**Wave Motion**

**Electromagnetic Waves**

λf = c = 2.998e8 m/s (speed of light in vacuum)

λ = Wavelength in meters

f = frequency in Hz

Speed of light

299,792,458 meters per second in vacuum (exactly)

299,704,600 meters per second in air

224,900,000 meters per second in water

124,000,000 meters per second in diamond

about 186,291.1 miles per second in vacuum

Energy equivalent to an EM photon

E = hf, where h = Plank's constant 6.626e-34 Js

E = hc/λ (alternate form)

E is the energy of the particle in Joules

f is frequency in Hertz, λ is wavelength in meters

c is speed of light in m/s, aprox 3e8 m/s

1 MeV = 1.6e-13 Joules or 160 fJ

1 Gev = 1.6e-10 Joules or 160 pJ

Momentum of an EM photon

p = hλ,

λ is the photon wavelength.

h = Plank's constant 6.626e-34 Js

Photon energy and momentum are related by

p = Ec

where E = hf = hcλ

The de Broglie equations relate the wavelength λ to the

momentum p, and frequency f to the total energy E of a particle:

λ = h/p

F = e/h

The equation can be equivalently written as

p = h'k

E = h'ω

where h' = h/2π is Dirac's constant

k = 2π/λ angular wavenumber

ω = 2πf angular frequency

Stefan–Boltzmann Law, energy radiated by a black body

P/A = σT⁴

T is absolute temperature in K

σ is the Stefan–Boltzmann constant, σ ≈ 5.67e−8 W/(m²K⁴)

P is energy per unit time, J/s or W

A is the surface area in m²

doppler Δf = v/λ = vf/c

v = velocity of source relative to listener

λ = wavelength of source, relative to source

f = frequency of source, relative to source

c is speed of light in m/s, aprox 3e8 m/s

z = v/c (redshift)

Relativistic doppler

β = v/c

z = Δf/f₀ = (fs – f₀)/f₀ = Δλ/λ₀ = (λ₀ – λs)/λ₀

Δf = f₀(√((1 + β) / (1 – β)) – 1)

z = (1 + β)γ – 1 (redshift)

z = √((1 + β) / (1 – β)) – 1

(only for motion in the direction of line of sight)

z = γ – 1

(only for motion at right angle to the line of sight)

Doppler Radar

fr = ft(c + v)/(c – v)

fr is shifted frequency

ft is original frequency

v is target velocity

c is speed of light

fd = ft – ft = 2vft//(c – v) ≈ 2vft/c

fd is beat frequency or doppler frequency

Color Spectrum

Color .. Frequency ....... Wavelength

UV ...... 780THz-30PHz . 10-380 nm

violet ... 668–780 THz .. 380–450 nm

blue ..... 631–668 THz .. 450–475 nm

cyan .... 606–630 THz .. 476–495 nm

green ... 526–606 THz .. 495–570 nm

yellow . 508–526 THz .. 570–590 nm

orange . 484–508 THz .. 590–620 nm

red ....... 400–484 THz .. 620–750 nm

IR ........ 300GHz-430THz 700nm-1mm

see also Refraction

**Sound**

λf = c (speed of the wave in m/s)

λ = Wavelength in meters

f = frequency in Hz

doppler Δf = v/λ = vf/s

v = velocity of source relative to listener

λ = wavelength of source, relative to source

f = frequency of source, relative to source

s = 343 m/s = speed of sound

s = 331.3√(1+T/273.15) m/s

T is temp ºC

sound is higher pitch on approach

lower at recession

string vibration

v = √(T/μ)

f = (1/(2L))√(T/μ)

v is velocity of propogation

T is tension in string

μ is linear density of the string (mass per unit length)

f is frequency of sound

L is length of string

Speed of sound in air vs temperature

s = (331 + 0.606T) m/s

T is temperature in ºC

A more accurate formula is

s = 20.0457√(T + 273.15)

speed of sound in water is 1484 m/s

speed of sound in salt water is 1533 m/s

speed of sound in copper is 3810 m/s

speed of sound in Hg liquid is 1450 m/s

speed of sound in methanol is 1143 m/s

speed of sound in diamond is 12000 m/s

speed of sound in pyrex glass is 5640 m/s

speed of sound in iron is 5130 m/s

speed of sound in al is 5100 m/s

speed of sound in brass is 4700 m/s

speed of sound in copper is 3560 m/s

speed of sound in gold is 3240 m/s

speed of sound in lead is 1322 m/s

speed of sound in rubber is 1600 m/s

speed of sound in H2 at 0ºC is 1286 m/s

speed of sound in He at 0ºC is 972 m/s

speed of sound in Air at 0ºC is 331 m/s

speed of sound in Air at 20ºC is 343 m/s

**Spacial wave**

wavenumber is the spatial frequency of a wave, either in
cycles per unit distance or radians per unit distance

(analogous to frequency)

(recommended terms: repetency and angular repetency)

For quantum mechanical waves, wavenumber multiplied by
Planck's constant is the canonical momentum.

k = 2π/λ

the number of radians per unit distance, sometimes
termed the angular wavenumber, in 1/meters

λ is wavelength in m