1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Relating Arc Length and Standing Wave Patterns

  1. Dec 11, 2013 #1
    1. The problem statement, all variables and given/known data

    I am currently reviewing the physics of 'standing waves on a string'. I know that for the nth harmonic, the length of the 'string' is [itex]\frac{n\lambda}{2}[/itex]. Instead of just memorizing these, I have been trying to apply my knowledge of Calculus to figure out why these numbers are what they are. However, I am having a bit of trouble. Here's what I have tried to do thus far:

    3. The attempt at a solution

    If you take the graph of Sin(x), the arc length of the curve between 0 and a can be calculated with the following integral:

    [itex]\int \sqrt{1+cos(x)^{2}}dx[/itex] (with limits 0 and a). So one wavelength would be equal to a distance of 2[itex]\pi[/itex]. The integral between 0 and [itex]\pi[/itex] [itex]\approx 3.82 [/itex] while the integral between 0 and 2[itex]\pi[/itex] [itex]\approx 7.64 [/itex]

    So, that means if we had a wave that was a sine function defined between 0 and [itex]\pi[/itex], the wavelength would be 1/2 that of a sin function between 0 and 2[itex]\pi[/itex] and it would have a length of 1/2 that of the function defined between 0 and 2[itex]\pi[/itex].

    What I am confused about is that I keep just getting:

    First Harmonic [itex]\frac{\lambda}{2}[/itex]=[itex]\frac{L}{2}[/itex] ([itex]\lambda[/itex]=L)

    Apart from it being the wrong answer, this would mean the wavelength, [itex]\pi[/itex], is equal to L which by the arc length formula is 3.82. Furthermore, would the wavelength have to be the same as the arc length (i.e. length of the string?)

    Also if I said I wanted to find the length of the sine wave between 0 and [itex]\pi[/itex] wouldn't I be able to say that the wavelength= [itex]\pi[/itex] and by definition L=wavelength/2 for first harmonic. Then I would get [itex]\pi[/itex]/2 = 3.82. 1.57≠3.82 :grumpy:

    Maybe by Length of the string they don't mean arc length?

    Am I just completely missing something here, or am I using Calculus in the wrong place.

    Thanks!


    (EDIT) I think I figure it out. I was making it WAY more complicated then it has to be. Since there are two nodes on each end, it is like fixing a string in two positions and pulling it up, so the length of the string can't change then.
     
    Last edited: Dec 11, 2013
  2. jcsd
  3. Dec 12, 2013 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Well done :)

    Basically the model is for those situations where the overall length of the string can change while the tension remains constant and the separation of the endpoints is a constant ... where there are not a lot of losses. If there are no losses, the standing wave does not even have to be a sine wave. (i.e. pluck it in the middle and you have a triangular standing wave.)

    The equation you start with for calculus is the wave equation in one dimension:
    http://en.wikipedia.org/wiki/Wave_equation
    ... the solution is generally not trivial (from scratch) but the upshot is that the solutions can be expanded in a trigonometric series, and you are back to counting nodes and antinodes between 0 and L.

    Aren't you glad you asked ?!
     
  4. Dec 12, 2013 #3

    CWatters

    User Avatar
    Science Advisor
    Homework Helper

    Correct. The wave length λ is the straight line distance between two nodes (or antinodes). The length of string refers to the length with no wave on it.

    Edit: oops my post crossed with Simons.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Relating Arc Length and Standing Wave Patterns
  1. Arc Length (Replies: 1)

  2. Arc Length (Replies: 5)

Loading...