Is sine*sine a form of a standing wave?

In summary, the equation describes a standing wave as a superposition of two waves with equal amplitude and wavelength, but I'm unsure as to what it is because the two waves will cancel each other out perfectly.
  • #1
deadlytrogdor
5
0
What kind of wave is Asin(kx)sin(wt)?

Using trig functions, I've rewritten it as

Bcos(kx-wt) - Bcos(kx+wt)

So it sort of looks like it's a standing wave in that it's a superposition of two waves traveling in opposite directions with equal amplitude and wavelength, yet I'm unsure since it seems like the two waves would be canceling each other out perfectly.

So what is it? :confused:
 
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  • #2
You seem like you're on the right track. What do you mean they would be canceling each other out perfectly? You already saw that the equation could be written in terms of w(x,t)=Asin(kx)sin(wt), which is not 0. Also, I guess you're using a new factor B, but be sure to keep track of your factors of 2.
 
  • #3
Okay. I guess I just wasn't thinking about this correctly graphically--I was thrown off by the minus sign instead of the plus sign.

So, just to be perfectly clear, it is then a standing wave, based off of what I said originally, correct?

Was it necessary for me to get it into the cos-cos form to see that it is a superposition of two waves, or is there a way to tell from the sin*sin equation that it is a standing wave? If it was like sin(kx)cos(wt) or something like that, I would have recognized it, but the sin*sin throws me off a bit.
 
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  • #4
Granted, standing waves are can be produced by interference, but I'd say the original equation describes it as a standing wave a bit better. You have a term that depends on frequency and time (wt), and you have a spatial term that will give nodes and antinodes. It's the spatial term that makes this a standing wave. It could be sin or cos.

sin(kx)cos(wt)=sin(kx)sin(wt+pi/2)=sin(2*pi/(lambda)*x)sin(wt+pi/2)

if x = lambda ± n*lambda/2 you have a node, I'll leave it to you to see how the antinodes would go

Maybe helpful
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html
 
  • #5
excellent, thanks!
 

1. What is a standing wave?

A standing wave is a type of wave that occurs when two waves with the same frequency and amplitude travel in opposite directions and interfere with each other. This results in stationary points of maximum and minimum amplitude.

2. How is a standing wave different from a traveling wave?

A traveling wave moves through space, while a standing wave appears to be stationary. In a traveling wave, the particles of the medium move in a continuous wave-like motion, while in a standing wave, the particles of the medium vibrate in place.

3. What is the relationship between sine and standing waves?

Sine is a mathematical function that describes the shape of a standing wave. The amplitude of the sine wave represents the maximum displacement of the particles in the medium, while the wavelength of the sine wave represents the distance between two consecutive nodes or antinodes in the standing wave.

4. Can any two sine waves form a standing wave?

No, in order to form a standing wave, the two sine waves must have the same frequency and amplitude and travel in opposite directions. If the waves do not meet these criteria, they will not interfere with each other in a way that creates a standing wave.

5. How are standing waves used in practical applications?

Standing waves are used in many practical applications, such as musical instruments, where they produce specific resonant frequencies that create pleasing sound. They are also used in telecommunications, such as in antennas, to transmit and receive signals efficiently. Additionally, standing waves are used in medical imaging, such as ultrasound, to produce clear images of internal structures.

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