Chasing the Wind You Made: Fixed-Point Iteration

Thirty seconds into my one good run on the DN yesterday, I hit the thing every new iceboater gets ambushed by: the wind moves. Not the true wind — that sat steady out of the northwest — but the wind I felt, which swung forward and strengthened the faster I went. Ease the sheet to match, the boat accelerates, the felt wind swings forward again, and you chase your own tail until either the trim settles or you skate off the hard-packed snow-ice. (I skated off.)

That chase has a name in numerical computing: fixed-point iteration. You have a quantity that depends on itself — speed sets the apparent wind, apparent wind sets the driving force, driving force sets the speed — and you find where it lands by feeding each guess back into the equation until the output stops moving.

On a beam reach the apparent wind is the hypotenuse of the true wind and your own velocity. Call the sail’s net efficiency k, and terminal speed is the v that satisfies v = k·√(w² + v²):

let w = 10.0, k = 0.9   // true wind (m/s), sail efficiency
var v = 0.0             // boat speed along the runners
for _ in 1...30 {
    v = k * (w*w + v*v).squareRoot()   // v = k · |apparent wind|
}
print(String(format: "terminal speed: %.2f m/s (%.1f× true wind)", v, v/w))

my ($w, $k) = (10, 0.9);   # true wind (m/s), sail efficiency
my $v = 0;                 # boat speed along the runners
$v = $k * sqrt($w**2 + $v**2) for 1 .. 30;
printf "terminal speed: %.2f m/s (%.1fx true wind)\n", $v, $v / $w;

Both print terminal speed: 20.65 m/s (2.1× true wind) — the boat settles above the wind that drives it, which is the whole reason iceboats are worth the frostbite.

Start at v = 0 and watch it climb: 9, 12.1, 14.1, 15.6… each step a smaller correction, because k < 1 makes the map a contraction and the gaps shrink geometrically. That inequality is doing real work. Nudge k toward 1 and the fixed point sprints out to four or five times wind speed before it converges; at k = 1 the iteration never settles at all — roughly what my body was reporting when it gave up on the turn.