Superposition Proof: Understanding Angle of Sin

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Discussion Overview

The discussion revolves around the mathematical understanding of superposition in the context of standing waves, specifically focusing on the angles of sine and cosine functions and their relation to nodes and amplitudes.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants question why the angle of sine is set to nπ, where n is an integer, to determine nodes in a standing wave.
  • It is noted that the sine function equals zero at multiples of π radians, which corresponds to nodes where the amplitude is zero.
  • There is a clarification that "y" represents displacement, while "A" denotes amplitude, leading to confusion about which term to use in the context of the discussion.
  • One participant proposes that the cosine function's argument could also be set to nπ/2 for odd integers to yield zero, prompting further inquiry about its implications.
  • Another participant suggests that the argument of sine relates to spatial positions along a string, while the cosine argument pertains to temporal positions, indicating a relationship between the two functions in determining nodes and antinodes.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the definitions of amplitude and displacement, and there is no consensus on the implications of setting the cosine argument to nπ/2. The discussion remains unresolved on these points.

Contextual Notes

There are limitations in the discussion regarding the definitions of amplitude and displacement, as well as the specific conditions under which the sine and cosine functions yield zero. The relationship between spatial and temporal aspects in standing waves is also not fully explored.

Neon32
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I don't get the first part. why did he make the angle of sin equal to n pi.

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A node is a location where the amplitude is zero. The sin function is zero when its argument is a multiple of π radians, nπ, where n = 0, 1, 2,...

In degrees, the sin is zero at 0, 180, 360, etc.
 
pixel said:
A node is a location where the amplitude is zero. The sin function is zero when its argument is a multiple of π radians, nπ, where n = 0, 1, 2,...

In degrees, the sin is zero at 0, 180, 360, etc.
Ok I understood this part but which one is the amplitude "Y" or "A"?

and can I take the angle of cos and make it equal to n pi/2 where n is odd number? It will also give me 0 in this case.
 
I probably shouldn't have used the word "amplitude" for y. y is the displacement for a given x,t, whereas the amplitude is the maximum value of y.

Those values of x that lead to the argument of sin being nπ will give y = 0 for all t. Will have to think about your question of setting the cos argument to nπ/2.
 
Neon32 said:
Ok I understood this part but which one is the amplitude "Y" or "A"?

and can I take the angle of cos and make it equal to n pi/2 where n is odd number? It will also give me 0 in this case.
An over view:
The argument of the sin includes an "x", leading to where (along the string) the function is zero.
The argument of the cos includes a "t" leading to when (in time) the function is zero.
The question related to where the nodes were, so work with the sin.
In a standing wave, even points of antinode are periodically at zero displacement - when that happens is found by playing with the cos function
 
PeterO said:
In a standing wave, even points of antinode are periodically at zero displacement - when that happens is found by playing with the cos function

That's shown in the simulation I referenced.
 

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