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BUILDING UP WAVES




  Throw a stone on the sheet of water in a pond.  Circular waves are created and spread throughout the water surface.  The waves travels as crests and troughs.  We are going to model this waves mathematically. 


     In a right angled triangle, opposite side divided by hypotenuse gives the sine of the angle x  as illustrated in the figure.

     Trigonometry says:  As the angle goes from zero degree to 90 degree, the sine value goes from 0 to 1.  When the angle increases to 180 degree, the sine goes down to 0.  As the angle further increases to 270 and to 360 degree,  the sine goes to -1 and then back to 0.  So when the angle makes one rotation, the sine makes one wavy motion as given in the figure.

      So sine function is very suitable to represent a wave.
  Y = sin x 
     As the x goes from 0 to 360 degree or in radian 0 to 2 pi, Y oscillates between +1 to -1 in wavy fashion.(PI = 3.14....)

 Y = A.sin x
     Here A represents the amplitude, the top of the crest or bottom of the trough.  When the angles are 90 degree or 270 degree: the sine value reach  +1 or -1 and Y touches the amplitude A.

 Angle can be written as 
  x = 2.pi.n.t
 Y = A.sin(2.pi.n.t)
    Instead of angle x, we have written '2pi n t'.  Where n represents the frequency of the wave and t the time.  This equation directly gives the displacement of the particle or state of the wave at any given instant of time t.  We can say this sine equation represents the wave in all its glory.  

    By adding or superimposing many sine wave suitably, we can create any pattern in time and space.  This process is called 'Fourier transform'.  For example we can create square waves, triangular waves or heart rhythms etc,.

      Fourier transform are used in image processing, compression of image data, storing finger print data  etc,.  

      Waves are used to represent oscillations of pendulum, sound, light, heat, alternating current and even subatomic particles like electrons (since they behave like waves).  So understanding the waves is essential to understanding the science. 

Comments

  1. sundar13 November 2017 at 02:25

    simple and excellent motivating article make science very interesting

    ReplyDelete

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