In the course of his work as a lighthouse and harbour engineer, Stevenson had made observations of wave heights at various locations in Scotland over a number of years. In 1852, he published a paper in which he suggested that waves increased in height by a ratio approximate to the square root of their distance from the windward shore. Stevenson developed this into the simple formula
H
=
1.5
F
{\displaystyle H=1.5{\sqrt {F}}}
, in which
H
{\displaystyle H}
is the wave height in feet and
F
{\displaystyle F}
is the fetch in miles.
Essential components for wave height prediction, most notably wind speed, are missing from Stevenson's formula. In 1852, mathematical analysis of the theory of water waves, and methods for numerical assessment of factors such as shoaling and surge, were in their infancy. Stevenson's analysis is possibly the first quantitative discussion of wave height as a (square root) function of fetch, and his paper is one of the first quantitative studies of wind speeds in the planetary boundary layer.
Modern analysis of Stevenson's formula indicates that it appears to conservatively estimate wave heights for wind speeds up to around 30 miles per hour, being based on his observations which most likely were taken for fetch lengths under 100 kilometres, without fully developed seas. The breakwater at Wick was exposed to a fetch length of approximately 500 kilometres, and wind speeds far in excess of 30 miles per hour, prior to its eventual destruction.
In 1965, the South African engineer Basil Wrigley Wilson proposed a method which can be used to approximate the significant wave height H1/3 and period T1/3 of wind waves generated by a constant wind of speed U blowing over a fetch length F. The units for these quantities are as follows:
H1/3 in metres (m)
T1/3 in seconds (s)
U in metres per second (m/s)
F in metres (m)
Wilson's formulae apply when the duration of the wind blowing is sufficiently long, as when the wind blows for only a limited time, waves cannot attain the full height and period corresponding to the wind speed and fetch length. Under conditions were the wind blows for a sufficiently long time, for example during a prolonged storm, the wave height and period can be calculated as follows:
g
H
1
/
3
/
U
2
=
0.30
{
1
−
[
1
+
0.004
(
g
F
/
U
2
)
1
/
2
]
−
2
}
{\displaystyle gH_{1/3}/U^{2}=0.30\left\{1-\left[1+0.004\left(gF/U^{2}\right)^{1/2}\right]^{-2}\right\}}
g
T
1
/
3
/
(
2
π
U
)
=
1.37
{
1
−
[
1
+
0.008
(
g
F
/
U
2
)
1
/
3
]
−
5
}
{\displaystyle gT_{1/3}/(2\pi U)=1.37\left\{1-\left[1+0.008\left(gF/U^{2}\right)^{1/3}\right]^{-5}\right\}}
In these formulae, g denotes the acceleration due to gravity, which is approximately 9.807 m/s2. The wind speed U is measured at an elevation of 10 metres above the sea surface. For conditions approximate to those for the Wick breakwater during a storm (fetch length of 500km, wind speed of around 75mph), the graph below shows that Wilson's method predicts a significant wave height (H1/3) of around 1.5 times that of Stevenson's.
Nonetheless, whilst Stevenson's formula is highly limited and unsuitable for engineering design application, it was notable for being an early attempt to apply mathematical theory to hydraulic engineering problems, and shows some limited agreement (albeit within a narrow range) with a more advanced formula developed by Ramón Iribarren in 1942. A major flaw in Stevenson's formula is the absence of consideration of wind speed, and comparison with Wilson's formula at 3 different wind speeds (30, 50 and 75mph) shows only a reasonable level of agreement for 50mph winds at fetch lengths up to around 100 metres.
Stevenson himself noted that the formula was an approximation, and actively encouraged further research into similar problems, imploring young engineers to redouble efforts in the advancement of coastal engineering during an 1885 address to the Institution of Civil Engineers in London. In addition to his work on wave growth, he also undertook research into the phenomenon of wave decay inside harbour basins.
