The equation of a travelling wave

Let me go to the beach. I'll walk out to the end of the pier and measure the height of the ocean at several places.

There appears to be a wave: the height of the water rises and falls, rises and falls, depending on the distance from the pier.

Let me take a movie of the ocean, so that I can measure the height of the water at each position ten times a second. (Click on the picture below to watch an animation)

Does the pattern move with time? No! It's the middle of winter -- the water is frozen solid! This is not a travelling wave; it's just a stationary wave.



  Q:  Can you write an equation which describes
      the height of the ocean surface 
      as a function of distance from the pier:


         y(x)  =    













                                  2 pi radians
         y(x)  =   (12 cm)  cos [ ------------  *  x (meters)  ]
                                    4 meters


                                        radians
         y(x)  =   (12 cm)  cos [ 1.57  ------- *  x (meters)  ]
                                         meter

Now, note that the peak height of the waves, 12 cm, occurs whenever the stuff inside the cosine term equals some multiple of 2 pi



                           radians
max amp when   cos [ 1.57  ------- *  x (meters)  ]   =  1
                            meter

                           radians
                   [ 1.57  ------- *  x (meters)  ]   =   0, 2 pi, 4 pi
                            meter

Let's go back to the beach in the early autumn, when the water is liquid and more interesting. Once again, I go out to the end of the pier and measure the water's height at different locations, AND at a series of different times. Look at how the ocean changes with time ...

This time, we see a real TRAVELLING WAVE. Watch the central crest:

what is the x-position of the crest at a time of t = 1 second ?



  Q:  Can you write an equation which describes
      the height of the ocean surface 
      as a function of distance from the pier  AND
      as a function of time?  It should look something
      like 

      either

         y(x)  =     A   *   cos [  k*x   +  ω*t  ]

      or

         y(x)  =     A   *   cos [  k*x   -  ω*t  ]

Remember, a crest will appear at every combination of x and t which cause



      cos [  k*x   +  ω*t  ]     =      1
  or
      cos [  k*x   -  ω*t  ]     =      1

so

             k*x   +  ω*t        =      0, 2 pi, 4 pi, ...
  or
             k*x   -  ω*t        =      0, 2 pi, 4 pi, ...

I suggest you consider the measurement taken at t = 0.1 seconds. Where is the central crest?



  Q:  Write an equation which describes the height of the 
      ocean surface as a function of distance from the pier  AND
      as a function of time?  It should look something
      like 

      either

         y(x)  =     A   *   cos [  k*x   +  ω*t  ]

      or

         y(x)  =     A   *   cos [  k*x   -  ω*t  ]