# 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.

- A wave is a disturbance that propagates through a medium.
- 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.

- Mechanical Waves
- 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.

- Transverse Waves
- Classifying waves by duration
*adj*. episodic;*noun*pulse- 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.

- Traveling Waves
- 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*= Δ*s*/Δ*t*the rate of change of distance with time by definition.*v*=*f*λ 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].

- Waves propagate with a finite speed (sometimes called the wave speed) that depends upon…

- Amplitude (
- One-dimensional wave equation
- linear version

where*f*(*x*,*t*) =*A*sin(2π(*ft*−*x*/λ) + ϕ)*A*=amplitude *f*=frequency λ = wavelength ϕ = phase - angular version

where*f*(*x*,*t*) =*A*sin(ω*t*−*kx*+ ϕ)*A*=amplitude ω = angular frequency (temporal frequency), ω = 2π *f**k*=wave number (spatial frequency), *k*= 2π/λϕ = phase

- linear version