Waves and oscillations




 

Introduction

Waves are everywhere:
                                            - Ocean
                                     - Earth
                                     - Matter (Probability waves, Predict behavior of e-, particles)
                                     - Sound -> What we hear
                                     - Sight -> What we see
                                     - Electromagnetic waves (Power to our homes, Radio & TV , Electronics)  

 Properties of waves:
                                               -Energy
                                      -Momentum
                                      -Velocity
                                      -Linearity

 Energy:
   - "Like moving objects, moving waves carry energy from one place to another" - J.R. Pierce
   - Energy is not transmitted by the media that support waves, but by the waves themselves.  
   - EM waves from sun to earth have power of 1KW/m2.
   - Plants are supported on this energy and we are supported by plants.


  Momentum:
     Moving waves have momentum when a wave is absorbed or reflected by and object, the wave pushes on that object.  

  Velocity:
     It takes time for a wave to go from one place to another. 
    Light: 186,000 Miles/Sec , 300,000 KM/Sec
    Sound: (Through air) ~ 1129 feet
/Sec

 Oscillations:  Motions that repeat themselves.

 Periodic Harmonic motion:   Motion that repeats itself at regular intervals.

Simple Harmonic Motion:
   Harmonic Motion that is sinusoidal. 

 


 

Oscillations and Periodic Harmonic Motion

Simple Harmonic Motion:

          


   Simple Harmonic Motion:  Harmonic Motion that is sinusoidal.

     A displacement X, that varies sinusoidally may be functionally represented as follows:

                                                              X = Xm Cos(wt + q)
                                                    X = Xm Sin(wt + q)


The amplitude, Xm, and the phase, q , depends on how motion was started (depend on initial conditions). The angular frequency, w, depends on intrinsic properties of the system. For example, when a mass on spring oscillates, the frequency of oscillation and phase of oscillation depend on how the motion was started (how far the mass was pulled, and how hard it was pushed when released).

 

        
        

Unit Circle:

    

 
Notice that the cosine function is positive in the first and forth quadrants of the unit circle diagram, while the sine function is positive
in the first and second quadrant of the unit circle diagram.

Harmoic motion, mass on a spring