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Waves & Optics Formulas

Complete collection of waves and optics formulas with detailed explanations, notation meanings, units, and real-world applications. Master wave phenomena and light.

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Wave Properties

Wave Equation

y(x,t)=Asin⁡(kx−ωt+ϕ)y(x,t) = A\sin(kx - \omega t + \phi)

General wave function describing displacement as a function of position and time.

Notation:

y(x,t)y(x,t):Displacement at position x and time t
AA:Amplitude
kk:Wave number
xx:Position
ω\omega:Angular frequency
tt:Time
ϕ\phi:Phase constant

Units:

y(x,t)y(x,t):m
AA:m
kk:rad/m
xx:m
ω\omega:rad/s
tt:s
ϕ\phi:rad

Applications:

  • •Sound wave propagation
  • •Light wave description
  • •Ocean wave modeling

Limitations:

Linear waves, small amplitude

Wave Speed

v=λf=ωkv = \lambda f = \frac{\omega}{k}

Wave speed equals wavelength times frequency, or angular frequency divided by wave number.

Notation:

vv:Wave speed
λ\lambda:Wavelength
ff:Frequency
ω\omega:Angular frequency
kk:Wave number

Units:

vv:m/s
λ\lambda:m
ff:Hz
ω\omega:rad/s
kk:rad/m

Applications:

  • •Sound speed calculations
  • •Light speed in media
  • •Wave propagation analysis

Limitations:

Linear waves

Wave Number

k=2πλk = \frac{2\pi}{\lambda}

Wave number equals 2π divided by wavelength.

Notation:

kk:Wave number
λ\lambda:Wavelength

Units:

kk:rad/m
λ\lambda:m

Applications:

  • •Wave phase calculations
  • •Interference patterns
  • •Wave packet analysis

Limitations:

Monochromatic waves

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Wave Phenomena

Doppler Effect

f′=fv±v0v∓vsf' = f\frac{v \pm v_0}{v \mp v_s}

Observed frequency equals source frequency times ratio of wave speed plus/minus observer speed to wave speed minus/plus source speed.

Notation:

f′f':Observed frequency
ff:Source frequency
vv:Wave speed
v0v_0:Observer speed
vsv_s:Source speed

Units:

f′,ff', f:Hz
v,v0,vsv, v_0, v_s:m/s

Applications:

  • •Radar speed detection
  • •Medical ultrasound
  • •Astronomical redshift

Limitations:

Non-relativistic speeds

Wave Interference

I=I1+I2+2I1I2cos⁡(Δϕ)I = I_1 + I_2 + 2\sqrt{I_1I_2}\cos(\Delta\phi)

Total intensity equals sum of individual intensities plus interference term.

Notation:

II:Total intensity
I1,I2I_1, I_2:Individual intensities
Δϕ\Delta\phi:Phase difference

Units:

I,I1,I2I, I_1, I_2:W/m²
Δϕ\Delta\phi:rad

Applications:

  • •Interference patterns
  • •Holography
  • •Wave superposition

Limitations:

Coherent sources

Standing Waves

y(x,t)=2Asin⁡(kx)cos⁡(ωt)y(x,t) = 2A\sin(kx)\cos(\omega t)

Standing wave function equals twice amplitude times sine of wave number times position times cosine of angular frequency times time.

Notation:

y(x,t)y(x,t):Displacement
AA:Amplitude
kk:Wave number
xx:Position
ω\omega:Angular frequency
tt:Time

Units:

y(x,t)y(x,t):m
AA:m
kk:rad/m
xx:m
ω\omega:rad/s
tt:s

Applications:

  • •Musical instruments
  • •Resonant cavities
  • •Wave reflection

Limitations:

Fixed boundary conditions

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Geometric Optics

Snell's Law

n1sin⁡(θ1)=n2sin⁡(θ2)n_1\sin(\theta_1) = n_2\sin(\theta_2)

Product of refractive index and sine of angle in first medium equals product in second medium.

Notation:

n1,n2n_1, n_2:Refractive indices
θ1,θ2\theta_1, \theta_2:Angles of incidence and refraction

Units:

n1,n2n_1, n_2:dimensionless
θ1,θ2\theta_1, \theta_2:rad or degrees

Applications:

  • •Lens design
  • •Prism optics
  • •Fiber optics

Limitations:

Smooth interface

Lens Formula

1f=1u+1v\frac{1}{f} = \frac{1}{u} + \frac{1}{v}

Reciprocal of focal length equals sum of reciprocals of object and image distances.

Notation:

ff:Focal length
uu:Object distance
vv:Image distance

Units:

f,u,vf, u, v:m

Applications:

  • •Camera lenses
  • •Microscopes
  • •Telescopes

Limitations:

Thin lens approximation

Magnification

M=−vu=h′hM = -\frac{v}{u} = \frac{h'}{h}

Magnification equals negative image distance divided by object distance, or image height divided by object height.

Notation:

MM:Magnification
vv:Image distance
uu:Object distance
h′h':Image height
hh:Object height

Units:

MM:dimensionless
v,uv, u:m
h′,hh', h:m

Applications:

  • •Optical instruments
  • •Image analysis
  • •Microscopy

Limitations:

Paraxial rays

✨

Wave Optics

Single Slit Diffraction

sin⁡(θ)=mλa\sin(\theta) = \frac{m\lambda}{a}

Sine of diffraction angle equals order times wavelength divided by slit width.

Notation:

θ\theta:Diffraction angle
mm:Order number
λ\lambda:Wavelength
aa:Slit width

Units:

θ\theta:rad or degrees
mm:dimensionless
λ\lambda:m
aa:m

Applications:

  • •Diffraction gratings
  • •Optical resolution
  • •Wave analysis

Limitations:

Small angles, far field

Double Slit Interference

dsin⁡(θ)=mλd\sin(\theta) = m\lambda

Slit separation times sine of angle equals order times wavelength.

Notation:

dd:Slit separation
θ\theta:Interference angle
mm:Order number
λ\lambda:Wavelength

Units:

dd:m
θ\theta:rad or degrees
mm:dimensionless
λ\lambda:m

Applications:

  • •Young's experiment
  • •Interference patterns
  • •Wave coherence

Limitations:

Coherent sources

Diffraction Grating

dsin⁡(θ)=mλd\sin(\theta) = m\lambda

Grating spacing times sine of angle equals order times wavelength.

Notation:

dd:Grating spacing
θ\theta:Diffraction angle
mm:Order number
λ\lambda:Wavelength

Units:

dd:m
θ\theta:rad or degrees
mm:dimensionless
λ\lambda:m

Applications:

  • •Spectroscopy
  • •Wavelength measurement
  • •Color analysis

Limitations:

Monochromatic light

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Wave Energy and Power

Wave Energy

E=12kA2E = \frac{1}{2}kA^2

Wave energy equals half spring constant times amplitude squared.

Notation:

EE:Wave energy
kk:Spring constant
AA:Amplitude

Units:

EE:J
kk:N/m
AA:m

Applications:

  • •Mechanical waves
  • •Energy transport
  • •Wave power

Limitations:

Simple harmonic motion

Wave Intensity

I=PA=12ρω2A2vI = \frac{P}{A} = \frac{1}{2}\rho\omega^2A^2v

Wave intensity equals power per unit area, or half density times angular frequency squared times amplitude squared times speed.

Notation:

II:Intensity
PP:Power
AA:Area
ρ\rho:Density
ω\omega:Angular frequency
vv:Wave speed

Units:

II:W/m²
PP:W
AA:m²
ρ\rho:kg/m³
ω\omega:rad/s
vv:m/s

Applications:

  • •Sound intensity
  • •Light intensity
  • •Energy flux

Limitations:

Plane waves

Wave Power

P=12μω2A2vP = \frac{1}{2}\mu\omega^2A^2v

Wave power equals half linear density times angular frequency squared times amplitude squared times speed.

Notation:

PP:Power
μ\mu:Linear density
ω\omega:Angular frequency
AA:Amplitude
vv:Wave speed

Units:

PP:W
μ\mu:kg/m
ω\omega:rad/s
AA:m
vv:m/s

Applications:

  • •String waves
  • •Power transmission
  • •Energy transport

Limitations:

One-dimensional waves

Practice Problems

  • 📝Calculate wavelength of 440Hz sound wave in air
  • 📝Find angle of refraction for light entering water at 30°
  • 📝Calculate Doppler shift for 1000Hz source moving at 50m/s

Try Interactive Calculators

Wave PropertiesWave SpeedBeat FrequencyDoppler EffectStanding WavesPhase DifferenceSound Intensity LevelOpen Pipe HarmonicsClosed Pipe HarmonicsWave PowerMach NumberSnell's LawYoung's Double SlitDiffraction Grating

Study Tips

  • 💡Remember wave speed depends on medium
  • 💡Use ray diagrams for geometric optics
  • 💡Pay attention to phase relationships