Another Standing Wave Tutorial:

http://hypertextbook.com/physics/waves/standing/index.shtml



On the atomic scale, it is usually more appropriate to describe the electron as a wave than as a particle. The square of an electron's wave equation gives the probability function for locating the electron in any particular region. The orbitals used by chemists describe the shape of the region where there is a high probability of finding a particular electron. Electrons are confined to the space surrounding a nucleus in much the same manner that the waves in a guitar string are constrained within the string. The constraint of a string in a guitar forces the string to vibrate with specific frequencies. Likewise, an electron can only vibrate with specific frequencies. In the case of an electron, these frequencies are called eigenfrequencies and the states associated with these frequencies are called eigenstates or eigenfunctions. The set of all eigenfunctions for an electron form a mathematical set called the spherical harmonics. There are an infinite number of these spherical harmonics, but they are specific and discrete. That is, there are no in-between states. Thus an atomic electron can only absorb and emit energy in specific in small packets called quanta. It does this by making a quantum leap from one eigenstate to another. This term has been perverted in popular culture to mean any sudden, large change. In physics, quite the opposite is true. A quantum leap is the smallest possible change of system, not the largest.

[Antonio Lao]

Russell,

Thanks. These are more than what I can chew in one setting. I have to take sometime going through the details. Again, thank you for your overwhelming response.