Math309 Assignment 1                                Name: Sandy Chan and Tomoko Kitagawa

Part 2 Wave function in 1D

 Except for A, any two parameters can determine the third.

eg) = cT

          = 2c /

       c  = / T

                       = 2c           ( II-1)

When velocity has a direction (i.e. in a certain number of dimensions like 2D, 3D), the equation ( II - I ) need modification.

    In this case, is interpreted as a vector between two crests.  

                 Then,

                    |c| = || / 2   where c is 'speed', not 'velocity'

                            

                a) 

                   

                    when graph is scaled horizontally by 1/c,

                    x → cx

                    y = cos cx

                    eg)    y = cos x

                           y = cos 2x

                b)  When graph is shifted left by a, 

                      x → x - a

                      eg)     y = cos x

                              y = cos ( x + a)

                   

                c)    When graph is scaled vertically by c,

                        f = cf

                        eg)    y = cos x

                               y = 2 cos x

                   

                d)    When graph is shifted vertically by a,

                        f = f + a

                        eg)    y = cos x

                    

  • Over, the equation of the wave in 1D is
  •            y = A cos ( 2/ ( x - ct))

               = A cos ( 2x / - 2ct / )       where = 2c /

               = A cos ( mx - t)                   where m = 2 /

     

    Part III    Affine Function in 2D and 3D

     

               A cos ( ax + by + c - dt) in 2D                              ( III-1)

               A cos ( ax + by + cz + d - et) in 3D                         ( III-2)

               A level line is a " curve" such as

                   Ax + By = constant                                                 ( III-3)

                   Ax + By + Cz = constant                                        ( III-4)

             for example, in the equation III-3,

                    Ax + By = constant

                    i) when B = 0,    Ax = constant

                                       x = constant / A

                       

                    ii) when B 0 ,         y = constant - Ax / B

                                               = -Ax / B + constant k

                      

               

                 Ax1 + By1 = constant k 

                 Ax2 + By2 = constant k

            So, A( x1 + x2) + B ( y2 - y1) = 0

                   [ A, B ] is perpendicular to [ x1 -x2, y2- y1]

             So, the level line Ax + By = k is perpendicular to [ A, B ]

                 

                  Ax + By = k                                    

                  Ax + By = l

                These two lines are parallel and also perpendicular to vector [ A, B ]

              

               p = [A, B ]

               || p || = 1/A² + B²

             || (k-l) p || = k-l / A² + B²

                Ax + By + Cz = k

                    Ax1 + By1 + Cz1 = constant k

                    Ax2 + By2 + Cz2 = constant k

                A( x1 - x2 ) + B( y1 - y2 ) + C( z1 - z2 ) = 0

                    so, [ A, B, C ] is perpendicular to [ x1 -x2, y1- y2, z1- z2 ]

                Therefore, the level line is perpendicular to [ A, B, C ]

             

               is also perpendicular to the level lines and,

                    = a p

                    so, || || = a / A² + B² + C²

                             = k A² + B² + C²                         where k = a / A² + B² + C²