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# Periodic Waves

## Summary

• 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; for example…
• meter [m] for displacement
• pascal [Pa] for pressure
• volt per meter [V/m] or newton per coulomb [N/C] for electric field, etc.
• Period (T)
• The time between successive cycles of a repeating sequence of events
• The ratio of time to the number of cycles  T = t n
• 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
• The ratio of the number of cycles to time f = n t
• 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]

• Wavelength (λ, the Greek letter lambda)
• 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
• the state of the medium.
• The rate of change of distance with time, by definition v = ∆s ∆t
• The product of frequency and wavelength for periodic waves

v = fλ

• Frequency and wavelength are inversely proportional. f ∝ 1 λ
• 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].
• Phase (φ, the Greek letter phi)
• 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 (symbol rad), which is itself a unitless ratio.

[rad = m/m = Pa/Pa = (V/m)/(V/m) = etc.]

• The degree is a non-SI unit that is acceptable for phase.

[360° = 2π rad = 1 cycle]

• One-dimensional, periodic wave function
• linear version

f(x,t) = A sin(2π(ft − x/λ) + φ)

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

f(x,t) = A sin(ωt − kx + φ)

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