Math309 Assignment 1                                Name: Sandy Chan and Tomoko Kitagawa

Part 2 Wave function in 1D

• simple periodic motion can be seen in cosine wave.
• Parameters used in here are (wave length): is the distance over which the wave pattern repeats, as measured at a fixed time.

• T(period): is the time for one complete wave cycle to pass a fixed position.

• (frequency): is the number of wave cycles passing a given point per unit time which is the inverse of the period. Also, it can be described as radians per seconds.

• c(velocity):  is the distance of the wave travel divided by time interval.

• A(amplitude): the maximum value of the disturance. It measures whatever physical quantity is affected by the wave.

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'

• at time t,

the graph is shifted right by ct

so y = cos (x - ct)

• Then,

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

• An Affine function of 2 variables is

Ax + By + C

also, an affine function of 3 variables is

Dx + Ey + Fz + G

• For the wave equation, phases can be expressed in affine functions.

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

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

• Level lines

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¡Ú 0 ,         y = constant - Ax / B

= -Ax / B + constant k

• For 2D level lines

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 ]

• For 2 different 2D level lines with different constant number

Ax + By = k

Ax + By = l

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

• Distance between two level lines are

p = [A, B ]

|| p || = 1/¡ÔA² + B²

|| (k-l) p || = k-l / ¡ÔA² + B²

• In 3D, the level line is

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 ]

• p = [ A, B, C ]

is also perpendicular to the level lines and,

= a p

so, || || = a / ¡ÔA² + B² + C²

= k ¡ÔA² + B² + C²                         where k = a / A² + B² + C²