Equation of the Tide Function

Oct 13, 2025 by talynt.org 124 views

Introduction

The tide in a harbor can be modeled by a sinusoidal function over a 24-hour period. This function represents the rise and fall of the water level in the harbor, providing valuable information for various applications such as navigation, coastal engineering, and environmental monitoring. In this article, we will explore the characteristics of the tide function and determine its equation.

The Tide Function

The tide function can be represented by a sinusoidal function of the form:

y = A sin(Bx) + C

where y is the height of the tide, A is the amplitude, B is the frequency, and C is the vertical shift.

Given Information

We are given the following information about the tide function:

The tide measures 5.15 ft at midnight.
The tide rises to a high of 10.2 ft.
The tide falls to a low of 0.1 ft.
The tide rises to 5.15 ft by the next midnight.

Determining the Equation

To determine the equation of the tide function, we need to find the values of A, B, and C. We can use the given information to set up a system of equations.

Midnight Tide Level

At midnight, the tide level is 5.15 ft. This corresponds to the point (0, 5.15) on the graph. We can substitute this point into the equation to get:

5.15 = A sin(B(0)) + C 5.15 = C

High Tide Level

The high tide level is 10.2 ft. This corresponds to the point (x, 10.2) on the graph. We can substitute this point into the equation to get:

10.2 = A sin(Bx) + C

Low Tide Level

The low tide level is 0.1 ft. This corresponds to the point (x, 0.1) on the graph. We can substitute this point into the equation to get:

0.1 = A sin(Bx) + C

Equation of the Tide Function

We can use the system of equations to determine the values of A, B, and C. We can start by solving for C:

C = 5.15

Next, we can substitute C into the other two equations:

10.2 = A sin(Bx) + 5.15 0.1 = A sin(Bx) + 5.15

We can subtract 5.15 from both sides of the equations to get:

5.05 = A sin(Bx) -5.05 = A sin(Bx)

We can add the two equations to get:

0 = 2A sin(Bx)

We can divide both sides by 2 to get:

0 = A sin(Bx)

We can conclude that A = 0, which means that the tide function is a horizontal line.

However, this is not possible since the tide function is a sinusoidal function. We can try another approach.

We can use the fact that the tide function is a sinusoidal function to write:

y = A sin(Bx) + C

We can substitute the given information into the equation to get:

5.15 = A sin(Bx) + C 10.2 = A sin(Bx) + C 0.1 = A sin(Bx) + C

We can subtract C from all three equations to get:

-0.05 = A sin(Bx) 5.1 = A sin(Bx) -5.05 = A sin(Bx)

We can add the first and third equations to get:

-0.05 - 5.05 = 2A sin(Bx)

We can simplify the equation to get:

-5.1 = 2A sin(Bx)

We can divide both sides by 2 to get:

-2.55 = A sin(Bx)

We can conclude that A = -2.55.

Finding the Frequency

We can use the fact that the tide function is a sinusoidal function to write:

y = A sin(Bx) + C

We can substitute the values of A and C into the equation to get:

y = -2.55 sin(Bx) + 5.15

We can use the fact that the tide function is periodic with a period of 24 hours to write:

B = 2π / 24

We can simplify the equation to get:

B = π / 12

Equation of the Tide Function

We can substitute the values of A, B, and C into the equation to get:

y = -2.55 sin(πx / 12) + 5.15

This is the equation of the tide function.

Conclusion

In this article, we have determined the equation of the tide function over a 24-hour period. The equation is:

y = -2.55 sin(πx / 12) + 5.15

This equation represents the rise and fall of the water level in the harbor, providing valuable information for various applications such as navigation, coastal engineering, and environmental monitoring.

References

[1] "Tides and Sea Levels". National Oceanic and Atmospheric Administration (NOAA).
[2] "Tidal Analysis and Prediction". United States Naval Observatory.

Glossary

Amplitude: The maximum displacement of a wave from its equilibrium position.
Frequency: The number of oscillations or cycles of a wave per unit time.
Period: The time required for one complete oscillation or cycle of a wave.
Sinusoidal function: A mathematical function that represents a wave or oscillation.
Tide: The periodic rise and fall of the water level in the ocean, caused by the gravitational pull of the moon and sun.
Tide Function Q&A =====================

Introduction

In our previous article, we determined the equation of the tide function over a 24-hour period. In this article, we will answer some frequently asked questions about the tide function.

Q: What is the amplitude of the tide function?

A: The amplitude of the tide function is -2.55 ft. This represents the maximum displacement of the tide from its equilibrium position.

Q: What is the frequency of the tide function?

A: The frequency of the tide function is π / 12. This represents the number of oscillations or cycles of the tide per unit time.

Q: What is the period of the tide function?

A: The period of the tide function is 24 hours. This represents the time required for one complete oscillation or cycle of the tide.

Q: What is the vertical shift of the tide function?

A: The vertical shift of the tide function is 5.15 ft. This represents the equilibrium position of the tide.

Q: How does the tide function change over time?

A: The tide function changes over time due to the gravitational pull of the moon and sun. The tide function is a sinusoidal function, which means that it oscillates or cycles over time.

Q: What are the applications of the tide function?

A: The tide function has various applications in fields such as navigation, coastal engineering, and environmental monitoring. It can be used to predict the rise and fall of the water level in the ocean, which is essential for safe navigation and coastal protection.

Q: How can I use the tide function in my work?

A: You can use the tide function in your work by substituting the values of x and y into the equation. For example, if you want to find the height of the tide at a given time, you can substitute the value of x into the equation and solve for y.

Q: What are some common mistakes to avoid when working with the tide function?

A: Some common mistakes to avoid when working with the tide function include:

Not accounting for the vertical shift of the tide function
Not using the correct values of amplitude and frequency
Not considering the effects of the moon and sun on the tide function

Q: How can I learn more about the tide function?

A: You can learn more about the tide function by reading books and articles on the subject, attending workshops and conferences, and practicing with real-world examples.

Conclusion

In this article, we have answered some frequently asked questions about the tide function. We hope that this information has been helpful in understanding the tide function and its applications.

References

[1] "Tides and Sea Levels". National Oceanic and Atmospheric Administration (NOAA).
[2] "Tidal Analysis and Prediction". United States Naval Observatory.

Glossary

Amplitude: The maximum displacement of a wave from its equilibrium position.
Frequency: The number of oscillations or cycles of a wave per unit time.
Period: The time required for one complete oscillation or cycle of a wave.
Sinusoidal function: A mathematical function that represents a wave or oscillation.
Tide: The periodic rise and fall of the water level in the ocean, caused by the gravitational pull of the moon and sun.