Why are sinusoids important?
Sinusoids occur often in nature.
Example: Pendulum
The pendulum moves more quickly at the center, and slows down and then stops as
it reaches the edges. What function is the horizontal displacement vs. time?
When the free-swinging pendulum is covering a small distance, its horizontal
displacement with time is approximately sinusoidal.
Example: Piano string
When the vibration of the piano string is small, a snapshot at one time is a
sinusoidal function of displacement along the string. If the displacement is
measured at one fixed location vs. time, it is a sinusoidal function of time.
Because of the boundary conditions (the ends of the string are stationary),
there must be a fixed number of cycles of the sinusoidal displacement along the
length of the string. This implies that there are discrete modes of oscillation.
In a later lecture, when we cover linear time-invariant systems, we will see
much more concretely why sinusoids are such important signals in signal
processing applications.
Up to Signals and Sinusoids