Sound energy flow per square cm

The energy flowing per second across unit area (intensity) for a progressive wave is I = ½ρω²a²v = 2π²n²a²ρv.

Given: n = 256 Hz, a = 0.50 cm = 5×10⁻³ m, v = 330 m/s, ρ = 1.29 kg/m³.

1. Energy density of the wave = ½ρω²a², and energy crossing unit area per second = (energy density) × v.
2. ω = 2πn = 2π × 256 = 1608.5 rad/s
3. Substituting:

I = 2π² n² a² ρ v
  = 2 × (9.8696) × (256)² × (5×10⁻³)² × 1.29 × 330
  = 1.377 × 10⁴ J s⁻¹ m⁻²

4. Convert to per cm²: 1 m² = 10⁴ cm², so I = 1.377×10⁴ / 10⁴ = 1.377 J s⁻¹ cm⁻²

Answer: energy flow ≈ 1.38 J per second across each square cm (1.38 W/cm²).