Frequency

FIELDS OF STUDY: Acoustics, Harmonics, Optics

ABSTRACT: This article discusses aspects of physical and electromagnetic wave phenomena, with regard to their frequency. The frequency of a wave is expressed as the number of cycles occurring in a specific amount of time. The time required for one cycle is the period of the wavelength. Phase relationships and interference are also discussed.

Principal Terms

• frequency: the number of complete waves or cycles that occur in one unit of time.
• harmonics: the study of the interaction of wave phenomena.
• hertz: a unit of frequency defined as one cycle per second.
• period: the length of time for one complete cycle of a wave or other cyclic property to occur.
• reciprocal: the inverse of a value, calculated as 1 divided by the value.
• sinusoidal: having a shape or pattern of behavior that can be described by a sine wave function.
• speed: the distance traveled per unit of time.
• wavelength: the distance from any point in a wave to the identical point in the next wave, usually measured from crest to crest.

Cyclic Phenomena

The term "cycle" generally indicates something that goes around in a circle. In physics, "cycle" indicates that a specific property or function has a value that progresses through a succession of other values and returns to the starting value in a precise manner that repeats. Phenomena that exhibit this behavior are associated with either circular or sinusoidal wave motions and properties. Such motions can be described by the same math functions, the sine and cosine.

The sine and cosine functions are themselves simple ratios of the lengths of the two sides of a right triangle at one vertex. The radius (plural: radii) of the circle can be rotated about the center by any amount to form the corresponding angle. A vertical line to the point on the circumference where it meets the displaced radius forms a right triangle with a base that is proportionately shorter than the length of the radius. In this right triangle, the displaced radius forms the hypotenuse and the vertical height of the triangle is the opposite. (The "opposite" is the side of the right triangle that is opposite the angle formed at the center of the circle.) The base of the triangle is called the "adjacent." The sine of the angle formed by the base and the hypotenuse is just the ratio of the length of the opposite to that of the hypotenuse (i.e., the radius). Likewise, its cosine is the ratio of the length of the adjacent to that of the hypotenuse. The reciprocal values of the sine and cosine are called the "secant" and "cosecant," respectively.

As the radius rotates, the angle that it forms at the center changes continuously. The value of the sine also changes accordingly. A graph of this variation produces the sideways S-shaped curve that is recognized as a sine wave. The value of the cosine follows the same pattern but is shifted from the sine values. The cosine at any angle has the value of the sine of an angle that is greater by 90 degrees.

There are two methods of describing the amount of rotation about the center, or axis of rotation. In one, the amount is stated in degrees of rotation, with one full revolution totaling 360 degrees. The other measurement of angles is in radians (rad). One radian is the angle formed by two radii when the length of the circumference they mark off is equal to the radius of the circle. There are 2π radians in one complete revolution.

Properties of Cyclic Phenomena

All cyclic phenomena, whether wavelike or circular, share several characteristics. The primary feature of all of them is that their behaviors or values repeat in the same regular way. The number of times that the cycle of any particular phenomenon repeats in a specific amount of time is its frequency. The most common of these is revolutions per minute (rpm) for rotational movements of physical objects, and cycles per second (cps) for wavelike properties. The conventional unit for cps is hertz (Hz), in honor of Heinrich Hertz (1857–94), for his contributions to the physics of electromagnetism (EM). The term is most often used to refer to EM waves.

The duration of just one cycle of the phenomenon is the period of the cycle. The period is calculated simply as the reciprocal value of the frequency. For example, a wave frequency of 100 cps has a corresponding period of 1/100, or 0.01, seconds per cycle (spc). If that same wave is progressing, or propagating, at a speed of 100 meters per second (m/s), each cycle will have moved it through a distance of 1 meter. Since this corresponds to the distance covered by just one complete cycle of the wave, it is the specific wavelength. Wavelength is only used to describe phenomena that travel through space or time. It is not applied to rotational motions.

Frequency, Wavelength, and Speed

EM waves such as visible light all travel at the same speed—the speed of light. Physical waves, such as sound, travel at different speeds determined by the density of the medium. The speed of sound in air, for example, is 331 meters per second (about 1,086 feet per second) at 0 degrees Celsius (32 degrees Fahrenheit), and 342 meters per second (about 1,122 feet per second) at 18 degrees Celsius (65 degrees Fahrenheit). In water, the speed of sound is about 1,140 meters per second (about 3,740 feet per second). The greater the density of the medium is, the more it can transmit sound waves. Another factor that affects the transmission of both physical and EM waves is the relative motion of the wave source and the observer or receiver of the emitted waves. When the two move toward each other, the apparent frequency of the waves increases and the apparent wavelength decreases, but if they move apart, the apparent frequency decreases and the wavelength increases. This is known as the "Doppler effect." It accounts for the apparent changes to the sound of a passing train, as well as the red shift and blue shift in the light observed from distant stars.

For physical waves, speed, frequency, and wavelength are all related. For EM waves, however, the constant speed of light requires that the frequency and wavelength are related in a manner that maintains the constancy of the speed of light.

Sample Problem

The crests of two successive waves passing a dock were observed to be 6 meters (about 20 feet) apart. It took 7.5 seconds for the crests to travel the 100 meter (about 328 feet) length of the dock. Calculate the frequency and period of the waves.

Answer

First, define the wavelength (λ) as the distance between any two successive wave crests. This is given as 6 m (20 ft). Next, determine the speed (s) by relating the distance that the waves travel in a certain amount of time. This is given as 100 m (about 328 ft) in a time of 7.5 s. The speed of the waves is therefore

s = 100 m ÷ 7.5 s = 13.33 m/s

The frequency (f) can then be calculated by dividing the waves’ speed by the wavelength, as

f = s/λ

f = 13.33 m/s ÷ 6 m

= 2.22 cps

(The calculation using feet per second for speed and feet for wavelength must yield the same answer for the frequency of the waves.)

The period (T) of the waves is calculated as the reciprocal of the frequency, or

T = 1/f = 1/2.22 cps

= 0.45 s

Frequency and Phase

The fundamental character of a wave is determined by its frequency. All wave phenomena have neutral points called "nodes" at which the value of the amplitude is zero. When two identical waves are synchronized such that their nodes and amplitudes coincide precisely, they are in phase. Their amplitudes do not matter. At all other times, they are out of phase by a particular amount. For example, two waves might be said to be 10 degrees out of phase. The amplitudes of waves that are in phase add together as constructive interference to produce a wave with greater amplitude. When they are out of phase, such that their positive and negative amplitudes overlap, they add together as destructive interference to produce a wave with less amplitude and loss of harmonic characteristics.

Waves of different frequencies can never be in phase. Instead, nodes and amplitudes may coincide occasionally in such a way that a harmonic beat frequency may arise. This is most apparent when one frequency is a whole-number multiple of the other. In such cases, nodes occur consistently when a number of wavelengths of one frequency span the same period of time as a different number of wavelengths of the other frequency.

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