19th Ave New York, NY 95822, USA

# 数学代写|偏微分方程作业代写Partial Differential Equations代考|The Wave Equation in Three and Two Space Dimensions

my-assignmentexpert™ 偏微分方程Partial Differential Equations作业代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。my-assignmentexpert™， 最高质量的偏微分方程Partial Differential Equations作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此偏微分方程Partial Differential Equations作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

my-assignmentexpert™ 为您的留学生涯保驾护航 在数学Mathematics作业代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的偏微分方程Partial Differential Equations代写服务。我们的专家在数学Mathematics代写方面经验极为丰富，各种偏微分方程Partial Differential Equations相关的作业也就用不着 说。

## 数学代写|偏微分方程作业代写Partial Differential Equations代考|Two Derivations of the 3D Wave Equation

In this section, we denote the coordinates of three-dimensional space by $(x, y, z)$. We give two derivations of the 3D wave equation:
$$u_{t t}=c^{2}\left(u_{x x}+u_{y y}+u_{z z}\right)=c^{2} \Delta u$$
The first is for the propagation of electromagnetic waves in a vacuum and is exact in the sense that no approximations are needed for the equation to be valid. The second one is for acoustics – the propagation of sound in air. Here, following the same path as for the vibrating string, we will derive the wave equation by making an assumption that disturbances are relatively small. Neither derivation will be given from “first principles”; rather, each will come from the following respective underlying systems of PDEs which we will take at face value:

• Maxwell’s equations for electromagnetic waves,
• the compressible Euler equations for sound waves.

## 数学代写|偏微分方程作业代写Partial Differential Equations代考|Three Space Dimensions: The Initial Value Problem and Its Explicit Solution

We now consider the initial value problem for the 3D wave equation. As in 1D, we must prescribe both the initial displacement and velocity. These are scalar functions of $\mathbb{R}^{3}$ and will be denoted, respectively, by $\phi(\mathbf{x})$ and $\psi(\mathbf{x})$. We assume throughout this chapter that $\phi$ and $\psi$ are smooth functions. For simplicity of notation, let us use $\mathbf{x}=\left(x_{1}, x_{2}, x_{3}\right)$ for spatial points instead of $(x, y, z)$. Our goal now is to solve the initial value problem:
IVP for the 3D Wave Equation in All of Space
$$\begin{cases}u_{t t}=c^{2} \Delta u, & \mathbf{x} \in \mathbb{R}^{3}, t>0 \ u(\mathbf{x}, 0)=\phi(\mathbf{x}), & \mathbf{x} \in \mathbb{R}^{3}, \ u_{t}(\mathbf{x}, 0)=\psi(\mathbf{x}), & \mathbf{x} \in \mathbb{R}^{3}\end{cases}$$
where $\phi$ and $\psi$ are smooth functions.
As we shall soon see, the key to solving (4.9) is to exploit the fact that the righthand side of the wave equation (the Laplacian) treats all the spatial directions (i.e., $x_{1}$, $x_{2}$, and $x_{3}$ ) equally. This is simply a consequence of the fact that the PDE models the propagation of a disturbance in a homogeneous medium wherein there is no preferred spatial direction.

• 麦克斯韦电磁波方程，
• 声波的可压缩欧拉方程。

## 数学代写|偏微分方程作业代写PARTIAL DIFFERENTIAL EQUATIONS代考|THREE SPACE DIMENSIONS: THE INITIAL VALUE PROBLEM AND ITS EXPLICIT SOLUTION

IVP for the 3D Wave Equation in All of Space
{在吨吨=C2Δ在,X∈R3,吨>0 在(X,0)=φ(X),X∈R3, 在吨(X,0)=ψ(X),X∈R3