Problem 43 - 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 depth; (b) 7.5 meters; (c) \(\approx 0.506\); (d) Phase shift (time of high tide from midnight).

Step 1: Understanding the Function

The given equation is a cosine function representing tidal depth over time, where:- \(D\) adjusts the vertical shift of the water level.- \(A\) represents the amplitude, or half the difference between high and low tides.- \(B\) affects the period of the tidal cycle.- \(C\) represents a phase shift, or the time at which the maximum or minimum occurs.

Step 2: Interpretation of D

The parameter \(D\) is the average water depth or the midpoint of the tidal range. This is because the function oscillates above and below \(D\) as the cosine function varies between -1 and 1.

Step 3: Finding the Value of A

Since the difference between high and low water levels is 15 meters, the amplitude \(A\) is half that difference. Therefore:\[ A = \frac{15}{2} = 7.5 \text{ meters} \].

Step 4: Determining B from the Tidal Period

The period of the cosine function is given by \(T = \frac{2\pi}{B}\). Given the time between high tides is 12.4 hours:\[ \frac{2\pi}{B} = 12.4 \]. Solving for \(B\), we have:\[ B = \frac{2\pi}{12.4} \approx 0.506 \].

Step 5: Interpretation of C

The parameter \(C\) determines the horizontal shift in the cosine function. Physically, it indicates the time offset from midnight when the first maximum (high tide) occurs. A zero or particular value of \(C\) would shift the graph left or right, indicating exactly when the high tide happens relative to an initial time point.

Key Concepts

Amplitude

In tidal functions, the amplitude is a crucial element that signifies half of the total fluctuation between high and low tides. It represents the height from the average or midpoint water level to either the high or low tide.
Understanding the amplitude helps in predicting tidal movements and water levels at any given time.

Here are some key points to note about amplitude in tidal functions:

• It is calculated as half of the total difference between the maximum and minimum tidal heights.
• Scientifically, it reflects the strength or intensity of the tidal cycle.
• For the Bay of Fundy example, the amplitude was computed as 7.5 meters, since the overall tidal difference is 15 meters.

Knowing the amplitude allows us to understand how significant the changes are in the tides, which can have profound implications on coastal activities. It is a straightforward yet vital number that informs how high the tides reach above (or fall below) the average sea level.

Period

The period in the context of tidal functions refers to the time it takes for the tide to complete one full cycle, returning to its original position. This is a repetitive process that follows the consistent pattern dictated by the function over time.
In the equation, the parameter that encodes the period is denoted as "B," which influences the function's wavelength.

To calculate the period, these steps are often followed:

• The formula is expressed as \( T = \frac{2\pi}{B} \), defining the duration of the cycle.
• For the Bay of Fundy, the period of the tidal cycle is observed to be 12.4 hours.
• By understanding this period value of 12.4 hours, it shows how regularly the tides rise and fall.
• Solving for \( B \) yields approximately 0.506 when considering the provided tidal period.

The period is particularly valuable in predicting when the tides reach specific levels. Managing coastal activities relies heavily on accurately knowing these cycles.

Phase Shift

Phase shift is an important concept when interpreting tidal functions as it defines the horizontal displacement of the tide's cycle relative to a reference point, often helping to pinpoint particular times when high or low tides occur.
In the equation, the phase shift is indicated by the parameter "C."

Key elements regarding phase shift include:

• It represents the time offset from the expected tidal pattern's starting point, which can be adjusted to match real-world observations.
• This offset allows the function to align with actual tide timings, catering for when the high tide is expected to peak in relation to a set base time, such as midnight.
• Without the correct phase shift, prediction of tides could be inaccurate, demonstrating its critical role in synchronizing the function with reality.

By understanding phase shift, we gain insight into accurate tidal predictions, ensuring we anticipate the exact timing of tides. Adjusting phase shift is similar to setting the watch correctly, so all other readings follow the appropriate timeline.