Construct Contour Plot in Mathematica for Quasi-Linear 1-D Wave Eq

In summary, the conversation is about how to construct a contour plot in Mathematica for the quasi-linear 1-D wave equation with piecewise constant initial conditions. The equation is shown to be $\frac{\partial\rho}{\partial t} + 2\rho\frac{\partial\rho}{\partial x} = 0$ and the initial conditions are given as $\rho(x,0) = \begin{cases}4, & x < -x_0\\3, & -x_0 < x < x_0\\1, & x > x_0\\\end{cases}$ with $x_0 = \pm 1$. The summary does not mention the contour plot being
  • #1
Dustinsfl
2,281
5
How can I construct a contour plot in Mathematica for
Consider the quasi-linear 1-D wave equation
$$
\frac{\partial\rho}{\partial t} + 2\rho\frac{\partial\rho}{\partial x} = 0
$$
with the piecewise constant initial conditions
When $x_0 = \pm 1$

$$
\rho(x,0) = \begin{cases}
4, & x < -x_0\\
3, & -x_0 < x < x_0\\
1, & x > x_0\\
\end{cases}
$$
 
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  • #2
dwsmith said:
How can I construct a contour plot in Mathematica for
Consider the quasi-linear 1-D wave equation
$$
\frac{\partial\rho}{\partial t} + 2\rho\frac{\partial\rho}{\partial x} = 0
$$
with the piecewise constant initial conditions
When $x_0 = \pm 1$

$$
\rho(x,0) = \begin{cases}
4, & x < -x_0\\
3, & -x_0 < x < x_0\\
1, & x > x_0\\
\end{cases}
$$

Would this be the correct contour plot?
View attachment 339
 

FAQ: Construct Contour Plot in Mathematica for Quasi-Linear 1-D Wave Eq

What is a contour plot in Mathematica?

A contour plot is a graphical representation of a 3-dimensional surface, where the points on the surface with the same value are connected by curves. In Mathematica, this is created using the ContourPlot function.

How do I construct a contour plot in Mathematica?

To construct a contour plot in Mathematica for a quasi-linear 1-D wave equation, you can use the ContourPlot function and specify the equation using ContourRegionFunction. You will also need to provide the appropriate boundary conditions and initial conditions for the wave equation.

What is a quasi-linear 1-D wave equation?

A quasi-linear 1-D wave equation is a partial differential equation that describes the propagation of a wave in one dimension. It is a special case of the general wave equation, where the coefficient of the second derivative term is a function of the dependent variable.

What are some applications of contour plots in Mathematica?

Contour plots are commonly used in many fields, such as physics, engineering, and mathematics, to visualize and analyze 3-dimensional data. They can be used to study functions, map out regions of different values, and identify critical points.

Can I customize the appearance of a contour plot in Mathematica?

Yes, you can customize the appearance of a contour plot in Mathematica by specifying various options in the ContourPlot function. These options include the color scheme, contour levels, axes labels, and more. You can also use Manipulate to dynamically change the plot based on different parameters.

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