Harmonic Wave Equation Calculator
Calculate displacement, velocity, and acceleration for harmonic waves using the wave equation
Calculate Harmonic Wave Parameters
Maximum distance from equilibrium position
Spatial period of the wave
Speed at which the wave travels
Phase offset at t=0, x=0
Location along the wave
Instant in time for calculation
Harmonic Wave Results
Frequency: 0.000 Hz
Period: 0.000000 s
Angular Frequency: 0.000 rad/s
Wave Number: 0.000 rad/m
Wave Velocity: 0.0 m/s
Wavelength: 0.000 m
Wave Equation:
y = 0.000 sin(2π/0.000 × (x - 0.0t) + 0.000)
Harmonic Wave Formulas
Example Calculation
Finding Wavelength from Displacement Measurements
Given: A = 14 mm, t = 1 s, φ = 0
At x = 0 mm: y = -7 mm
At x = 10 mm: y = 7 mm
From first point: -7 = 14 sin(-2πv/λ)
From second point: 7 = 14 sin(2π(10-v)/λ)
Solution: v = 5 mm/s, λ = 60.24 mm
Standard Wave Example
Given: y = 0.07 sin(2π/0.4 × (x - 320t) + π/6)
Amplitude: A = 0.07 m
Wavelength: λ = 0.4 m
Velocity: v = 320 m/s
Phase shift: φ = π/6 rad
Common Wave Types
Sound Waves (Air)
v ≈ 343 m/s, f: 20 Hz - 20 kHz
Water Waves
v varies, λ: 1 cm - 100 m
Light Waves
v = 3×10⁸ m/s, λ: 400-700 nm
Radio Waves
v = 3×10⁸ m/s, f: kHz-GHz
Wave Properties
Wave Equation Tips
Displacement is sinusoidal function of position and time
Phase determines wave's starting position
Particle velocity leads displacement by π/2
Wave speed equals frequency × wavelength
Maximum particle speed = A × 2πf
Understanding Harmonic Wave Equations
What is a Harmonic Wave?
A harmonic wave is a sinusoidal wave that propagates through space and time. It represents the simplest form of wave motion where particles oscillate in simple harmonic motion as the wave passes through them.
Wave Properties
- •Amplitude (A): Maximum displacement from equilibrium
- •Wavelength (λ): Distance between successive peaks
- •Frequency (f): Number of oscillations per second
- •Phase (φ): Initial phase offset
Harmonic Wave Equation
y = A sin(2π/λ × (x - vt) + φ)
- y: Displacement at position x, time t
- A: Amplitude (maximum displacement)
- λ: Wavelength (spatial period)
- x: Position along the wave
- v: Wave velocity (propagation speed)
- t: Time
- φ: Initial phase angle
Key Insight: The term (x - vt) represents the wave traveling in the positive x direction. For waves traveling in the negative x direction, use (x + vt).
Applications & Examples
Sound Waves
Air pressure oscillations that propagate as longitudinal waves, carrying audio information through compression and rarefaction.
Water Waves
Surface waves on water bodies where particles move in circular or elliptical paths as the wave energy propagates.
Electromagnetic Waves
Light, radio, and other EM waves where electric and magnetic fields oscillate perpendicular to propagation direction.