and if the liquid was incompressible, the depth at pressure p would be (p − p0)/p0, so that the lowering of the surface due to compression is
kez/k − k − z = ½z2/k, when k is large.
(17)
For sea water, λ is about 25,000 atmospheres, and k is then 25,000 times the height of the water barometer, about 250,000 metres, so that in an ocean 10 kilometres deep the level is lowered about 200 metres by the compressibility of the water; and the density at the bottom is increased 4%.
On another physical assumption of constant cubical elasticity λ,
dp = λdρ/ρ, (p − p0)/λ = log (ρ/ρ0),
(18)
| dp | = | λ | dρ | = ρ, λ ( | 1 | − | 1 | ) = z, 1 − | ρ0 | = | z | , λ = kρ0, | |
| zd | ρ | dz | ρ0 | ρ | ρ | k |
(19)