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,
$1.5 = 2.0 – 1.0 \cos (\dfrac{2 \pi}{12}t)$
$\cos (\dfrac{2 \pi}{12}t) = 0.5$
$t = 2.0 h$