Why depth rating is not a linear scale

Why doubling the wall does far more than doubling the depth.

Depth rating is not a linear scale because a cylinder's buckling pressure grows with the cube of its wall thickness while the pressure of the surrounding water grows only in direct proportion to depth. Doubling how deep a housing needs to go requires nowhere near doubling the wall, and doubling the wall buys far more than double the depth, the two figures moving at very different rates.

A straight line meets a cube

Water pressure rises steadily with depth, by about one atmosphere for every ten metres of sea water, so twice the depth means twice the pressure with no surprises anywhere along the way. It would be reasonable to expect the wall needed to resist that pressure to scale just as simply. Buckling resistance refuses to cooperate, because the relationship governing it raises the wall thickness to the third power. A modest increase in wall delivers a disproportionately large increase in the pressure the housing survives, and a modest reduction gives up far more capability than the reduction itself suggests.

Put numbers on it and the gap is stark. To double a housing's buckling pressure, and so its depth rating, the wall has to grow by the cube root of two, about 26 percent. Doubling the wall raises the buckling pressure eightfold. Taking a housing from a hundred metres to a thousand metres, ten times the pressure, needs a wall only a little over twice as thick, provided the diameter and material stay the same and the wall stays thin enough to fail by buckling. Sizing by simple proportion, assuming a rating twice as deep needs a wall twice as thick, therefore over-builds a housing by a wide margin, and the error widens with every extra step of depth demanded because the exponent is three.

Snowballs and cubes

Rolling a snowball across a lawn until its diameter doubles takes about eight times as much snow, even though the ball only looks twice as large from across the garden, because the snow needed to fill a ball grows with the cube of its size. The same mismatch links wall thickness to buckling pressure, which is why a wall that looks only slightly thicker on a drawing can hide a buckling pressure that has risen by a much larger proportion.

The cube cuts both ways. A designer trimming the wall to save weight loses collapse resistance far faster than the saved weight suggests: a wall made ten percent thinner keeps only about 73 percent of its buckling pressure, a loss of more than a quarter. Weight does not share the cubic generosity either, since a thicker wall costs weight in direct proportion to its extra thickness, so the payoff is specifically in collapse resistance per millimetre of wall.

Rounding ratings up, stretching old designs

The asymmetry shapes how a depth rating is set. A housing is rarely built to the exact minimum wall its target depth calculates to, because a little extra thickness buys a disproportionately large margin against buckling for a small weight cost, a trade a designer working from linear intuition would never think to make so cheaply. The same reasoning explains why quoted depth ratings so often sit at round, rounded-up numbers, since the last small step of wall needed to reach one costs little weight and a rating that has been rounded up carries its extra margin almost for free.

Upgrades benefit too. A housing rated for a shallower depth can often be pushed meaningfully deeper with a modest increase in wall thickness, far less than the proportional increase a linear guess would predict, which helps anyone stretching an existing design to a new mission instead of starting from a blank sheet.

The crossover into crushing

The cubic relationship holds only within the buckling-dominated regime this set has been describing. Once a wall grows thick enough, or a diameter small enough, that the housing fails by simple material crushing, the relationship reverts to something close to the straight line ordinary intuition expects, and wall added past that crossover buys steadily less per unit of weight. Most working pressure housings sit well inside the buckling regime, since a wall thick enough to crush first is usually far heavier than the job calls for, so the cubic sensitivity governs almost every real depth rating. The articles on windows and on testing come back to it from other angles, since it makes some of the hardest parts of a housing hard for reasons that trace straight back to this cube.