Wave Equation Calculator

Calculate wave properties and visualize waveforms

Wave Parameters

Enter characteristics of the wave

m
Amplitude must be a positive number
Maximum displacement
m
Wavelength must be a positive number
Distance between peaks
Hz
Frequency must be a positive number
Oscillations per second
m/s
Wave speed must be a positive number
Speed of wave propagation
rad
Starting angle
s
Time must be a non-negative number
Time since wave began
Wave Number (k)
3.14
rad/m
Angular Frequency (ω)
6.28
rad/s
Period (T)
1.00
s
Wave Equation
y = sin(3.14x - 6.28t)
Wave function
Understanding Wave Equations

The wave equation describes how waves propagate through different media. For a sinusoidal wave traveling in the x-direction, the wave function is:

  • Wave Function: y(x,t) = A sin(kx - ωt + φ)
  • Wave Number: k = 2π/λ (rad/m)
  • Angular Frequency: ω = 2πf (rad/s)
  • Wave Speed: v = fλ = ω/k
  • Period: T = 1/f
  • Waves transfer energy without transferring matter
  • Wave speed depends on the medium properties
Wave Equation Formulas
y(x,t) = A sin(kx - ωt + φ)
k = 2π/λ
ω = 2πf
v = fλ = ω/k
T = 1/f
Where:
y = Wave displacement (m)
A = Amplitude (m)
k = Wave number (rad/m)
ω = Angular frequency (rad/s)
t = Time (s)
x = Position (m)
φ = Phase constant (rad)
λ = Wavelength (m)
f = Frequency (Hz)
v = Wave speed (m/s)
T = Period (s)
Wave Examples
Wave Type Frequency Wavelength Speed
Ocean wave 0.1 Hz 100 m 10 m/s
Sound wave (air) 440 Hz 0.78 m 343 m/s
Light wave (red) 4.3×10¹⁴ Hz 700 nm 3×10⁸ m/s
Radio wave (FM) 100 MHz 3 m 3×10⁸ m/s
Seismic wave 0.05 Hz 20 km 1000 m/s

Wave Behavior and Phenomena

1

Reflection: Waves bouncing off surfaces. Examples: Echoes, mirrors.

2

Refraction: Waves changing direction when entering different media. Examples: Lens focusing, mirages.

3

Diffraction: Waves spreading out when passing through openings. Examples: Sound around corners, CD rainbow patterns.

4

Interference: Waves combining constructively or destructively. Examples: Noise-canceling headphones, soap bubbles.

Frequently Asked Questions

Frequency and wavelength are inversely proportional through the wave equation:

v = f × λ

Where v is the wave speed. This means:

  • Higher frequency → Shorter wavelength
  • Lower frequency → Longer wavelength

For electromagnetic waves in vacuum, v = c (speed of light), so:

c = f × λ

This relationship explains why radio waves have long wavelengths (meters to kilometers) while gamma rays have extremely short wavelengths (picometers).

Wave speed depends on the properties of the medium:

  • Sound Waves:
    • v = √(B/ρ) for liquids and gases
    • v = √(Y/ρ) for solids
    • Where B = bulk modulus, Y = Young's modulus, ρ = density
  • Light Waves:
    • v = c/n
    • Where c = speed in vacuum, n = refractive index
  • String Waves:
    • v = √(T/μ)
    • Where T = tension, μ = linear density
  • Water Waves:
    • v = √(gλ/2π) for deep water
    • Where g = gravity, λ = wavelength

Generally, wave speed increases with stiffness and decreases with density.

Phase velocity and group velocity describe different aspects of wave propagation:

Parameter Phase Velocity Group Velocity
Definition Speed of a single wave crest Speed of wave packet envelope
Formula v_p = ω/k v_g = dω/dk
Relation v_g = v_p + k(dv_p/dk)
Dispersion Can exceed c in some media Speed of information transfer
Application Monochromatic waves Wave packets, signals

In non-dispersive media (like vacuum for light), v_p = v_g. In dispersive media (like glass for light), v_p ≠ v_g.

Standing waves form when two waves of the same frequency and amplitude travel in opposite directions and interfere:

  • Resulting wave pattern appears stationary
  • Characterized by nodes (no displacement) and antinodes (max displacement)
  • Formation requires:
    • Wave reflection at boundaries
    • Specific frequencies (resonance)

Standing wave equation:

y(x,t) = 2A sin(kx) cos(ωt)

Applications:

  • Musical instruments (strings, wind instruments)
  • Laser cavities
  • Microwave ovens
  • Quantum wavefunctions