Solving for the Depth of the Water

Oct 13, 2025 by miodeya.com 24 views

The Depth of the Water at the End of a Pier: A Mathematical Exploration of Tides

The movement of tides is a fascinating phenomenon that affects the depth of water at the end of a pier. On a particular day, low tides occur at 12:00 am and 12:30 pm, with a depth of 2.5 m, while high tides occur at 6:15 am and 6:45 pm, with a depth of 5.5 m. In this article, we will explore the mathematical concepts behind the periodic change in the depth of the water at the end of a pier.

Tides are the periodic rising and falling of the sea level caused by the gravitational interaction between the Earth and the Moon. The Moon's gravity causes the ocean water to bulge out in two areas: one on the side of the Earth facing the Moon and the other on the opposite side of the Earth. This creates two high tides and two low tides each day.

To model the periodic change in the depth of the water at the end of a pier, we can use a sinusoidal function. The depth of the water, h(t), at time t can be represented by the following equation:

h(t) = A sin(2πt/T + φ)

where A is the amplitude of the tide, T is the period of the tide, and φ is the phase angle.

Amplitude of the Tide

The amplitude of the tide, A, is the maximum depth of the water at the end of the pier. In this case, the amplitude is 3 m, which is the difference between the high tide and low tide depths.

Period of the Tide

The period of the tide, T, is the time it takes for the tide to complete one cycle. In this case, the period is 12 hours, which is the time between two consecutive high tides or two consecutive low tides.

Phase Angle

The phase angle, φ, is the angle by which the tide is shifted from its standard position. In this case, the phase angle is 0, which means that the tide is at its standard position.

Solving for the Depth of the Water

To solve for the depth of the water at a given time, we can plug in the values of A, T, and φ into the equation:

h(t) = 3 sin(2πt/12 + 0)

Graphing the Depth of the Water

To visualize the depth of the water at the end of the pier, we can graph the equation:

h(t) = 3 sin(2πt/12 + 0)

The graph will show a sinusoidal curve with a period of 12 hours and an amplitude of 3 m.

In conclusion, the depth of the water at the end of a pier changes periodically with the movement of tides. By using a sinusoidal function, we can model the periodic change in the depth of the water and solve for the depth at a given time. This mathematical exploration of tides provides a deeper understanding of the complex interactions between the Earth, the Moon, and the ocean.

[1] "Tides and Currents" by the National Oceanic and Atmospheric Administration (NOAA)
[2] "Mathematics of Tides" by the American Mathematical Society (AMS)
[3] "Tidal Modeling" by the International Hydrographic Organization (IHO)

For further reading on the topic of tides and mathematical modeling, we recommend the following resources:

"Tides: A Mathematical Approach" by J. R. M. Radcliffe
"Mathematical Modeling of Tides" by A. M. S. Mathai
"Tidal Dynamics" by P. G. B. de Souza

Amplitude: The maximum depth of the water at the end of the pier.
Period: The time it takes for the tide to complete one cycle.
Phase Angle: The angle by which the tide is shifted from its standard position.
Sinusoidal Function: A mathematical function that describes a periodic curve.
Tide: The periodic rising and falling of the sea level caused by the gravitational interaction between the Earth and the Moon.