Oscillating functions: amplitude, period and frequency

1 Oscillating functions: amplitude, period and frequency

Particular types of periodic functions ( HELM booklet 2.2) that are especially important in engineering are the sine and cosine functions. These are possible choices when modelling behaviour that involves oscillation or motion in a circle. The usefulness of these functions is rather limited if we confine our attention only to sin $(x)$ and cos $(x)$ . Use of functions such as 3sin $(2 x)$ , 5cos $(3 x)$ and so on, and other functions made up of sums of functions of this type, enables the modelling of a great variety of situations where the quantity being modelled is known to change in a periodic way. Here we will examine the behaviour of sine and cosine functions and consider a modelling context where choice of a sine function is appropriate. Figure 6 shows how the terms amplitude , period and frequency are defined with respect to a general sinusoid (the name for any general sine or cosine function).

Figure 6 :

{ Defining amplitude and period for a sinusoid}

The amplitude represents the difference between the maximum (or minimum) value of a sinusoidal function and its mean value (which is zero in Figure 6). The frequency represents the number of complete cycles of the function in each unit change in $x$ . The period is such that $f (x + T) = f (x)$ for all $x$ , e.g. for $sin x$ , $T = 2 π$ .

Example 2

Sketch the sinusoids:

$y = sin x$
$y = 2 sin x$
$y = cos x$
$y = cos \frac{x}{2}$

Solution

Figure 7

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Figure 8

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Task!

Using the graphs in Figures 7 and 8, state the amplitude, frequency and period of

$sin x$
$2 sin x$
$cos x$
$cos \frac{x}{2}$
Give frequency and period in terms of $π$ .

amplitude = 1, frequency = $1 ∕ 2 π$ , period = $2 π$ .
amplitude = 1, frequency = $1 ∕ 2 π$ , period = $2 π$ .
amplitude = 2, frequency = $1 ∕ 2 π$ , period = $2 π$ .
amplitude = 1, frequency = $1 ∕ 4 π$ , period = $4 π$ .
See Figure 7 for the sine functions and Figure 8 for the cosine functions.

Note that (2) has twice the amplitude of (1) and (4) has half the frequency and twice the period of (3).

Note that the cosine functions $cos n x$ have the same shape as the sine functions $sin n x$ but, at $x = 0$ , the cosine functions have a peak or maximum, whereas the sine functions have the value zero, which is the mean value for both of these functions. Indeed the graph of $y = cos x$ is exactly like that for $y = sin x$ with all the $x$ values displaced by $π ∕ 2$ .

More general forms of sine and cosine function are given by $y$ = $a sin (b x)$ , and $y$ = $a cos (b x)$ where $a$ and $b$ are arbitary constants. These are functions with frequency $\frac{b}{2 π}$ , period $\frac{2 π}{b}$ and amplitude $a$ . The peak values of the sine functions occur at $x$ values equal to $\frac{π}{2}$ , $\frac{5 π}{2}$ , $\frac{9 π}{2}$ etc. The minimum values occur at $x$ values equal to $\frac{3 π}{2}$ , $\frac{7 π}{2}$ , $\frac{11 π}{2}$ etc.

When the period is measured in seconds, frequency is measured in cycles per second or Hz which has units of 1/time.

Exercises

Figure 7 on page 37 shows on the same axes the graphs of y = sin x and y = 2 sin x .
1. State in words how the graph of $y = 2 sin x$ relates to the graph of $y = sin x$
2. Sketch the graphs of (i) $y = \frac{1}{2} sin x$ , (ii) $y = \frac{1}{2} sin x + \frac{1}{2}$
Figure 8 on page 37 shows on the same axes the graph y = cos x and y = cos x 2
1. State in words how the graph of $y = cos x$ relates to the graph of $y = cos \frac{x}{2}$
2. Sketch graphs of (i) $y = cos 2 x$ , (ii) $y = 2 cos x$

$y = sin 2 x$ has the same form as $y = sin x$ but all the $y$ values are doubled. The graph is ‘stretched’ vertically.
$y = cos \frac{x}{2}$ has the same form as $y = cos x$ but all the $y$ values are halved. The graph is ‘shrunk’ vertically.