Waves in 1
Using our knowledge of general function transformations, we can change
simple functions into more complex ones. In this case, we are going to
use a cosine curve as a model for wave behaviour in one dimension. Let's
take a look at some of the key features of a standard cosine curve.
A cosine curve
is a periodic function. Over a given amount of time, the function will
repeat itself into identical copies, called periods. As you can see, the
function resembles a wave. The standard cosine function is based on the
unit circle. There are two ways to consider the behaviour of the
Assume that the above graphic has position in the y axis, and the
x-axis represents the position of curve. Time is held constant in this
equation; imagine that the graphic represents a snapshot of a moving
wave at any point in time. Starting from x
= 0, the
graph takes the value of the x-coordinate as the radius moves around in
a counter-clockwise fashion until it returns to its initial starting
Assume the above graphic has position in the y axis, and time in the
x-axis (as time increases, the graph moves to the right). The position
of interest is held constant in this case. A good example would a Ferris
Wheel. Imagine that you are sitting in the same spot in the Ferris
Wheel (as is usually the case); your position is now fixed in your
seat. As time progresses, the chair begins to move up and down relative
to the ground as Ferris Wheel rotates. The above graphic tracks your
progress as you oscillate up and down. Your position remains the same,
but your height is changing with respect to time.
In any case, the curve that is traced out is the same. This
co-dependence allows the wave equation to be properly expressed as
y(x,t). However, we will begin our look with the analysis of the wave
equation in 1D. Because it is easier to visualize, we will use the time
independent equation. There are a few ways to change the visual
appearance of a standard wave.
Current Formula: y = cos x
The easiest way to
alter the appearance of a cosine curve is to modify its amplitude. By
multiplying the entire function by a constant, A, the height of the
function is scaled. When A is greater than 1, the maximum height
increases; when A is less than 1, the maximum height decreases. In the
above picture, the pink/salmon coloured curve (the tallest one)
corresponds to A = 2. The purple cosine curve (medium sized) is when A =
1, and the grey-blue curve (smallest) occurs when A = 0.5. We can add
this factor into our wave equation.
Current Formula: y = A cos x
That strange looking symbol above the curve is lambda ( λ ). λ is used to
represent the wavelength of a function. The wavelength of a function
refers to the distance (length)
that defines one period of the function. For the standard cosine curve,
its value of lambda is 2π. Over the course of 2π radians, the
function will repeat itself. In order to modify the wave equation, we
can multiply the x argument 2π. This would cause
the period of a cosine curve to go from being [0,2π] to [0,1]. This is
since each value of [0,1] maps to a unique value in [0,2π] so the function
remains the same general shape, but scaled inward.
Current Formula: y = A (cos [2πx]
further modify this by dividing the value of the argument of function by
dividing by any desired λ. This
forces the range from [0,1] to [0,λ]. This means that
each value of [0,λ] will now map to a
value in the interval of [0,1]. This effectively generates
the desired curve for any given wavelength λ.
Current Formula: y = A [cos (2πx/λ)
As with other translations, we may wish to translate the curve along
the x-axis. This keeps the original shape of the curve, but translates
it by c units to the left (if c is positive) or to the right (if c is
negative). The value of c is known as the time shift because it changes
the graph by varying the distance
between the two curves over a varying amount of time. The value
of the shift can be seen by finding the same point in the two graphs and
measuring the distance between them. In the above graphic, the
difference between the two crests can be used to find the value of c.
Current Formula: y = A [cos 2π/λ (x-ct) ] OR y = A [cos (2π/λx-2πc/λt)]
There is an interesting quantity that
arises from this formula. The constant portion of the argument of the
current formula (2πc/λ)
units in frequency (Time)-1
that when multiplied by
time yields a constant (which makes sense since we are taking the cosine
of this quantity). A common measure of this unit is Hz or s-1
which is commonly used to express the number of occurrences per second.
The value of this constant 2πc/λ is known as the radian frequency, and
is typically denoted by the Greek symbol lowercase omega, ω
. It turns out that the quantities
c and λ have units that will cancel out and become 1/Time.
2πc/λ = [(constant) (distance/time)] /
(distance) = (constant)/Time ... which is proportional to Time-1
The radian frequency is a useful
quantity because it allows one to more accurately describe the rate of
rotation of an object. For example, when one quotes a velocity of an
object, it does not give a notation of an object's rate of rotation.
Take the given example below.
From the diagram, there are two arcs (red and blue) that sweep out an
angle (θ) over a given amount of time. While both angles sweep out the
same angle, an object travelling along the red path must have a higher
velocity than an object on the blue path. The distance contained in the
red arc is greater than that of the blue arc, so for a fixed amount of
time, the red object must travel faster. Radian frequency allows one to
describe how fast an object is rotating or how many periods it completes
in a given amount of time.
Now that we have established the wave equation in one dimension, we
extend it into higher dimensions.
On To Wave
Equations In Higher Dimensions