The Outdoor Sound Shadow: A Refraction Bound on How Far an Outdoor Source Can Be Heard

How far away an outdoor sound can be heard is often estimated from spherical spreading alone. That estimate omits refraction: in air whose sound speed varies with height, sound rays curve, and when the sound speed decreases with height the rays curve upward and leave a shadow zone near the ground into which no direct ray reaches. This paper computes the shadow boundary, and the resulting limit on audible range, for stated temperature and wind gradients. For a source 3.0 m above the ground and a listener at ear height 1.7 m, a strong sunny-afternoon near-surface gradient of -0.15 (m/s)/m gives a ray curvature radius of 2,287 m and a shadow boundary at 205 m. A 100 dB(A) source (level at 1 m) that would reach the 40 dB(A) level at 677 m in still, neutral air is therefore cut off at 205 m - a factor of 3.30 reduction. Upwind propagation in a 5 m/s wind shortens the boundary to 162 m, and the two effects combined shorten it to 127 m. The shadow distance contains no source-power term: raising the source level to 110, 120, or 130 dB(A) - 10, 100, and 1,000 times the acoustic power - leaves the sunny-afternoon boundary at 205 m in every case. Beyond the shadow boundary, extra loudspeaker power buys almost nothing under lapse conditions. Numerical ray tracing reproduces the closed-form shadow distance to within 0.1% in every upward-refracting case computed.