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Chapter 21: Problem 3

Find the characteristic curves for the two-dimensional wave equation and the two-dimensional diffusion equation.

Short Answer

Expert verified
For the wave equation, the characteristic curves are lines (in 2D) or planes (in 3D) perpendicular to the wave fronts. The diffusion equation does not possess real characteristic curves.

Step by step solution

01

Differentiating the two-dimensional wave equation

The two-dimensional wave equation can be written as: \(u_{tt} = c^2(u_{xx} + u_{yy})\). To find its characteristic curves, we consider a path in the xy-plane given by the function \(Φ(t) = (X(t), Y(t))\). To simplify the problem, we assume there are characteristics for each of \(Φ(t)\) and that the derivative is not zero. Hence, \(Φ'(t) ≠ 0\).
02

Using the chain rule to differentiate

Using the chain rule, we differentiate the wave equation and it turns out to be a family of straight lines whose slope is either \( c\) or \(-c\). This gives us \(Φ’(t) = cN\), where \(N\) is a unit normal. Therefore, in the direction perpendicular to the wave fronts, we are either going directly with the wave at speed \( c\) or directly against it at speed \( c\). The characteristic curves are hence lines (in 2D) and planes (in 3D) perpendicular to the wave fronts.
03

Differentiating the two-dimensional diffusion equation

The exact same procedure will be used for the diffusion equation, which can be written as: \(u_t = α(u_{xx} + u_{yy})\). Using the chain rule on this equation, we won't be able to achieve a form in terms of \( N \), as it does not possess real characteristic curves.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Two-Dimensional Wave Equation
The two-dimensional wave equation is crucial in physics and engineering as it describes how waves, such as sound or light, propagate through a medium. This equation is expressed as: \[ u_{tt} = c^2(u_{xx} + u_{yy}) \]Where:
  • \(u_{tt}\) represents the second partial derivative of \(u\) with respect to time \(t\), indicating how the wave changes over time.
  • \(c\) is the wave speed, a constant showing the velocity at which the wave travels through the medium.
  • \(u_{xx}\) and \(u_{yy}\) are the second partial derivatives of \(u\) with respect to the spatial variables \(x\) and \(y\), respectively.
The wave equation suggests that changes in time of the wave field (on the left) are directly proportional to its spatial curvature (on the right), scaled by the square of the wave speed. When analyzing characteristic curves, the approach involves visualizing paths along which the wave has constant amplitude. By applying the chain rule, these curves are identified, often taking the form of straight lines or planes depending on the wave directions and speeds. These straight lines illustrate how wave fronts propagate through space, making them a fundamental concept in understanding wave behaviors in 2D media.
Exploring the Two-Dimensional Diffusion Equation
The two-dimensional diffusion equation is vital in modeling how substances such as heat or particles spread over time. It is given by the following expression:\[ u_t = \alpha(u_{xx} + u_{yy}) \]Here:
  • \(u_t\) is the partial derivative of \(u\) with respect to time, representing how the quantity changes over time.
  • \(\alpha\) is the diffusivity constant, representing how quickly the substance spreads through the medium.
  • \(u_{xx}\) and \(u_{yy}\) again denote the second partial derivatives with respect to the spatial dimensions.
Unlike the wave equation, the diffusion equation does not possess real characteristic curves. This implies that the spreading does not occur along distinct paths but instead disperses more evenly over time. The application of the chain rule reveals that in diffusion, the relation does not simplify to a manageable "normal" path. This reflects how diffusion is inherently a smoothing process, gradually eliminating sharper features as substance or heat diffuses throughout the available space.
Utilizing the Chain Rule in Partial Differential Equations
The chain rule is a fundamental calculus tool that helps differentiate composite functions. In the context of partial differential equations, the chain rule becomes essential for identifying how a function changes along curved or linear paths. When examining characteristic curves in solutions like the wave or diffusion equation, we often use the chain rule as follows: Given a function \( \Phi(t) = (X(t), Y(t)) \), we differentiate a composite function by applying the chain rule to each component, capturing the changes over time and space.
  • In the wave equation, differentiating using the chain rule helps identify paths of specific linear qualities, leading to characteristic straight lines where wave fronts travel at speed \(c\).
  • For the diffusion equation, the chain rule helps delineate how spreads occur, highlighting the lack of distinct linear paths and underscoring the nature of diffusion as a process that evens out over time.
In practice, the chain rule simplifies complex differential relationships, turning them into accessible forms for analysis, providing clarity when looking for exact or approximate solutions to physical phenomena represented by these equations. Understanding its application is key to navigating deeper mathematical concepts linked with waves and diffusion processes.

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Most popular questions from this chapter

Solve the Cauchy problem for the two-dimensional Laplace equation subject to the Cauchy data \(u(0, y)=0,(\partial u / \partial x)(0, y)=\epsilon \sin k y\), where \(\epsilon\) and \(k\) are constants. Show that the solution does not vary continuously as the Cauchy data vary. In particular, show that for any \(\epsilon \neq 0\) and any preassigned \(x>0\), the solution \(u(x, y)\) can be made arbitrarily large by choosing \(k\) large enough.

Consider the operator \(\mathbf{L}_{\mathbf{x}}=\nabla^{2}+\mathbf{b} \cdot \boldsymbol{\nabla}+c\) for which \(\left\\{b_{i}\right\\}_{i=1}^{m}\) and \(c\) are functions of \(\left\\{x_{i}\right\\}_{i=1}^{m}\). (a) Show that \(\mathbf{L}_{\mathbf{x}}^{\dagger}[v]=\nabla^{2} v-\boldsymbol{\nabla} \cdot(\mathbf{b} v)+c v\), and $$ \mathbf{Q}\left[u, v^{*}\right]=\mathbf{Q}[u, v]=v \nabla u-u \nabla v+\mathbf{b} u v $$ (b) Show that a necessary condition for \(\mathbf{L}_{\mathbf{x}}\) to be self- adjoint is \(2 \mathbf{b} \cdot \nabla u+\) \(u(\boldsymbol{\nabla} \cdot \mathbf{b})=0\) for arbitrary \(u .\) (c) By choosing some \(u\) 's judiciously, show that (b) implies that \(b_{i}=0\). Conclude that \(\mathbf{L}_{\mathrm{x}}=\nabla^{2}+c(\mathbf{x})\) is formally self-adjoint.

Find the characteristic curves for \(\mathbf{L}_{x}[u]=\partial u / \partial x\).

Use \(J \delta(\mathbf{x}-\mathbf{a})=\delta(\boldsymbol{\xi}-\boldsymbol{\alpha})\) and the coordinate transformation from the spherical coordinate system to Cartesian coordinates to express the \(3 \mathrm{D}\) Cartesian delta function in terms of the corresponding spherical delta function at a point \(P=\left(x_{0}, y_{0}, z_{0}\right)=\left(r_{0}, \theta_{0}, \varphi_{0}\right)\) where the Jacobian \(J\) is nonvanishing.

Show that the integral equation associated with the damped harmonic oscillator DE \(\ddot{x}+2 \gamma \dot{x}+\omega_{0}^{2} x=0\), having the \(\mathrm{BCs} x(0)=x_{0}\) \((d x / d t)_{t=0}=0\), can be written in either of the following forms. (a) \(x(t)=x_{0}-\frac{\omega_{0}^{2}}{2 \gamma} \int_{0}^{t}\left[1-e^{-2 \gamma\left(t-t^{\prime}\right)}\right] x\left(t^{\prime}\right) d t^{\prime}\). (b) \(x(t)=x_{0} \cos \omega_{0} t+\frac{2 \gamma x_{0}}{\omega_{0}} \sin \omega_{0} t-2 \gamma \int_{0}^{t} \cos \left[\omega_{0}\left(t-t^{\prime}\right)\right] x\left(t^{\prime}\right) d t^{\prime}\). Hint: Take \(\omega_{0}^{2} x\) or \(2 \gamma \dot{x}\), respectively, as the inhomogeneous term.
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