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WAVES: SINE CURVE REVIEW
Math topic
: periodic functionsKey terms: amplitude, period, angular frequency, phase shift, vertical offset
In the preceding applet you were given the opportunity to change the four parameters and to observe the result on the graph.
The Math behind the Observations:
Many of your observations and conclusions in experimenting with the applet are characteristic of periodic functions in general. Sine functions belong to this class. The graph of a periodic function repeats itself over and over. This behavior was evident in the sine curve applet, where the graph was continuously oscillating about a horizontal axis. The period of the function is the distance (measured on the horizontal axis) between successive cycles. It is worth noting that there are numerous applications of periodic functions. For example, periodic functions are used to describe oscillatory motion such as a vibrating guitar string or a pendulum swinging back forth. In the following we present an illustration and we determine specific parameters for the generalized sine function.
We use the same parameters a, b, c, and d in the corresponding equation
y = a sin [b(t-c)] + d.
The specific parameters for any generalized sine curve are presented below. Try to relate your observations from the applet to these constants.
a is the amplitude - the height of each peak above the baseline (or the distance between the maximum value of the function and its average value).
p is the period or wavelength – the length of each cycle.
b is the angular frequency, given by b= 2p /p.
c is the phase shift – the horizontal offset of the basepoint.
d is the vertical offset – height of the baseline.
USNA Mathematics Department
Comments to: Professor Carol G. Crawford, at cgc@nadn.navy.mil or Professor Mark D. Meyerson, at mdm@nadn.navy.mil