Amplitude

FIELDS OF STUDY: Electromagnetism; Acoustics; Harmonics

ABSTRACT: This article describes the basic properties of different types of waves and relates them to a wave’s amplitude. Waves are characterized by the basic properties of amplitude and frequency. Amplitude describes the distance between the neutral value of a wave to its maximum displacement.

Principal Terms

• crest: the highest point of a wave from its neutral value; the distance between the crest or trough of a wave and the wave’s neutral value is called the amplitude.
• displacement: the upward or downward extent to which the amplitude differs from the neutral value of a wave.
• frequency: the number of complete wavelengths that occur within one unit of time, typically expressed as hertz (Hz; cycles per second).
• loudness: the intensity of sound waves, which depends on the wave’s amplitude; measurements of loudness or volume are expressed in decibels.
• oscillation: a variation between maximum and minimum values of displacement from a neutral value.
• peak amplitude: the value of the amplitude at its maximum displacement from the neutral value of the wave.
• peak-to-peak amplitude: the absolute value of the sum of the peak positive and peak negative amplitudes; the distance between the crest and the trough of a wave.
• root-mean-square amplitude: for sinusoidal wave systems, the square root of the sum of squared amplitude values divided by the number of amplitude values.
• trough: the lowest point of a wave from its neutral value.
• wavelength: in any wave system, the distance from one point in a wave to the equivalent point in the next wave, typically measured between successive peak values.

Properties of Waves

A wave is any physical phenomenon that can be described as an oscillation, or an upward and downward displacement that travels through a medium, such as water or air, or through space. Waves in water and the vibrations of a guitar string are just two examples of waves. Visible light and all other wavelengths of light are electromagnetic wave phenomena. Sound is another physical phenomenon that can be described in terms of wave properties. An essential feature of wave systems is their specific wavelength. Wavelength describes the distance between two identical points in successive waves. Wavelengths are typically measured between successive peaks, or crests, of a wave system. Another property of waves is their frequency, or the number of times waves repeat in a single unit of time.

Wave properties can be clearly visualized in water. For example, if the crests of a series of waves approaching a shore are separated by a distance of 3 meters (10 feet) then the wavelength of those waves is 3 meters. If six waves pass the same point in a span of three seconds, then their frequency is two waves per second. Frequency is normally described in units of cycles per second called hertz (Hz). The neutral value of water waves is at the level of perfectly smooth, undisturbed water. The difference between this level and any part of a wave is the displacement. The maximum displacement either above or below the neutral level is the amplitude of the wave. As each wave approaches, half of the wave is above the neutral level and half is below the neutral level. The maximum upward displacement of the wave is the crest, and the maximum downward displacement of the wave is the trough. The terms "crest" and "peak amplitude" are often considered synonyms.

Simple Harmonic Oscillations

The pendulum in a grandfather clock is one of the most common examples of a simple harmonic oscillator. The motion of a pendulum can be described in terms of wave functions. At rest, a pendulum hangs straight down, which is its neutral value. The distance that the pendulum is swung away from the neutral value is its displacement. The maximum displacement in either direction is the amplitude. The motion of the pendulum as its swings between its maximum displacement on either side demonstrates a sinusoidal relationship. Sinusoidal motion is a pattern of behavior that can be described by a sine wave function. Once the pendulum reaches its maximum displacement, its motion reverses and it falls toward the neutral position once again. The distance from the peak amplitude in one direction to the peak amplitude in the other direction is called the peak-to-peak amplitude.

Sound Waves

Sound waves propagate or move through a medium such as air or water when a force is exerted against the matter that makes up the medium, compressing the atoms or molecules together. This compression travels through the medium as the compressed molecules push against the molecules next to them. At the same time, the molecules of matter behind the compression are pulled farther apart. The compression is the positive part of the sound wave. The rarefied portion that follows is the negative part of the sound wave. The compression and rarefaction of the particles in the medium displace those particles from their neutral position. The loudness of the sound is determined by the extent that the particles are displaced. The greater the amplitude of a sound wave, the greater the loudness of the sound.

When the compression and the rarefaction stages have passed, the molecules return to their neutral position. The motion of the matter involved in a sound wave can therefore be described by the same sinusoidal wave functions as other waves. The displacement of the medium from its neutral position describes the amplitude of the sound wave. The number of sound waves per unit time defines the frequency of the sound (its pitch). The distance between equivalent points in successive sound waves defines the wavelength of the sound.

Sinusoidal Motion and Amplitude

The amplitude of any wave is the maximum displacement of the wave from its neutral, or equilibrium, value. Smooth and cyclic wave functions can be described by a sine wave function, or sinusoidal motion. In simple terms, sinusoidal motion follows the value of the sine of an angle about a fixed center, just like a point on a rotating wheel. Sine values cycle between a range of 1 and −1, beginning at an angle of 0 degrees, or 0 radians. A radian is the arc length along a circle that is equal to the length of the radius of the circle. An angle in radians is equal to the arc length divided by the radius, which is equal to approximately 57.3 degrees. As the point travels around the circle, it reaches an angle of 90 degrees (or π/2 radians) relative to its start point. As it travels further and again reaches the neutral level, the angle it makes with its starting point is 180 degrees (or π radians). At the lowest point the angle is 270 degrees (or 3π/2 radians). The full circle is completed at the starting point, after moving through an angle of 360 degrees (or 2π radians) about the center of the circle.

A graph of the sine function exhibits the characteristic "sideways S" shape of a sine curve. In a sine wave, the amplitude is equal to the sine value of 90 degrees, which is 1, or 270 degrees, which is −1. Note that the trigonometric values of sine, cosine, and tangent have no units, as they are simply ratios and not measurements.

Sample Problem

On a windy day at the beach, waves roll in regularly along a pier that rises 3 meters (10 feet) from the level seabed. The crest of each wave reaches the top of the pier, and the trough reaches only halfway to the top of the pier. Determine the peak amplitude of the waves.

Answer

The piles of the pier mark a distance of 3 meters (10 feet) from the seabed to the top of the pier. The trough of each wave reaches one-half of that distance, or 1.5 meters (5 feet). The maximum difference between the crest and the trough of a wave, or the peak-to-peak amplitude of the waves, is therefore 1.5 meters (5 feet), corresponding to the distance from the top of the pier to the trough of the wave. The peak amplitude of the waves is equal to the difference in wave height from the neutral point to the crest, or one-half of the peak-to-peak amplitude, which in this case is 0.75 meters (2.5 feet).

A = (AP−P) / 2 = 1.5 m / 2 = 0.75 m

Sines and Cosines

Both the sine and the cosine function are used to describe wave motion. They have the same behavior, but the value of the cosine is shifted 90 degrees (π/2 radians) from the value of the sine for the same angle. Whereas sine begins at 0, cosine begins at the peak amplitude of 1.

Bibliography

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