Equivalence of Derivations in the Wave Equation from Maxwell's Equations

In summary, the conversation is about the derivation of the wave equation from Maxwell's equations. The student is struggling with one step and is unsure if their final result is equivalent to the given result in the book. The necessary equations and identities are provided in an attached file. The student's final result (5) differs from the book's result (4) by one term. They are wondering if (4) and (5) are equivalent and how to get from (5) to (4). The solution is to use the chain rule, as (4) and (5) are indeed equivalent.
  • #1
Valeria
2
0

Homework Statement


I am actually following the derivation of the wave equation from Maxwell equations. And I do not understand one step, because in the task for the derivation I get a slightly different result (maybe they are equivalent, but I am not sure).

Homework Equations


In the attached file:
(1) - starting equation; (2) and (3) - necessary identities; (4) - final result from the book; (5) - my final result.

The Attempt at a Solution


(5) is the equation I got. As you can see, only one term is different. Is equation (5) equivalent then to equation (4)? If yes, how to get from (5) to (4)? If no, what could be incorrect?
 

Attachments

  • equations.png
    equations.png
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  • #2
(4) and (5) are equivalent. Use the chain rule.
 

What is the wave equation derivation?

The wave equation derivation is a mathematical process used to derive the equation that describes the behavior of waves. It is based on the principles of conservation of energy and momentum.

What are the applications of the wave equation derivation?

The wave equation derivation is used in various fields, such as physics, engineering, and mathematics, to study and understand the behavior of different types of waves, including sound waves, electromagnetic waves, and water waves. It is also used in the development of technologies, such as communication systems and medical imaging.

What are the basic steps involved in the wave equation derivation?

The basic steps involved in the wave equation derivation include defining the physical properties of the wave, such as amplitude, frequency, and wavelength, applying the principles of conservation of energy and momentum, and solving differential equations to obtain the final equation.

What are the key assumptions made in the wave equation derivation?

The wave equation derivation is based on certain assumptions, such as the wave is traveling in a single direction, the medium is uniform and continuous, and there are no external forces acting on the wave. These assumptions help simplify the mathematical process and make it easier to derive the equation.

What are the limitations of the wave equation derivation?

The wave equation derivation has some limitations, such as it only applies to linear waves, and it does not take into account the effects of dispersion, nonlinearity, and dissipation. In some cases, these limitations can be overcome by using more advanced mathematical techniques and models.

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