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# Water Wave Mechanics: Euler's Equations of Motion

Who? What? Where? When? Why?

First off, I want to say that in developing myself as a Ph.D. student, I have realized the importance of utilizing multiple resources when learning a topic. This has been especially true in the process of preparing for my comprehensive exam that will be held this August 2022. In the midst of studying, I have found happiness in sharing and creating simple posts that help explain what exactly I am learning as a multitude of friends, family, and even colleagues wonder.

This is the first post of many that will flow in the following month as I grasp hold of the key topics that will be necessary to pass into the next stage of my Ph.D. journey. Leonhard Euler was a key contributor to the category of water wave mechanics, which is one of the classes I took this previous spring 2022. He was born in Switzerland and was an early user of applied mathematics. He heavily contributed to the development of ordinary and partial differential equations.

His bachelor degree in Arts was from the University of Basel and his master degree is in Philosophy.

In 1761, he publlished "Principia motus fluidorum." This was the foundation of the Euler and continuity equations, both critical in the study of water wave mechanics (Dean & Dalrymple, 1991).

Euler's Equation of Motion is derived from Newton's 2nd law.

"The Euler equation is based on Newton's second law, which relates the change in velocity of a fluid particle to the presence of a force. Associated with this is the conservation of momentum, so that the Euler equation can also be regarded as a consequence of the conservation of momentum(Tec-Science, 2021)."

I decided to share my notes in regards to the derivation of the Euler Equation from Newton's 2nd law.

Hopefully you enjoy this and look forward to future shares!

Euler equation example
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Citations:

Dean, R. G., & Dalrymple, R. A. (1991). Water wave mechanics for engineers and scientists. World Scientific.

-, T.-S., By, -, & Tec-Science. (2021, December 5). Derivation of the Euler equation of motion (conservation of momentum) - tec-science. tec. Retrieved June 27, 2022, from https://www.tec-science.com/mechanics/gases-and-liquids/derivation-of-the-euler-equation-of-motion-conservation-of-momentum/#:~:text=The%20Euler%20equation%20is%20based,of%20the%20conservation%20of%20momentum.

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