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Bounding curves in harmonic motion

Harmonic motion graphs may be enclosed by bounding curves. When a function has a varying amplitude    , such that the amplitude rises and falls multiple times within a period, we can determine the bounding curves from part of the function.

Graphing an oscillating cosine curve

Graph the function f ( x ) = cos ( 2 π x ) cos ( 16 π x ) .

The graph produced by this function will be shown in two parts. The first graph will be the exact function f ( x ) (see [link] ), and the second graph is the exact function f ( x ) plus a bounding function (see [link] . The graphs look quite different.

Graph of f(x) = cos(2pi*x)cos(16pi*x), a sinusoidal function that increases and decreases its amplitude periodically.
Graph of f(x) = cos(2pi*x)cos(16pi*x), a sinusoidal function that increases and decreases its amplitude periodically. There is also a bonding function drawn over it in red, which makes the whole image look like a DNA (double helix) piece stretched along the x-axis.
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Access these online resources for additional instruction and practice with trigonometric applications.

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Key equations

Standard form of sinusoidal equation y = A sin ( B t C ) + D or y = A cos ( B t C ) + D
Simple harmonic motion d = a cos ( ω t )   or   d = a sin ( ω t )
Damped harmonic motion f ( t ) = a e c t sin ( ω t ) or f ( t ) = a e c t cos ( ω t )

Key concepts

  • Sinusoidal functions are represented by the sine and cosine graphs. In standard form, we can find the amplitude, period, and horizontal and vertical shifts. See [link] and [link] .
  • Use key points to graph a sinusoidal function. The five key points include the minimum and maximum values and the midline values. See [link] .
  • Periodic functions can model events that reoccur in set cycles, like the phases of the moon, the hands on a clock, and the seasons in a year. See [link] , [link] , [link] and [link] .
  • Harmonic motion functions are modeled from given data. Similar to periodic motion applications, harmonic motion requires a restoring force. Examples include gravitational force and spring motion activated by weight. See [link] .
  • Damped harmonic motion is a form of periodic behavior affected by a damping factor. Energy dissipating factors, like friction, cause the displacement of the object to shrink. See [link] , [link] , [link] , [link] , and [link] .
  • Bounding curves delineate the graph of harmonic motion with variable maximum and minimum values. See [link] .

Section exercises

Verbal

Explain what types of physical phenomena are best modeled by sinusoidal functions. What are the characteristics necessary?

Physical behavior should be periodic, or cyclical.

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What information is necessary to construct a trigonometric model of daily temperature? Give examples of two different sets of information that would enable modeling with an equation.

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If we want to model cumulative rainfall over the course of a year, would a sinusoidal function be a good model? Why or why not?

Since cumulative rainfall is always increasing, a sinusoidal function would not be ideal here.

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Explain the effect of a damping factor on the graphs of harmonic motion functions.

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Algebraic

For the following exercises, find a possible formula for the trigonometric function represented by the given table of values.

x y
0 4
3 1
6 2
9 1
12 4
15 1
18 2

y = 3 cos ( π 6 x ) 1

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x y
0 5
2 1
4 3
6 1
8 5
10 1
12 3
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x y
0 2
π 4 7
π 2 2
3 π 4 3
π 2
5 π 4 7
3 π 2 2

5 sin ( 2 x ) + 2

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x y
0 2
π 4 7
π 2 2
3 π 4 3
π 2
5 π 4 7
3 π 2 2
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x y
0 1
1 3
2 7
3 3
4 1
5 3
6 7

4 cos ( x π 2 ) 3

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x y
0 2
1 4
2 10
3 4
4 2
5 4
6 10
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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