Mathematics 309 - Wave Motion in 1D

 

 

Terms and Concepts

  • A = Amplitude
  • ג = Wavelength
  • C = Velocity
  • T = Period
  • ω = Frequency
    		• A = The maximum distance from the rest position (0).
    ג = The spatial distance between two crests of the wave.
    C = The velocity at which crests travel.
    T = The time require for one compete up/down cycles or the time to complete one wavelength.
    ω = The number of cycles per second or radians per second.

     

    Fundamental Wave Equations

    T = 1 / ω (cycles)
       = 1 / ( ω / 2∏ ) (radians)

    C = Distance/ Time
        = ג / T
        = ω / ג
        = ג ω / 2∏

    This equation holds in any number of dimensions if C = speed as opposed to velocity, which has direction.

    For example :
       |C| = |ג| ω / 2∏    holds in any number of dimension.

     

    Displacement of Wave

    At time 0:
      Y = Cos (X)

    At time t:
    The graph is shifted to the right by the amount ct.
    According to the basic idea of shifting and scaling, the new function is
      Y = Cost (X - Ct)

    We are going to scale 2∏ ---> ג .
    Scaling by ג / 2∏.
    The new function is Y = Cos (X / (ג /2∏ ))
         = Cos ( 2∏X/ג )