There is an interesting quantity that
arises from this formula. The constant portion of the argument of the
current formula
(2πc/λ) has
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.
ω =
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
Back to
Composite Functions
Back to
the Introduction