More precisely, engineers assume that the solid satisfies a linear integro-differential
equation, the Cummins equation. It was empirically derived in [34], in which the
author dealt with ship motions using a linear model, supposing the conservation
of the energy and the balance of forces. The experimental data coming from the
return to equilibrium problem, called decay test in the hydrodynamical engineering
context, are then used to identify the coefficients of this linear equation.
John in [75] studied the problem in shallow water in one horizontal dimension for
an object with flat bottom: he considered the linearized fluid equations for small
amplitude waves and he wrote an explicit expression for the solid motion under
linear approximation. Ursell in [120] and Maskell and Ursell in [90], using like John
the linear approach, obtained an explicit solution in integral form for the vertical
displacement of the object. From Wehausen and Laitone [123] we know that also
Sretenskii, several years before Cummins, obtained an integro-differential equation
for the vertical displacement which he solved numerically. The Cummins equation
for the vertical displacement reads
ˆ t
(m + a∞ ) δ̈G (t) = −cδG (t) −
K(τ )δ̇G (t − τ )dτ,
(III.13)
0

where δG (t) = zG (t) − zG,eq is the displacement from the equilibrium position of
the vertical position of the center of mass, m is the mass of the structure, a∞ is
the added mass at infinity frequency, c is the hydrostatic coefficient and K is the
impulse response function (also known as retardation function and fluid memory).

### Language ###
English

### Assessment ###
The text discusses an equation called "Cummins equation" and who derived it first, which could be a natural point of historical inquiry.

### Query ###
Who first discovered the Cummins equation?
Who first discovered the Cummins equation?