The Physics
Hypertextbook
Opus in profectus

# The Nature of Waves

## Summary

• Properties
• A wave is a disturbance that propagates through a medium.
• verb propogate; noun propogation refers to the transmission of a disturbance from one location to another.
• sing. medium plu. media refers to the intervening substance(s) through which a disturbance is transmitted.
• Waves transfer energy, momentum, and information, but not mass.
• Classifying waves by medium
• Mechanical Waves
• Matter is the medium.
• Sound is a mechanical wave.
• Electromagnetic Waves
• Electric and magnetic fields are the media.
• Light is an electromagnetic wave.
• Gravitational Waves
• The gravitational field is the medium.
• Classifying waves by orientation
• Transverse Waves
• The disturbance is perpendicular to the direction of propagation.
• All electromagnetic waves are transverse. This includes light.
• A crest is a point of maximal change in the positive direction.
• A trough is a point of maximal change in the negative direction.
• Longitudinal Waves
• The disturbance is parallel to the direction of propagation.
• Sound is a longitudinal wave.
• A compression or condensation is a region where the medium is under compression.
• A rarefaction or dilation is a region where the medium is under tension.
• Surface Waves or Complex Waves
• A combination transverse-longitudinal wave that forms near the surface of some media.
• Deep water waves are surface waves.
• Torsional Waves
• The disturbance causes the medium to twist.
• Classifying waves by duration
• The disturbance is momentary and sudden.
• adj. periodic, harmonic; noun wave train
• The disturbance repeats at regular intervals.
• Classifying waves by appearance
• Traveling Waves
• Appear to move.
• Standing Waves
• Do not appear to move.
• Characteristics of periodic waves
• Amplitude (A)
• The maximum absolute value of a periodically varying quantity.
• Amplitude has the unit of the quantity that is changing (ex. displacement, pressure, field strength, etc.)
• Period (T)
• The time between successive cycles of a repeating sequence of events.
• T = t/n (time per number of cycles)
• The SI unit of period is the second [s].
• Frequency (f)
• The number of cycles of a repeating sequence of events in a unit interval of time.
• f = n/t (number of cycles per time)
• Frequency and period are reciprocals (or inverses) of one another: f = 1/T.
• The SI unit of frequency is the hertz [Hz = 1/s = s−1].
• Phase (ϕ)
• The stage of development of a periodic process.
• Two points on a wave with the same phase have the same…
• quantity of disturbance (ex. displacement) and
• rate of change of disturbance (ex. velocity).
• Phase is an angular quantity.
• Adjacent points in phase are separated by one complete cycle.
• Adjacent points out of phase are separated by half a cycle.
• The SI unit of phase is the radian, which is itself a unitless ratio [rad = m/m = Pa/Pa = (V/m)/(V/m) = etc.].
• Wavelength (λ)
• The distance between any point on a periodic wave and the next nearest point corresponding to the same portion of the wave.
• Wavelength is measured between adjacent points in phase.
• The SI unit of wavelength is the meter [m].
• Speed (v)
• Waves propagate with a finite speed (sometimes called the wave speed) that depends upon…
• the type of wave,
• the composition of the medium, and
• the state of the medium
• v = Δst the rate of change of distance with time by definition.
• v = fλ the product of frequency and wavelength for periodic waves.
• Frequency and wavelength are inversely proportional.
• Lower frequency waves have longer wavelengths.
• Higher frequency waves have shorter wavelengths.
• The speed of a wave is sometimes known as its wave speed.
• The SI unit of speed is the meter per second [m/s].
• One-dimensional wave equation
• linear version

f(xt) = A sin(2π(ft − x/λ) + ϕ)

where A = amplitude f = frequency λ = wavelength ϕ = phase
• angular version

f(xt) = A sin(ωt − kx + ϕ)

where A = amplitude ω = angular frequency (temporal frequency), ω = 2πf k = wave number (spatial frequency), k = 2π/λ ϕ = phase