Toby Lewis - 53790028
ClaytonArnall - 74310996
Sukh Bajwa - 774422994

Mathematics 309 - the elementary geometry of wave motion

Part I - scaling and shifting

One way of obtaining one graph from another is by scaling and shifting either x or y or both, so that the graph y=f(x) becomes the graph y = af(bx + c) + d. The way to see what happens to the graph is to understand what happens in simple steps with just one change at a time.
y = f(x)



y = 2f(x)
All y-values on the graph are scaled by 2.
In subsituting cf for f,
vertical distances are scaled by c.
y = f(x)

y = f(x)+1
1 is added to all y values.
y = f(x)






y = f(2x)
The height at x on the new graph is
equal to the height at 2x on the old one.
The new graph is obtained by compressing
the old one horizontally by 2.
In substituting cx for x,
horizontal distances are scaled by 1/c.
y = f(x)

y = f(x+1)
The height at x on the new graph
is equal to the height at x+1 on the old one.
The new graph is obtained by shifting
the old one 1 to the left.
y = f(x) y = f(2x-1)+1

Only the last one is tricky, since it involves a sequence of substitutions. In order to see which ones, we unravel the process.

  • The function f(2x-1)+1 is obtained from f(2x-1) by adding 1.
  • The function f(2x-1) is obtained from f(x-1) by substituting 2x for x.
  • The function f(x-1) is obtained from f(x) by substituting x-1 for x.
To get the graph, we go backwards. We start with the graph of f(x).
  • Substitute x-1 for x, y = f(x-1):
  • Substitute 2x for x, y = f(2x-1):
  • Add 1 to y, y = f(2x-1)+1:


Wave Functions in 1D

First, we will define the key terms used in simple periodic motion

Simple Periodic Motion - Motion that repeats itself. The motion that we will concern ourselves with assumes that the waves have constant velocity, period, frequency and wavelength.

Amplitude (A) - The maximum distance the crest of the wave travels from its equilibrium position.

Period (T) - The time it takes for one complete cycle of the wave to pass by a fixed point. By one cycle, we mean one complete oscillation of the wave.

Wave Length (&lambda) - The spatial distance between two adjacent crests of the wave

Frequency) - The number of cycles per unit of time that pass a fixed point.

  • Frequency and period have a reciprocal relationship; one is the inverse of the other.

    Example:

      If the frequency is 2 cycles per second, then the period will be:

      T = 1/ω

      T = 1/2 seconds

  • ω can also be measure in radians per second. To accomplish this change in units(ie. scale), simply divide ω by 2π As a result:

      T = 1/(ω/2π)

      T = 2π/ω

  • From simple physics, we know that velocity = distance/time. In our case, λ(wavelength) represents the distance, c represents velocity, and T(period) represents time. Given any two of these variables, the third can easily be determined. For example:

      c = λ/T

      c = λ/(2π/ω) = λω/2π


  • What if we want to see what this graph looks like over a time t? Since λ = c*t, if we want to know the phase shift at time t, we shift it right by ct.
  • y = cos(x - ct)
    y = cos(x)

    		y = cos(x+π)
    The height at x on the new graph
    is equal to the height at x+π on the old one.
    The new graph is obtained by shifting
    the old one π to the left.

  • Up until now, we've assumed the wave has a wavelength of 2π. What if we want to scale the wavelength to something other than 2π? We have to adjust x in the function f(x)= cos(x) by a factor of λ/2π. That is:

      y = cos(x)(λ/2π)) = cos(2πx/λ)

      y = cos(x)

      		cos(2x)
      The height at x on the new graph
      is equal to the height at 2x
      on the old one. The new graph is obtained
      by compressing the old one horizontally by 2.


  • If we combine this with the previous formula regarding the graph at time t, we will get the fundamental wave equation

      y = Acos[(2π/λ)*(x - ct)], where A is the amplitude of the wave.


Wave Functions in Several Dimensions

  • Waves in several dimensions are composed of the same elements as waves of one dimension. These elements hold the same relationship as well.
    • c = Velocity
    • λ = "Wave Length" λ vector between crests.
    • ω = time frequency


  • When working with waves in several dimensions, our wavelength(λ) is a vector. That is, a wave can now travel in any direction within the dimension(rather than just to the right in 1D). The image above is a representation of the crests of a wave(as level curves) and the wavelength vector(perpendicular to the level curves).

  • Similar to one dimension, in multiple dimensions the wave height is calculated as follows:

      y = Acos(Ax + By - ct)

  • Following from this, to understand the wavelength vector in multiple dimensions, we must recall affine functions in 2D and 3D. An affine function in 2D can be represented by:
      Ax + By + c = 0

  • The dot product of the vectors [x,y] and [A,b] effectively portray the orthogonal relationship between λ and the level lines of the "curve" Ax + By + c = 0, where c = 0(vectors must originate from the origin).

  • As proven by vector algebra, if the dot product of two vectors is zero, then the vectors are perpendicular to each other.

      Ax + By = 0
      [A,B] • [x,y] = 0
      [A,B]⊥[x,y]

    The result of this is that the "curve" Ax + By = 0 is the line through the origin perpendicular to [A,B]

  • The level lines of Ax + By = 0 are all perpendicular to the vector [A,B]


  • Example:


  • What we are interested in determining is the wavelength vector λ. A vector is composed of a distance and a direction. To determine the signed distance between Ax + By = c and Ax + By = 0, we take the constant c(which represents the particular level line we are dealing with), and divide it by the magnitude of the vector [A,B]. The result of this is a ratio that represents the magnitude of λ. That is:

      d = c/(sqrt(A² + B²))

  • To find the distance between the two level lines (1)Ax + By = c1 and (2)Ax + By = c2, subtract the distance from (1) to the origin from (2) to the origin. That is:

      d = (c2 - c1)/(sqrt(A² + B²))

  • The previous equations allow us to determine the magnitude of λ, but to explicitly describe a vector, the equation must also include its direction. Since λ is in the same direction as [A,B], it must be a multiple of [A,B]. This can be expressed by:

      λ = k * [A,B]

  • This expression leads to an explicit description of λ as a wavelength vector. The magnitude of λ is a multiple of the magnitude of [A,B]. This can be expressed by:

      λ = k * (sqrt(A² + B²))

  • As previously shown, the magnitude of λ can also be described by the distance

      c/(sqrt(A² + B²))

  • If we equate these two expressions, the square root's cancel out and we are left with an expression for the scalar k, and this takes us to an explicit description of λ as a vector:

      λ =(c/(A² + B²))[A,B]

  • To extend these equations past two dimensions, simply add consecutive terms to the original affine function. For example:

      y = Sin(x) in 2D would become

      z = Sin[xy] in 3D

    Sin[xy] in 3D for postive values
    of x and y

    		Sin[xy] in 3D showing the saddle at the origin