Mechanical Wave Energy Calculator
Enter medium density, angular frequency, amplitude, wave speed and volume to compute wave energy density E=½ρω²A², intensity I=E·v and total energy. ρ=1.2, ω=1, A=1, v=340 → E=0.6 J/m³, I=204 W/m².
Input Data
Results
At a glance:Mechanical wave energy density: for a traveling wave in an elastic medium, the sum of kinetic and potential energy density per unit volume is E=½·ρ·ω²·A², where ρ is the medium density (kg/m³), ω the angular frequency (rad/s, ω=2πf) and A the amplitude (m). The wave intensity (energy flux density) I=E·v=½·ρ·ω²·A²·v, in W/m², is the energy passing per unit time through unit area perpendicular to propagation, with v the wave speed. The total energy in a volume V is E_total=E·V. History: Lord Rayleigh's 1877 'Theory of Sound' systematically derived wave-energy formulas; the vector form of energy flux is the Poynting vector (electromagnetic) or acoustic intensity vector (sound). Classic example: ρ=1.2 kg/m³ (air), ω=1 rad/s, A=1 m, v=340 m/s → E=0.6 J/m³, I=204 W/m². The human hearing threshold is 1e-12 W/m² (A≈1e-11 m), the pain threshold 1 W/m² (A≈0.01 m). Applications: (1) acoustics — decibels, noise control; (2) seismology — seismic wave energy; (3) ultrasound — medical imaging, cleaning; (4) water waves — wave-power generation; (5) musical instruments — string/pipe energy.
Formula
Energy density: E = ½·ρ·ω²·A²
Wave intensity: I = E·v = ½·ρ·ω²·A²·v
Total energy: E_total = E·V
Angular frequency: ω = 2π·f
Wave speed: v = λ·f = ω/k
$$E = \tfrac{1}{2}\rho\omega^{2}A^{2}, \quad I = E v = \tfrac{1}{2}\rho\omega^{2}A^{2}v, \quad E_{\text{total}} = E V$$How to Use
- Enter medium density ρ (kg/m³), angular frequency ω (rad/s), amplitude A (m), wave speed v (m/s) and volume V (m³).
- The tool computes E=½ρω²A², I=E·v and E_total=E·V.
- Typical: ρ=1.2, ω=1, A=1, v=340 → E=0.6 J/m³, I=204 W/m²; hearing threshold I=1e-12 W/m².
Case Studies
Sound intensity in air
ρ=1.2, ω=1, A=1, v=340 → E=0.6 J/m³, I=204 W/m².
Hearing threshold I=1e-12 W/m² corresponds to A≈1e-11 m.
Pain threshold I=1 W/m² corresponds to A≈0.01 m.
Ultrasound imaging
Higher ω and A raise intensity sharply (I∝ω²A²).
Medical ultrasound uses MHz frequencies for fine resolution.
Intensity limits protect tissue from heating.
FAQ
What is wave energy density?
The energy per unit volume a traveling wave carries: E=½ρω²A², with ρ the medium density, ω the angular frequency and A the amplitude.
How is intensity related to energy density?
Intensity I=E·v is the energy crossing unit area per unit time, equal to the energy density times the wave speed v.
Why does intensity depend on ω² and A²?
Both the kinetic energy (∝v_particle², and v_particle∝ωA) and potential energy density scale with ω²A², so I∝ρω²A²v.
What is the hearing threshold?
About 1e-12 W/m² (for 1 kHz sound), corresponding to a displacement amplitude of ~1e-11 m — remarkably small, yet detectable by the ear.
Related Tools
References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.