1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Help with understanding the linear wave equation

  1. Aug 29, 2010 #1
    1. The problem statement, all variables and given/known data

    Suppose an element of a string, called [tex]\[\triangle x\][/tex] with T being the tension.
    The net force acting on the element in the vertical direction is

    [tex]\[\sum F_{y} = Tsin(\theta _{B}) - Tsin(\theta _{A}) = T(sin\theta _{B} - sin\theta _{A})\]

    I know what small-approximation is, but I suspect there is a definitive reason to why we choose sin ~= tan and not sin ~= delta. But y/x is arctan.. if we are talking about that.. So what is it?

    To be more clear, the reason we use partial is because the function contains two variables, x and t, right?

    Any help is appreciated! Thank you!
  2. jcsd
  3. Aug 29, 2010 #2


    User Avatar
    Homework Helper

    Well, for small-angle approximations you can set sin(θ) ≈ tan(θ) or sin(θ) ≈ θ, depending on which one is more useful for the particular calculation you're doing. In this case it appears that they want to use the derivative dy/dx, which is equal to the tangent of the angle, so it's more useful to choose tangent.
    Right, and because you want to take the derivative of y with respect to only x, leaving t constant.
  4. Aug 30, 2010 #3
    LOL I am so stupid. tan = sin/cos, and I always thought x/y. It was opposite / adjacent, which makes dy/dx.

    After reading a bit on derivative on Wiki,
    According to the book
    1. So why do the physicists imagine this "infinitesimal displacement"?

    2. So in essence, the rate of change, dy/dx gives the rate of change. If we interpret it in dy/dx form, we have the slope of a tangent line. I see the relationship between dy/dx and tan, but how do I see the relationship between the slope of the tangent line and tan?

    Thank you! I hope I don't sound dumb :)
  5. Aug 30, 2010 #4


    User Avatar
    Homework Helper

    1. As opposed to a finite displacement or something? I'm not sure I understand what you're confused about here. Generally speaking, that's just one way to think about a derivative, you move an infinitesimal amount in the x direction and see how much your function changes in the y direction. (If it were a finite displacement, you might have different slopes at different points within the interval.)

    2. Well why do you think they call it the tangent function? :wink: Try this: just draw a straight line on a graph, and draw a triangle to figure out the slope as [itex]\Delta x/\Delta y[/itex]. Then using the same triangle, find the angle between the line and the x-axis.
  6. Sep 1, 2010 #5
    Thank you!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook