Logarithmic spiral beach

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 shaped like a logarithmic spiral when seen in a map, plan view, or aerial photograph. These beaches are also commonly referred to as half heart beach, crenulate-shaped bay, or headland-bay beach.

Logarithmic spiral relation

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

r

=

e

θ

cot

⁡

α

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

where:

θ

{\displaystyle \theta }

= the angle of rotation, is located between two lines drawn from the origin to any two points on the spiral.

r

{\displaystyle r}

= the ratio of the lengths between two lines that extend out from the origin. The two lines are given as

R

O

{\displaystyle R_{O}}

and

R

{\displaystyle R}

.

So

r

{\displaystyle r}

also equals the ratio

R

/

R

O

{\displaystyle R/R_{O}}

.

α

{\displaystyle \alpha }

= the angle between any line

R

{\displaystyle R}

from the origin and the line tangent to the spiral which is at the point where line

R

{\displaystyle R}

intersects the spiral.

α

{\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.