Problem 33 - Algebra and Trigonometry

Question

Tides The Bay of Fundy in Nova Scotia has the highest tides in the world. In one 12 -hour period the water starts at mean sea level, rises to 21 \(\mathrm{ft}\) above, drops to 21 \(\mathrm{ft}\) below, then returns to mean sea level. Assuming that the motion of the tides is simple harmonic, find an equation that describes the height of the tide in the Bay of Fundy above mean sea level. Sketch a graph that shows the level of the tides over a 12 -hour period.

Step-by-Step Solution

Answer

The equation is \( h(t) = 21 \cos(\frac{\pi}{6}t) \).

Step 1: Understand the Problem and Parameters

The problem involves tides modeled as simple harmonic motion. The tidal cycle completes in 12 hours, indicating a periodic nature. The amplitude (maximum deviation from mean) is 21 ft above and below mean sea level.

Step 2: Identify Equation Form

Simple harmonic motion can be modeled by the equation for sine or cosine waves: \( h(t) = A \sin(Bt + C) + D \). Here, \(A\) is the amplitude, \(B\) affects the period, \(C\) is the phase shift, and \(D\) is the vertical shift.

Step 3: Determine Amplitude and Vertical Shift

For our tide, the amplitude \(A\) is 21 ft since the tide rises 21 ft above and falls 21 ft below the mean sea level (which is the equilibrium, or \(D = 0\)).

Step 4: Calculate the Period

A full cycle (crest to crest or trough to trough) takes 12 hours. The standard period of \(2\pi\) is adjusted by \(B\) such that \(\frac{2\pi}{B} = 12\). Solving for \(B\), we get \(B = \frac{\pi}{6}\).

Step 5: Choose and Adjust the Function

We choose a cosine function that starts at the mean sea level rising: \( h(t) = 21 \cos(\frac{\pi}{6}t) \). Since cosine usually starts at its maximum, no horizontal shift (\(C = 0\)) is necessary to model starting at mean sea level with the cosine function due to its symmetry.

Step 6: Graph the Function

A graph of the function \( h(t) = 21 \cos(\frac{\pi}{6}t) \) over the interval \(t = 0\) to \(t = 12\) starts at 0 ft (mean sea level), reaches 21 ft at \(t = 3\) (peak), returns to 0 ft at \(t = 6\), falls to -21 ft at \(t = 9\) (trough), and returns to 0 ft at \(t = 12\).

Key Concepts

Amplitude in Harmonic Motion

In the context of harmonic motion, the amplitude is a key component that measures how far the oscillating quantity deviates from its equilibrium, or mean, position. Amplitude is essentially the maximum distance from the mean level that the wave reaches.
This concept is crucial in understanding the tides in the Bay of Fundy, where the amplitude is given as 21 feet. This means that the tide can rise 21 feet above the mean sea level or fall 21 feet below it. Visualizing this helps us see that amplitude dictates the extreme points of the wave
and highlights the range of motion of the tidal waves.

• The equilibrium position is the middle point, or average height, which the tide returns to periodically.
• With an amplitude of 21 feet, the wave reaches its highest and lowest points at 21 feet above and below this equilibrium.

Understanding amplitude helps in visualizing the scope and strength of tidal movements
as they influence the coastal regions affected by these harmonic motions.

Understanding the Cosine Function in Tidal Equations

A cosine function is often used in simple harmonic motion due to its properties and symmetry. In mathematics, the cosine function starts at its maximum, descends to the minimum, and returns to the maximum
in a symmetric pattern over its period. The general form of a cosine function is given by:\[ h(t) = A \cos(Bt + C) + D \]where:

• \(A\) represents the amplitude, which in our example is 21 feet.
• \(B\) influences the period of the wave and is calculated based on the frequency of the cycles (\( B = \frac{\pi}{6} \) in this case).
• \(C\) is the phase shift, but here it is zero because no horizontal adjustment is needed.
• \(D\) is the vertical shift; in tidal levels, the mean sea level is the equilibrium, thus \(D\) is zero.

Cosine functions make modeling tides straightforward because they naturally align with scenarios where the starting position is the mid-level of the wave, like the mean sea level for tides. This way, it complements the natural rhythm of tides as seen in places like the Bay of Fundy.

Periodic Functions and Their Relevance

Periodic functions are mathematical functions which repeat their values in regular intervals or periods. They are central to understanding phenomena like tides, which are natural cycles.
The coastal tide in the Bay of Fundy exhibits periodic behavior because it completes one full cycle of rising and falling, every 12 hours.
This repeating nature matches with the periodic function definition.

• The function, \( h(t) = 21 \cos\left(\frac{\pi}{6}t\right) \), models this scenario.
• The period of the function can be determined by \( \frac{2\pi}{B} \), and in this exercise, it equates to 12 hours, representing a complete tidal cycle.
• This period determines how often the wave pattern repeats itself at set intervals and is crucial for predicting future tides.

Periodicity helps predict tidal movements, thus aiding in navigation, fishing, and even generating energy. Recognizing this periodic nature allows us to anticipate and plan for the fluctuations caused by tides, contributing to our broader understanding of wave patterns.