Finding Density as a Function of Space and Time for 1D Wave Equation Problem

You are given the pressure at time 0 and a constant pressure gradient, so you can use the ideal gas law to find the density at any point and time.In summary, the problem involves a sealed 1D pipe with a given pressure and pressure gradient, and the goal is to find the density as a function of position and time. The ideal gas law can be used to relate pressure and density and determine the density at any given point and time. The initial conditions and boundary conditions can also be used to find a relation between velocity and density, but it is unclear how to use this to find the density as a function of (x,t).
  • #1
jlee07
1
0

Homework Statement


Hello-

I'm having trouble understanding a problem:

Consider a sealed 1D pipe of length L. At t=0, v=0 everywhere and the pressure is given by: P=P_0 +δP

and δP = (p-bar)x/L

P_0 and (p-bar) are both constants.

and I'm supposed to find density (ϱ) as a function of x and t.

I don't understand why I'm given the pressure, and how to find δϱ.

Homework Equations

The Attempt at a Solution


Using the boundary conditions, I have an expression for the velocity with sin terms. But I don't know how to use that to find the density as a function of (x,t).
 
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  • #2
The gas laws give you a relation between pressure and density.
 

FAQ: Finding Density as a Function of Space and Time for 1D Wave Equation Problem

What is the 1D wave equation problem?

The 1D wave equation problem is a mathematical model used to describe the propagation of a wave through a medium in one dimension. It takes into account factors such as the wave's amplitude, frequency, and initial conditions to determine its behavior over time.

What are the applications of the 1D wave equation problem?

The 1D wave equation problem has applications in various fields such as physics, engineering, and geology. It can be used to understand and predict the behavior of sound waves, electromagnetic waves, and seismic waves. It is also used in the design of structures and in analyzing the effects of vibrations on buildings and bridges.

What are the key components of the 1D wave equation problem?

The key components of the 1D wave equation problem are the wave's amplitude, frequency, and initial conditions. These factors, along with the properties of the medium, determine the shape, speed, and direction of the wave as it propagates through the medium.

What are the boundary conditions for solving the 1D wave equation problem?

The boundary conditions for solving the 1D wave equation problem depend on the specific problem being solved. However, in general, the boundary conditions specify the behavior of the wave at the boundaries of the medium. This could include fixed or free ends, or the reflection or transmission of the wave at the boundary.

What are the methods used to solve the 1D wave equation problem?

The 1D wave equation problem can be solved using various methods, including analytical and numerical techniques. Analytical solutions involve finding an exact mathematical expression for the wave's behavior, while numerical solutions use algorithms and computational methods to approximate the wave's behavior. Examples of numerical methods include the finite difference method and the finite element method.

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