### 物理代写|光学工程代写Optical Engineering代考|CET824

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|光学工程代写Optical Engineering代考|WAVES AND THE WAVE EQUATION

Sound is the propagation of pressure variations or density changes of air. Light is the propagation of variation in the electric and magnetic fields.

Consider that the vibration $u$ propagates in the $z$ direction with the speed $v$, as shown in Fig. 1.1. Assuming the shape $f$ of the wave variation $u$ propagating in the $z$ direction at time $t=0$, we have
$$u(z, t=0)=f(z) .$$
The wave $u$ moves a distance $v t$ at time $t$ but its shape is not changed, so we have
$$u(z, t)=f(z-v t) .$$
This means that the wave variance does not change independently with variables of time $t$ and position $z$, but only as a function of $z-v t$. The relationship among variation $u$, position $z$ and time $t$ exists but does not depend on the shape of variation $f$. Using
$$\tau=z-v t$$
we have
\begin{aligned} & \frac{\partial u}{\partial z}=\frac{\partial u}{\partial \tau} \cdot \frac{\partial \tau}{\partial z}=\frac{\partial u}{\partial \tau} \ & \frac{\partial u}{\partial t}=\frac{\partial u}{\partial \tau} \cdot \frac{\partial \tau}{\partial t}=-v \frac{\partial u}{\partial \tau} \end{aligned}

Differentiating these again, we have
\begin{aligned} & \frac{\partial^2 u}{\partial z^2}=\frac{\partial}{\partial \tau}\left(\frac{\partial u}{\partial \tau}\right) \frac{\partial \tau}{\partial z}=\frac{\partial^2 u}{\partial \tau^2} \ & \frac{\partial^2 u}{\partial t^2}=\frac{\partial}{\partial \tau}\left(\frac{\partial u}{\partial \tau}\right) \frac{\partial \tau}{\partial t}=v^2 \frac{\partial^2 u}{\partial \tau^2} . \end{aligned}
Therefore we have
$$\frac{\partial^2 u}{\partial z^2}=\frac{1}{v^2} \frac{\partial^2 u}{\partial t^2} .$$
This equation describes the wave propagating in the $+z$ direction with the velocity $\pm v$. This is called the wave equation. ${ }^1$

In general, by extending Eq. (1.8), the wave equation in three dimensions $(x, y, z)$ can be written as
$$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}=\frac{1}{v^2} \frac{\partial^2 u}{\partial t^2},$$
or by using the Laplacian operator
$$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2},$$
the 3-D wave equation is rewritten as
$$\nabla^2 u=\frac{1}{v^2} \frac{\partial^2 u}{\partial t^2}$$

## 物理代写|光学工程代写Optical Engineering代考|PLANE WAVE

The most simple solution of the wave equation is a sinusoidal wave. The sinusoidal wave propagation to the $z$ direction with the velocity $v$ is written as
$$u(z, t)=A \cos [k(z-v t)+\phi] .$$
The equation evidently satisfies the wave equation (1.8). The maximum of variation $A$ is called the amplitude, and $k(z-v t)+\phi$ is the phase. The sinusoidal wave is a periodic function both in space and time, as shown in Fig. 1.2. The period in space is called the wavelength, denoted by $\lambda$. If the wave propagates a distance of $\lambda$, the phase of the wave in Eq. (1.12) changes by $2 \pi$;
$$k \lambda=2 \pi,$$
so we have
$$k=2 \pi / \lambda .$$
Since $k$ means a number of $\lambda$ in the length of $2 \pi$ and $k$ is called the wave number or propagation constant, in the field of optical wave guides, ${ }^2 \phi$ is initial phase and can be 0 , if the spatial coordinate $z$ and the time coordinate $t$ are arbitrary values.

The period $T$ in time is given by
$$T=\lambda / v$$
and the reciprocal of $T$ is frequency
$$v=1 / T .$$
Finally, we have the frequency
$$v=v / \lambda,$$
which means frequency is a number of waves per unit distance of $v$ (a propagation distance per unit time). The angular frequency is defined as
$$\omega=2 \pi v .$$
The light velocity in vacuum is a physical constant $c$. In the case of light wave, $v$ is light velocity in a medium. The ratio of $v$ and $c$ is refractive index $n$.
$$n=c / v .$$
The frequency is written as
$$v=c /(n \lambda)=c / \lambda_0,$$
where $\lambda_0$ denotes the light wavelength in vacuum. Therefore, $\lambda$ is wavelength in a medium, which is written as
$$\lambda=\lambda_0 / n$$

# 光学工程代考

## 物理代写|光学工程代写Optical Engineering代考|WAVES AND THE WAVE EQUATION

$$u(z, t=0)=f(z)$$

$$u(z, t)=f(z-v t)$$

$$\tau=z-v t$$

$$\frac{\partial u}{\partial z}=\frac{\partial u}{\partial \tau} \cdot \frac{\partial \tau}{\partial z}=\frac{\partial u}{\partial \tau} \quad \frac{\partial u}{\partial t}=\frac{\partial u}{\partial \tau} \cdot \frac{\partial \tau}{\partial t}=-v \frac{\partial u}{\partial \tau}$$

$$\frac{\partial^2 u}{\partial z^2}=\frac{\partial}{\partial \tau}\left(\frac{\partial u}{\partial \tau}\right) \frac{\partial \tau}{\partial z}=\frac{\partial^2 u}{\partial \tau^2} \quad \frac{\partial^2 u}{\partial t^2}=\frac{\partial}{\partial \tau}\left(\frac{\partial u}{\partial \tau}\right) \frac{\partial \tau}{\partial t}=v^2 \frac{\partial^2 u}{\partial \tau^2}$$

$$\frac{\partial^2 u}{\partial z^2}=\frac{1}{v^2} \frac{\partial^2 u}{\partial t^2}$$

$$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}=\frac{1}{v^2} \frac{\partial^2 u}{\partial t^2}$$

$$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2},$$
3-D波动方程重写为
$$\nabla^2 u=\frac{1}{v^2} \frac{\partial^2 u}{\partial t^2}$$

## 物理代写|光学工程代写Optical Engineering代考|PLANE WAVE

$$u(z, t)=A \cos [k(z-v t)+\phi]$$

$$k \lambda=2 \pi,$$

$$k=2 \pi / \lambda$$

$$T=\lambda / v$$

$$v=1 / T$$

$$v=v / \lambda,$$

$$\omega=2 \pi v .$$

$$n=c / v \text {. }$$

$$v=c /(n \lambda)=c / \lambda_0,$$

$$\lambda=\lambda_0 / n$$

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## MATLAB代写

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