Why don't we use arc length formula to calculate wavelength?

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Discussion Overview

The discussion revolves around the question of why the arc length formula is not used to calculate the wavelength of a sine wave. Participants explore the relationship between the mathematical representation of waves and their physical interpretations, particularly focusing on the definitions of wavelength and amplitude.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the use of the arc length formula for calculating wavelength, suggesting that it could represent the length of the sine function between points of the same phase.
  • Another participant argues that measuring wavelength as the length of the sine function would not be useful, as the current definition of wavelength has significant physical meaning.
  • A different viewpoint suggests that using the arc length formula could combine wavelength and amplitude, referencing musical notes on stringed instruments as an analogy.
  • Some participants express that a misunderstanding of the mathematical function of waves can lead to misinterpretations of wavelength.
  • There is a mention that the visual representation of electromagnetic waves in textbooks can contribute to misconceptions about the relationship between period and wavelength.

Areas of Agreement / Disagreement

Participants express differing views on the appropriateness of using the arc length formula for wavelength calculations. There is no consensus on whether this approach is valid or useful, indicating ongoing debate and exploration of the topic.

Contextual Notes

Participants highlight the importance of understanding the physical representation of waves versus their mathematical functions, suggesting that definitions and visual representations may lead to confusion.

LLT71
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can you please explain me why don't we use arc length formula to calculate wavelength? seems a bit confusing...
 
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Could you please be more specific?
 
Wavelength of a sine wave, λ, can be measured between any two points with the same phase, such as between crests, or troughs, or corresponding zero crossings as shown.
Why don't we treat that as length of sine function between that two points instead, cause that would give us full length of that function in that interval? This way it seems to me like "length of a line" between two points (crests etc.)=wavelength.
 
Well for one, that measurement wouldn't be useful. Wavelength as it is currently defined has a lot of physical meaning. Some sort of length along the curve does not.
 
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LLT71 said:
Why don't we treat that as length of sine function between that two points instead, cause that would give us full length of that function in that interval?
If you have a sin function usually the vertical axis is something like E field strength or pressure, not distance. So it usually wouldn't make sense unit-wise to use the arc length formula.
 
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LLT71 said:
Wavelength of a sine wave, λ, can be measured between any two points with the same phase, such as between crests, or troughs, or corresponding zero crossings as shown.
Why don't we treat that as length of sine function between that two points instead, cause that would give us full length of that function in that interval? This way it seems to me like "length of a line" between two points (crests etc.)=wavelength.

It's an interesting idea. That would actually be a combination of the wavelength and the amplitude. If you think about music, and notes on a stringed instrument, you can see why the traditional definitions of wavelength and amplitude are important.
 
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thank you all! I think seeing a wave just in a way of math function and not understanding what that function actually represents led me to misinterpret wavelength.
 
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LLT71 said:
thank you all! I think seeing a wave just in a way of math function and not understanding what that function actually represents led me to misinterpret wavelength.
You are welcome! It is a pretty common misunderstanding for electromagnetic waves just because of the way books draw EM waves.
 
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Dale said:
You are welcome! It is a pretty common misunderstanding for electromagnetic waves just because of the way books draw EM waves.
you are totally right! for ex. one can easily conclude from the way pictures represent them (particulary, basic wave function) that T[period]=lambda[wavelength], which at first seems very obvious and unit-wise not.
 

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