Understanding the behavior of many physical systems involves understanding how
they vibrate. We are usually interested in the frequency and the
amplitude of a vibration.
A car driving across a bumpy road will bounce up and down. The spacing and
height of the bumps, the speed of the car, and the makeup of the car's
suspension system determines the frequency and amplitude at which the car
bounces. Here, the frequency is the number of bounces per second, and the
amplitude is the maximum amount up or down by which the car bounces. (Most
passengers would prefer a low amplitude bounce!)
Oscillations such as this are traditionally modeled by scientists and engineers
with functions of the form
A cos(2 Pi f t)
where A is the amplitude of the oscillation, f is the frequency of the
oscillation, and t is time. Take some time now to use Mathematica to generate a
variety of plots for this function for different values of A and f. What
will the two axes of the plot be?
Click here for the answer
Do your first plot by using 1 for both the amplitude and the frequency. You
should notice that the resulting curve oscillates between a maximum of 1 and
and a minimum of -1, which reflects the amplitude. You should also notice that
the curve goes through an entire oscillation once every second, which reflects
the frequency. Frequency is typically measured in units of Hz, where 1
Hz means 1 complete oscillation per second.
Try some more visualization. What happens if you double A or cut it in half?
What happens if you double or halve f? What happens if you double A and
f simultaneously? Double one and halve the other?
Be sure that you obtain a good physical intuition for this basic
single-frequency oscillation function before you move on. You will rapidly
become lost in the rest of the lesson if you don't develop a good intuition
here. Before you continue, you should discover that the function is quite
simple to understand. Hopefully, you will also conclude that visualizing this
function was instrumental in understanding it.