Logarithmic spiral beaches

A logarithmic spiral beach is a type of beach which develops in the direction under which it is sheltered by a headland, in an area called the shadow zone. It is characterized as a logarithmic spiral because if you look at it in plan view or aerially, it represents the same shape that is created from the logarithmic spiral relation. These beaches are also commonly referred to as ‘half heart’ or ‘crenulate’ shaped bays, or ‘headland bays’.

Logarithmic spiral relation

Logarithmic Spiral

The logarithmic spiral can be determined using the equation (written in polar coordinates):

${\displaystyle r=e^{\theta \cot \alpha }}$

where:

${\displaystyle \theta }$ = the angle of rotation, is located between two lines drawn from the origin to any two points on the spiral.

${\displaystyle r}$ = the ratio of the lengths between two lines that extend out from the origin. The two lines are given as ${\displaystyle R_{O}}$ and ${\displaystyle R}$. So ${\displaystyle r}$ also equals the ratio ${\displaystyle R/R_{O}}$.

${\displaystyle \alpha }$ = the angle between any line ${\displaystyle R}$ from the origin and the line tangent to the spiral which is at the point where line ${\displaystyle R}$ intersects the spiral. ${\displaystyle \alpha }$ is a constant for any given logarithmic spiral.

Spiral development

This type of beach forms due to the refraction of approaching waves and their diffraction by an upcoast headland. The approaching wave front curves as a result of wave diffraction at the headland, which in turn causes the shoreline to bend and yield a log spiral shape. Log spiral beaches are often on swell-dominated coasts where waves generally approach the shoreline from one main direction at an oblique angle. The oblique approaching waves refract and diffract into the "shadow zone" which can be considered a relatively sheltered hook of beach behind the headland. Increase in sediment size, wave height, berm height, and swash zone gradient from the up coast headland generally characterizes the concave seaward curved part of the beach.

Famous logarithmic spiral beaches

Half Moon Bay State Beach, California

References

• Chapman, D. M., Dr. "Zetaform of Logarithmic Spiral Beach." Australian Geographer 14.1 (1978): 44-45.
• Kimberley, M. M. "Fitting a Logarithmic Spiral to the Shoreline of a Headland-Bay Beach" Computers & Geoscience 15 No. 7 (1989): 1089-1108.
• LeBlond, Paul H. "An Explanation of the Logarithmic Spiral Plan Shape of HeadlandBay Beaches." Journal of Sediment Petrology 49.4 (Dec. 1979): 1093-1100.
• Meeuwis, June, and P.A.J. Van Rensburg. "Logarithmic Spiral Coastlines: The Northern Zululand Coastline." The South African Geographical Journal 68.1 (1986): 18- 43.