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# 物理代写|声学代写Acoustics代考|SIO190 The Five Most Useful Math Techniques

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## 物理代写|声学代写Acoustics代考|The Five Most Useful Math Techniques

Below is a list of the five most useful mathematical techniques for the study of acoustics and vibration based on my experience. Techniques number one and number five are self-explanatory. The other three will be introduced in more detail in this section.

• Substitution
• Taylor series
• The product rule or integration by parts
• Logarithmic differentiation
• Garrett’s First Law of Geometry: “Angles that look alike are alike.”

## 物理代写|声学代写Acoustics代考|Taylor Series

Acoustics and vibration are the “sciences of the subtle.” Most of our attention will be focused on small deviations from a state of stable equilibrium. For example, a sound pressure level ${ }^1$ of $115 \mathrm{~dB}_{\mathrm{SPL}}$ is capable of creating permanent damage to your hearing with less than $15 \mathrm{~min}$ of exposure per day [2]. That acoustic pressure level corresponds to a peak excess pressure of $p_1=16 \mathrm{~Pa}\left(1 \mathrm{~Pa}=1 \mathrm{~N} / \mathrm{m}^2\right)$. Since “standard” atmospheric pressure is $p_m=101,325 \mathrm{~Pa}$ [3], that level corresponds to a relative deviation from equilibrium that is less than 160 parts per million ( $\mathrm{ppm})$ or $p_1 / p_m=0.016 \%$.

If we assume that any parameter of interest (e.g., temperature, density, pressure) varies smoothly in time and space, we can approximate the parameter’s value at a point (in space or time) if we know the parameter’s value at some nearby point (typically, the state of stable equilibrium) and the value of its derivatives evaluated at that point. ${ }^2$ The previous statement obscures the true value of the Taylor series because it is frequently used to permit substitution of the value of the derivative, as we will see throughout this textbook.

Let us start by examining the graph of some arbitrary real function of position, $f(x)$, shown in Fig. 1.1. At position $x_o$, the function has a value, $f\left(x_o\right)$. At some nearby position, $x_o+d x$, the function will have some other value, $f\left(x_o+d x\right.$ ), where we will claim that $d x$ is a small distance without yet specifying what we mean by “small.”

The value of $f\left(x_o+d x\right)$ can be approximated if we know the first derivative of $f(x)$ evaluated at $x_o$.
$$f\left(x_o+d x\right) \cong f\left(x_o\right)+\left.\frac{d f}{d x}\right|_{x_0} d x$$
As can be seen in Fig. 1.1, the approximation of Eq. (1.1) produces a value that is slightly less than the actual value $f\left(x_o+d x\right)$ in this example. That is because the actual function has some curvature that happens to be upward in this case. The differential, $d x$, is used to represent both finite and infinitesimal quantities, depending upon context. For approximations, $\mathrm{d} x$ is assumed to be small but finite. For derivation of differential equations, it is assumed to be infinitesimal.

We can improve the approximation by adding another term to the Taylor series expansion of $f(x)$ that includes a correction proportional to the second derivative of $f(x)$, also evaluated at $x_o$. For the example in Fig. 1.1, the curvature is upward so the second derivative of $f(x)$, evaluated at $x_o$, is a positive number, so $\left(d^2 f / d x^2\right)_{x_o}>0$.

## 物理代写|声学代写声学代考|五种最有用的数学技巧

• 代换
• 泰勒级数
• 乘积法则或部分积分
• 对数微分
• 加勒特几何第一定律:“看起来相似的角是相似的。”

## 物理代写|声学代写Acoustics代考|Taylor Series

$$f\left(x_o+d x\right) \cong f\left(x_o\right)+\left.\frac{d f}{d x}\right|_{x_0} d x$$从图1.1中可以看出，Eq.(1.1)的近似产生的值略小于实际值 $f\left(x_o+d x\right)$ 在这个例子中。这是因为实际的函数有一个向上的曲率。微分， $d x$，用于表示有限和无穷小的量，具体取决于上下文。对于近似值， $\mathrm{d} x$ 假设是小而有限的。对于微分方程的推导，假设它是无穷小的 我们可以通过在$f(x)$的泰勒级数展开中加入另一项来改进近似，该项包含与$f(x)$的二阶导数成比例的修正，也在$x_o$处求值。对于图1.1中的例子，曲率是向上的，因此$f(x)$的二阶导数在$x_o$处的值是一个正数，因此$\left(d^2 f / d x^2\right)_{x_o}>0$。

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。