Math 309 First Assignment

by Perry Cheng
st:88237011

Topics Covered:

1. Scaling and Shifting Graph
2. Wave Functions in 1D
3. Affine Functions in 2D and 3D
4. Waves in 2D and 3D

Scaling and Shifting Graph

Wave Functions in 1D

All waves in the first dimension are all classified with these basic parameters: , , c, A. They are the fundamentals of a wave or else it would not be classified as a wave. They'll be explain in more detail.

• Wavelength ()
The Wavelength is the spatial distance between 2 crests of a wave as seen below

• Frequency ()

The Frequency is the number of complete wave cycles occurring per unit time.  But we are instead switching from cycles per second to radians per second.  Since 1 cycle in a wave, usually takes 2p in order to complete one cycle, so in order to change it from cycles per seconds to radians per second, we need to multiply a 2p to the original 1/T where T is the period it takes to complete a cycle:

• Velocity (c)

The Velocity c is the velocity at which the crests travel.

• Amplitude (A)

The Amplitude is just the Height of the wave.

The wave equation has all the basic parameters of a wave as we have discussed before. Within the wave equation it has a height of A, shifted by ct (*distance = velocity x time), with the width of the wave scale by a multiple of . A sample picture:

Recall the Affine Function in 2D is: Ax + By + C. An affine function in 2D is basically a line, that has a normal vector [A, B]. The normal vector is perpendicular to the level lines of Ax + By + C = 0 or Ax + By = C {* y = (C - Ax)/B where B not equal to zero}.  It is perpendicular because when Ax + By = 0, it is just the dot product between the vectors [A, B] and [x, y], and any dot product that is equal to zero is orthogonal (or perpendicular). A sample picture:

From the sample picture, the line Ax + By = C is perpendicular to [A, B] and are at a signed distance apart: so in our example, its from 0 to C:

In 3D, the function behaves the same way as in 2D, but instead its has an additional variable: Ax + By + Cz + D. Furthermore, it's a plane in 3D instead of a line with a normal vector [A, B, C]. But as in 2D, the normal vector is perpendicular to the plane Ax + By + Cz + D, and signed distance is the same with an added variable const C.

Much like waves in 1D, the behavior of the waves stays relative the same in the 2D and 3D system. The wavelength in 2D becomes a wavelength vector which is perpendicular to the level lines or the crests of the plane. The wave length vector is a multiple of some vector [a, b]. So let the constant multiple be k such that = k [a, b].  Furthermore like the waves in 1D, the wavelength is the distance between 2 crests, in the 2D case it becomes the signed distance ( |||| ) apart between 2 level lines or crests of the plane. So the signed distance from level 0 to 2p is:
Furthermore since = k [a, b] and the then the constant k is . Therefore our wavelength vector is

Also our wave height changes as our wave equation changes.  Since our wave equation is related to the wave length, then as wavelength changes into vector form, that means the wave equations must changes in some proportion to it.  The wave equation is now: where the [a, b] is related to , the wave length vector, t is the how the wave height changes with respect to time, and is the radian frequency which is the same in 1D.  A visual representation at time = 0:

If you already notice from the previous sections that systems in 2D, are very similar in 3D, and that is the case for waves too.  everything has an additional variable since the vector increase by one variable ex. [a, b, c].  So that means = k [a, b, c] (the wave length vector is still a multiple of some vector but now in 3D) the signed distance from level 0 to 2p is: and similarly the wavelength vector is: and as well the wave height  is:

Course Web Page:
http://www.math.ubc.ca/people/faculty/cass/courses/m309-03a

Others:
http://www.physicsclassroom.com/Class/waves/wavestoc.html
http://www.mathpages.com/home/kmath251/kmath251.htm
All equations are done using Microsoft Equation 3.0 and transported to Adobe Photoshop and converted into jpeg files.