Docking with Tides

Did this simple interactive upon request by a colleague who is teaching the JC1 topic of Oscillations.

Based on the following question, this is used as a quick visual to demonstrate why there must be a minimum depth before the boat approaches harbour.

The rise and fall of water in a harbour is simple harmonic. The depth varies between 1.0 m at low tide and 3.0 m at high tide. The time between successive low tides is 12 hours. A boat, which requires a minimum depth of water of 1.5 m, approaches the harbour at low tide. How long will the boat have to wait before entering?

The equation of the depth of water H based on the amplitude of the tide a can be given by $H=H_o+a\cos \omega t$ where $H_o$ is the average depth of the water.

$H=H_o+a\cos \omega t$

When H = 1.5m,

$1.5=2.0–1.0\cos ({\displaystyle \frac{2\pi }{12}}t)$

$\cos ({\displaystyle \frac{2\pi }{12}}t)=0.5$

$t=2.0h$