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Average power of wave on a string

Short answer

Average power = ½ μ v ω² A². Wave speed v = √(50/0.625) = 8.94 m/s, ω = 2π×160 = 1005 rad/s, giving P_avg ≈ 282 W.

A string has a linear density of 625 gm/m and is stretched with a tension 50N. A wave, whose frequency and amplitude are 160Hz and 10mm respectively, is travelling along the string. At what average rate is the wave transporting energy along the string?

Given: μ = 625 g/m = 0.625 kg/m, T = 50 N, f = 160 Hz, amplitude A = 10 mm = 0.010 m.

Formulas

v = √(T/μ)        (wave speed)
ω = 2πf
P_avg = ½ μ v ω² A²

Steps

  1. Wave speed: v = √(50 / 0.625) = √80 = 8.944 m/s.
  2. Angular frequency: ω = 2π × 160 = 1005.3 rad/s, so ω² = 1.0106 × 10⁶ rad²/s².
  3. A² = (0.010)² = 1.0 × 10⁻⁴ m².
  4. Power:
P_avg = ½ × 0.625 × 8.944 × 1.0106×10⁶ × 1.0×10⁻⁴
      = 0.3125 × 8.944 × 101.06
      = 282.5 W

Answer: the wave transports energy at an average rate of about 2.8 × 10² W (≈ 282 W).

✓ Answer checked by Claude Opus 5.5 (high effort)

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