Stonebridge said:
It is not necessary for waves to be traveling in opposite directions to obtain constructive or destructive interference;
[EDIT] this is for HOOKE'S LAW MATERIALS [/EDIT]
I don't think so homie in terms of what he's learning. baby steps first :)
This is why that is wrong from Hooke's law materials and
if you're just learning about standing waves then this is what you're talking about.
If they aren't traveling in opposite directions in the medium (assuming transverse waves on a string) then they are traveling in the same direction. If that is so then they can't ever overlap because that would necessitate a slower mechanical transverse wave in the medium with a faster mechanical transverse wave behind it. This is not possible because the wave speed is determined by the qualities of the medium only. (mechanical waves in a medium following Hooke's law.)
Stonebridge said:
and a standing wave pattern is made up of places of constructive (antinodes) and destructive (nodes) interference.
Maybe you could rephrase the question?
Let me try to elaborate and clarify this
OriginalPost; said:
To create a standing wave, two progressive waves must be traveling in opposite directions along the same line. They also must have the same frequency, wavelength and therefore speed.
This is correct. The speed of wave propagation (wave speed) is determined entirely by the properties of the medium. wave speed (v) = (wavelength)(frequency) which is constant because if you increase frequency wavelength decreases keeping v constant for each material.
Destructive waves are similar only they travel in opposite directions.
Why does it make a difference what way the waves are moving?
see above where i said why stonebridge is incorrect.
The best/easiest way to think of producing a standing wave is to quite literally tie one end of a long string (or slinky is classic ;) ) to something around belly height or have a friend hold it with you in the air so he/she can learn this also.
hold it somewhat taught (which is kinda delicate to do with a slinky without destroying it but it can be done carefully!). Tell your friend to hold their end fixed. "shake" your end vertically a significant distance (once at first and observe the behavior of the wave you send and get back).
Now do this repeatedly at (roughly) the same frequency (until you get a nice standing wave) increasing this frequency carefully can produce more places of maximum displacement (anti-nodes) and more places with zero displacement on the string (nodes). This is a standing wave. Hence the name, it is not traveling like the single pulse you sent and reflected first, the wave seem to be "standing".
What is happening here is the waves you send across the string (traveling wave) by repeatedly shaking up and down (incident pulses) produces a sinusoidal wave traveling across the string.
Each incident wave hits your friend's hand and reflects back up the string toward you but inverted or "upside-down" (reflected wave). *If your friends end was "free" the single wave wouldn't invert. Fixed or free ends are called the boundary conditions.
anyway, there is a property of transverse waves (in mediums that obey Hooke's law) called the principle of linear superposition. Simply put its really just a fancy way of saying the wave amplitudes are added when they occur at the same place on the string.
Be-careful with the simple definition of what i just said and be certain to understand truly what is happening here. The reflected wave is traveling back toward you and it meets the next incident wave about to be reflected. This is meeting and the following overlap of the waves is called interference and there is two kinds.
Destructive interference (At nodes) and
constructive interference (at anti-nodes). So both are occurring in your standing wave and superposition is occurring as well to build your standing wave. This is where the principle of linear superposition is demonstrated.
Lets say each wave has an amplitude of magnitude A. When the two waves start to overlap, his makes your string a "different shape" than the initial wave you sent. The shape at any instant of the resulting wave, due to the princ. of superposition, depends on at what point are the two waves at in the overlap. The displacement of the string itself is achieved by taking the displacement of the two waves (which must be traveling in opposite directions because of what i stated up top which is ONLY TRUE FOR MATERIALS THAT OBEY HOOKE'S LAW and the linear wave equation that results) individually as if they were a singularity in the medium and adding them.
The function of the resulting y displacement of the string (a graph of what you see when looking at the string in real life) is y(x,t)=y1(x,t) +y2(x,t) where y1 is the equation of a wave traveling in one direction and y2 is the eq. of the wave traveling in the opposing direction. Both dependent on the x coordinate of each point in question and time.
This results in additions (and "subtractions" at points in terms of adding a negative)
There are the two names for certain cases of these additions:
-
Constructive Interference is what happens at anti-nodes: the displacements of the two waves are the same and produce a larger displacement than each would individually (additive)
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Destructive Interference- Occurs at nodes: the displacements of the opposing waves are exactly equal in magnitude and opposite in direction canceling each other out.
and everything in between the nodes and anti-nodes follows the same idea (superposition)
PHEWWWW
sorry that was so long but it should straighten out most confusion surrounding this topic ;)
and I hope someone reads this as it took me some time to type this out so I hope it helps someone!
-GreenLantern
