Logarithmic Spiral Beaches

Beaches which can be modeled by a logarithmic equation are often called log spiral beaches. They have also historically been described as ‘zeta cure bays’, ‘half heart’ or ‘crenulate’ shaped bays, or ‘headland bays’.
The curve can be modeled using the equation:
r = e^θcotɑ
where:
r = radius,
θ = the angle of rotation, and
ɑ = the angle between radius and tangent (constant for any given logarithmic spiral)

In order for this type of beach to form you must have a stretch of coast made up of alternating bedrock headlands and sandy bays backed by weak sediment. Log spiral beaches are often on swell-dominated coasts and are established by the refraction of approaching waves and their diffraction by the rocky headlands. The headlands alter the prevailing direction of swell approach and interrupt the movement of sediment being carried by longshore currents. Waves must generally approach the shoreline from one main direction and the angle of approach of the waves must be oblique. Increase in sand size, wave height, berm height, and swash zone gradient from the up coast headland generally characterize the concave seaward curved part of the beach. The ‘tangent section’ forms so that it is parallel with the refracted dominate swell. Examples of log spiral beaches include Birch Bay in Washington, Half Moon Bay in California and Pearl Beach in New South Wales.

References

Chapman, D. M., Dr. "Zetaform of Logarithmic Spiral Beach." Australian Geographer
14.1 (1978): 44-45.LeBlond, Paul H. "An Explanation of the Logarithmic Spiral Plan Shape of Headland
Bay 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.