PDE Wave Equation and Energy Conservation

In summary, the conversation revolved around energy conservation and the different forms it can take. Specifically, there was a discussion about the differential and integral forms of energy conservation, and the differences between global and local forms. The main symbols used were μ, c2, ut, and ux. There was also a question about the age of one of the participants.
  • #1
Maybe_Memorie
353
0

Homework Statement



Just looking back through my notes and it looks like I'm missing some. Just a few questions.

For one example in the notes I have the wave utt-c2uxx + u3 = 0 and that the energy density 1/2u2t + c2/2u2x + 1/4u4

I have that the differential form of energy conservation is εt = μx + (wave equation)μt = μx for solutions.

First question, what is the integral form of energy conservation?

The notes then say that here μ = c2utux. Why is this?

Also what is the difference between global and local form of energy conservation? Is it just that local has finite limits?

Are there any websites or online notes that might have some of this info? I searched a lot but can't find much

Thanks!
 
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  • #2
Hey Maybe_Memorie! :smile:

I think that before we can give an answer, we first need some definitions.
What does each of your symbols represent?
And what do they mean?
To be honest, I felt a bit reluctant to respond, since I have no idea what I'm getting into with these fragments of information.

Btw, are you still 17? (Your blog says so. :wink:)
 

Related to PDE Wave Equation and Energy Conservation

1. What is the PDE wave equation?

The PDE (partial differential equation) wave equation is a mathematical equation that describes the behavior of waves in a given system. It is commonly used in physics and engineering to model various phenomena such as sound waves, electromagnetic waves, and water waves.

2. How does the PDE wave equation relate to energy?

The PDE wave equation includes a term for energy, known as the energy density, which represents the amount of energy per unit volume in the system. This term is essential in studying the propagation and behavior of waves, as it allows us to analyze how energy is transferred and conserved within the system.

3. What is the significance of the energy conservation property in the PDE wave equation?

The energy conservation property in the PDE wave equation ensures that the total energy in the system remains constant. This means that as the wave travels through the system, the total amount of energy remains the same, even if it is redistributed among different forms (e.g., potential energy, kinetic energy) or transferred to other parts of the system.

4. How is the energy density calculated in the PDE wave equation?

The energy density in the PDE wave equation is calculated using the formula: energy density = (1/2) * (wave speed)^2 * (amplitude)^2. This formula takes into account the wave speed, which is a measure of how fast the wave is traveling through the medium, and the amplitude, which is the maximum displacement of the wave from its equilibrium position.

5. Can the PDE wave equation be applied to real-world systems?

Yes, the PDE wave equation is a widely applicable mathematical model that can be used to study a variety of physical systems. It has been successfully applied in fields such as acoustics, optics, and fluid dynamics, and has been instrumental in understanding and predicting the behavior of waves in real-world scenarios.

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