Problem 28 - Applied Calculus

Question

The Bay of Fundy in Canada has the largest tides in the world. The difference between low and high water levels is 15 meters (nearly 50 feet). At a particular point the depth of the water, \(y\) meters, is given as a function of time, \(t,\) in hours since midnight by $$y=D+A \cos (B(t-C))$$ (a) What is the physical meaning of \(D ?\) (b) What is the value of \(A ?\) (c) What is the value of \(B ?\) Assume the time between successive high tides is 12.4 hours. (d) What is the physical meaning of \(C ?\)

Step-by-Step Solution

Answer

(a) Average height; (b) 7.5 meters; (c) \( \frac{2\pi}{12.4} \); (d) Time of first high tide.

Step 1: Understanding D - Mean Water Level

In the equation \( y = D + A \cos(B(t-C)) \), \( D \) represents the average or midline depth of the water level. It is the water level when the tide is neither at its high point nor at its low point. Physically, it is the halfway point between the high and low tides.

Step 2: Calculating A - Amplitude of the Tide

The amplitude \( A \) represents the maximum deviation from the mean water level \( D \). Given the difference between the high tide and low tide is 15 meters, the amplitude is half of this, as it accounts for the deviation from the mean. Thus, \( A = \frac{15}{2} = 7.5 \) meters.

Step 3: Calculating B - Frequency of the Tide Cycle

The parameter \( B \) determines the frequency of the tidal cycle. Given the time between successive high tides is 12.4 hours, this is the period of the cosine function. The mathematical relationship is \( B = \frac{2\pi}{ ext{Period}} \), so \( B = \frac{2\pi}{12.4} \).

Step 4: Understanding C - Phase Shift of the Tide

The parameter \( C \) indicates the phase shift, representing the horizontal shift of the cosine function along the time axis. Physically, it indicates when, in terms of hours from midnight, the first high tide occurs.

Key Concepts

Amplitude of Tides

The amplitude of tides, represented by the symbol \( A \) in the tidal motion equation, is the measure of how much the water level rises or falls compared to the average sea level. In mathematical terms, the amplitude indicates the maximum extent of variation from the mean water level.

For the Bay of Fundy, where the difference between high tide and low tide is 15 meters, the amplitude is half of this total height difference because the tide goes equally high above and below the average sea level. Imagining a wave, the amplitude is from the center of the wave to the tallest or deepest point. Therefore, the amplitude in this case is \( 7.5 \) meters.

Understanding amplitude helps in planning maritime activities. Knowing the change in tide heights assists in navigation, port operations, and even environmental protection.

Tide Frequency

Tide frequency is determined by how often a complete cycle of tides occurs. This cycle is marked by the pattern of moving from one high tide to the next. In mathematical equations involving tides, such as \( y = D + A \cos(B(t-C)) \), the frequency is represented by \( B \).

In the case of the Bay of Fundy, we know that successive high tides occur every 12.4 hours. The frequency \( B \), can be calculated using the formula \( B = \frac{2\pi}{\text{Period}} \). Plugging in our period of 12.4 hours, we can see how frequently the tides repeat. This information is crucial for predicting tides and managing time-sensitive activities like fishing or operations at coastal facilities.

Predicting tide frequency allows communities to mitigate potential impacts of extreme tides, safeguard infrastructures and plan marine resource utilizations effectively.

Phase Shift

The phase shift, represented by \( C \) in the equation \( y = D + A \cos(B(t-C)) \), denotes the horizontal shift of the tide cycle along the time axis. Essentially, it tells us when the high tide happens, relative to a standardized time, in this case, midnight.

Understanding \( C \) aids in timing the peak tides in a given location. If you were observing how the tide varies throughout the day or planning activities around specific water levels, knowing the phase shift is crucial. It shows how many hours after midnight the tide reaches its high point, helping adjust expectations and activities accordingly.

Monitoring and utilizing phase shift data ensures safety, efficient use, and adaptation to the natural rhythm of tidal changes. This component of the tidal equation allows accurate tide predictions which are significant for coastal management and planning.