Periodic Functions

Resources

A periodic function is a function whose values repeat at regular intervals. Given an interval of length t, and a function ƒ, if the value of the function at x + t is equal to the value of the function at x then f is a periodic function. In standard function notation this is written ƒ (x + t) = ƒ (x) (read “f of x plus t equals f of x”). The shortest length t for which the function repeats is called the period of the function. The number of times a function repeats itself within a fixed space or time is called its frequency. The maximum value of the function is called the amplitude of the function. When the graphs of two functions having the same period and frequency repeat at different values of the independent variable (x), they are said to be phase shifted or out of phase, and the difference is called the phase angle.

A function may be represented by a graph, which is a picture of how the value of the function (dependent variable) changes when the independent variable changes. Some of the more common periodic functions include the sawtooth, the square wave, and the trigonometric functions (sine, cosine, and tangent) (Figure 1).

Many natural phenomena can be understood in terms of the repeating patterns of waves. For instance, sound travels in waves, energy changes propagate on the surface of liquids in the form of waves, light behaves like both a particle and a wave, radio signals (a form of “light”) travel as waves, and alternating current electricity behaves like a wave. All of these phenomena are described by periodic functions, sometimes called wave functions. The sine and cosine functions derive from the lengths of adjacent sides of a right triangle (sides that meet to form a 90^° angle), and describe how the lengths of these sides change when the hypotenuse (the side of a right triangle that is opposite the 90^° angle), taken to be the radius of a circle with a length of 1 unit, is rotated through 360^°. Because the hypotenuse can be rotated a full 360^° as many times as desired, the length of each side will repeat itself as the hypotenuse is rotated twice, then three times, then four times, and so on. The sine function (ƒ(x) = sin x) describes how the length of the vertical side changes and the cosine function (ƒ(x) = cos x) describes how the length of the horizontal side changes (Figure 2).

It is interesting to see how the graphs of these functions change when the amplitude, period, and phase are changed (Figure 3).

For both functions, the amplitude is changed by changing the radius of the circle, and the period is adjusted by multiplying the angle of rotation by a constant before determining the length of either side. Adding the value of a fixed angle to the angle of rotation before determining the length of a side, adjusts the phase. In general form, then, these functions are written ƒ(x) = A sin (Bx + C), or ƒ(x) = A cos (Bx + C) where x is the angle of rotation, A determines the amplitude, B determines the period, and C determines the phase.

With physical phenomena, it is often the case that the independent variable of interest is time. The period (t), then, has units of seconds, and the frequency (v) is the inverse (1/t) of the period. The wavelength (l) is the distance the wave travels in the time (t) that it takes to complete one period, and depends on the velocity with which the wave travels. Radio waves, for instance, travel at the speed of light (300,000 km/s). If your favorite radio station is 98.7 FM, then you can calculate the wavelength of the radio waves it broadcasts, since the call number corresponds to the frequency of the broadcast waves in MHZ (1 Hz is the equivalent of 1 cycle/s). The wavelength is given by the formula l = v/v, where v is the velocity of the wave. The station at 98.7 on the FM dial is broadcasting radio waves that are approximately 9.8 ft (3 m) long.

KEY TERMS

Amplitude— The maximum value of a periodic function.

Frequency— The frequency of a periodic function is the number of times that it repeats itself in a fixed space or time.

Period— The shortest length over which a periodic function does not repeat itself.

Trigonometric functions— Angular functions which can be described as ratios of the sides of a right triangle to each other.

Wavelength— The distance between two consecutive crests or troughs in a wave.

Resources

BOOKS

Abbot, P. and M.E. Wardle. Teach Yourself Trigonometry, 2nd ed. New York: McGraw-Hill, 2003.

McKeague, C.P. Trigonometry, 5th ed. Pacific Grove, CA: Brooks/Cole, 2003.

Pierce, John R. Almost All About Waves. Cambridge, MA: MIT Press, 1981.

J. R. Maddocks