Rain Cape Geometry: A Sizing Rule for a Walking Cape and the Wind Speed at Which No Practical Cape Keeps the Lower Legs Dry

A rain cape protects by geometry: its hem is the edge of a shadow, and everything the wearer needs kept dry must stay inside the shadow that wind-driven rain casts from that edge. This paper derives a sizing rule from that observation. Rain is taken to fall at a terminal speed of 8 m/s, tilted by a headwind and by the walker's own motion of 1.3 m/s, giving a rain angle whose tangent is the apparent horizontal rain speed divided by 8. The walking leg is modelled as a straight segment from hip to ankle, with a reference stride of 0.75 m placing the target point — the leg at 0.25 m above the ground, just above boot top — 0.307 m ahead of the body centre line for a wearer 1.75 m tall. The required hem radius is the target offset plus the horizontal drift of rain falling from hem height to target height, with a clearance condition at hem height also enforced. Length is the dominant defence: a waist-height cape with its hem at 1.00 m already requires a hem radius of 0.616 m in a 2 m/s wind and is impractical at any wind above calm, while the longest practical cape, hem at 0.45 m, needs a radius of 0.440 m (hem circumference 2.76 m) at 4 m/s wind and 0.490 m (circumference 3.08 m) at 6 m/s. Against a stated practical radius limit of 0.55 m, the rule gives a plain negative result: at a stride of 0.75 m, no practical cape keeps the lower legs dry above a wind speed of 8.5 m/s. Shortening the stride to 0.55 m raises that limit to 11.8 m/s; lengthening it to 0.90 m lowers it to 6.0 m/s. Above the limit the failure is geometric, not a matter of fabric: no waterproofing improvement can repair a shadow that no longer covers the leg.