What has come to be known as the Burgers equation was derived earlier in this book as Eq. (54) of Chapter 3, and is repeated here for convenience:
$$ \begin{equation} \frac{\partial p}{\partial x} - \frac{\delta}{2c_{0}^{3}}\frac{\partial^{2}p}{\partial \tau^{2}} = \frac{\beta p}{\rho_{0}c_{0}^{3}}\frac{\partial p}{\partial\tau},\tag{189}\label{eq:189} \end{equation} $$
where
$$ \begin{equation} \delta = \rho_{0}^{-1}\left[\frac{4}{3}\mu + \mu_{B} + \kappa\left(c_{v}^{-1} - c_{p}^{-1}\right)\right]\tag{190}\label{eq:190} \end{equation} $$
is referred to as the diffusivity of sound (Lighthill, 1956), $\mu$ is shear viscosety, $\mu_{B}$ is bulk viscosity, $\kappa$ is thermal conduct1vity, and $c_{v}$ and $c_{p}$ are the specific heat at constant volume and constant pressure, respectively. Equation $\eqref{eq:189}$ accounts explicitly for the effect of thermoviscous diesipalion on the propagation of finite-amplitude sound.
The present section discusses the properties of the Burgers equation and some of its solutions. Also, since the historical introduciion in Chapter1 stops with the beginning of World War II, the discussion here begins with a historical sketch of the origins of the Burgers equation, with an attempt to explain why this partial differential equation, which has played a central role in nonlinear acoustics over the second half of the twentieth century, happened to be called the Burgers equation.
本书第 3 章的公式 (54) 早先已经推导出 Burgers 方程，为方便起见，在此重复一遍:
$$ \begin{equation} \frac{\partial p}{\partial x} - \frac{\delta}{2c_{0}^{3}}\frac{\partial^{2}p}{\partial \tau^{2}} = \frac{\beta p}{\rho_{0}c_{0}^{3}}\frac{\partial p}{\partial\tau},\tag{189} \end{equation} $$
其中
$$ \delta = \rho_{0}^{-1}\left[\frac{4}{3}\mu + \mu_{B} + \kappa\left(c_{v}^{-1} - c_{p}^{-1}\right)\right]\tag{190} $$
被称为声扩散系数（Lighthill，1956 年），$\mu$ 是剪切粘度，$\mu_{B}$ 是体积粘度，$\kappa$ 是热导率，$c_{v}$ 和 $c_{p}$ 分别是恒定体积和恒定压力下的比热。方程 $\eqref{eq:189}$ 明确考虑了热粘模数对有限振幅声音传播的影响。
本节将讨论 Burgers 方程的性质及其部分解法。此外，由于第一章的历史介绍以第二次世界大战开始为起点，这里的讨论从 Burgers 方程的起源开始，试图解释为什么这个在二十世纪下半叶非线性声学中发挥核心作用的偏微分方程被称为 Burgers 方程。
History
Workers in applied mathematics and fields related to fluid mechanics loosely refer to any partial differential equation of the form
$$ \begin{equation} a\frac{\partial\theta}{\partial\xi_{2}} + b\frac{\partial\theta}{\partial\xi_{1}} + c\theta\frac{\partial\theta}{\partial\xi_{1}} + d\frac{\partial^{2}\theta}{\partial\xi_{1}^{2}} = 0\tag{191}\label{eq:191} \end{equation} $$
(generally with the coefficient $b$ set to zero) as the Burgers equation. Here the dependent variable $\theta(\xi_{1},\xi_{2})$ is a function of two independent coordinates, one usually identified as a time and the other as a distance variable. The partial differential equation is quasilinear, because the nonlinear term involves only a first derivative, but the highest-order derivative, appearing in a linear term, is linear and second order.
应用数学和流体力学相关领域的工作人员将任何如下形式的偏微分方程
$$ \begin{equation} a\frac{\partial\theta}{\partial\xi_{2}} + b\frac{\partial\theta}{\partial\xi_{1}} + c\theta\frac{\partial\theta}{\partial\xi_{1}} + d\frac{\partial^{2}\theta}{\partial\xi_{1}^{2}} = 0\tag{191} \end{equation} $$
统称为(一般将系数 $b$ 设为零) 伯格斯方程。这里的因变量 $\theta(\xi_{1},\xi_{2})$ 是两个独立坐标的函数，其中一个通常是时间变量，另一个是距离变量。该偏微分方程是准线性方程，因为非线性项只涉及一阶导数，但出现在线性项中的最高阶导数是线性的二阶导数。
The earmarks of the Burgers equation are these two features, plus the feature that the equation has an intrinsic parabolic nature, analogous to the partial differential equation governing time-dependent heat conduction in one dimension. This parabolic nature is embodied in the fact that the second-derivative term involves diffeientiation with respect to only one of the two coordinates. The quantities $a$, $b$, $c$, and $d$ are often taken as constants, but in some discussions. one or more are taken as functions of one or both of the independent coordinates.
Burgers 方程的特征就是这两个特点，再加上该方程具有内在的抛物线性质，类似于在一维中控制含时热传导的偏微分方程。 这种抛物线性质体现在二次导数项只涉及两个坐标中一个坐标的微分。$a$、$b$、$c$ 和 $d$ 通常被视为常数，但在某些讨论中，一个或多个常数被视为一个或两个独立坐标的函数。
The first appearance of such an equation in the archival literature, at least in a fluid-dynamic context, is generally considered to be in a 1915 paper published by Bateman in a meteorological journal not normally read by research workers in fluid dynamics. (The authors have been told that such an equation may hnve appeared in a purely mathematical context in one of Forsyth’s books or papers, written sometime between 1895 and 1910, but they have not yet checked to confirm this.) Bateman’s paper was largely review and cited an impressive list of fundamental research papers by his contemporaries and predecessors. The emphasis was on the question of the existence of discontinuous motion in fluids, primarily in the context of wakes behind moving bodies, for which one model is discontinuous potential flow (with the fluid taken as incompressible), and where the discontinuity was one of tangential fluid velocity, and not what we call a shock.
在档案文献中，至少是在流体动力学背景下，首次出现这样的方程，一般认为是在 Bateman 1915 年发表的一篇论文中，该论文发表在流体动力学研究人员通常不会阅读的气象学期刊上。(作者被告知，在 1895 年至 1910 年期间，这样一个方程可能曾以纯数学的形式出现在 Forsyth 的某本著作或某篇论文中，但他们尚未对此进行核实)。Bateman 的论文主要是综述性的，引用了他同时代以及前辈的大量基础研究论文。论文的重点是流体中存在不连续运动的问题，主要是在运动物体后方的湍流中，其中一个模型是不连续势能流（流体不可压缩），不连续是流体切向速度的不连续，而非我们所说的冲击。
It was understood that this discontinuity would not be abrupt were the viscosity taken into account, and to show a relatively simple example of an equation having some resemblance to the equations of fluid mechanics and where the limit of zero viscosity led to a discontinuity, Bateman wrote down the partial differential equation
$$ \begin{equation} \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \nu\frac{\partial^{2}u}{\partial x^{2}}.\tag{192}\label{eq:192} \end{equation} $$
This can be viewed as a one-dimensional version of the Navier—Stokes equation, which appears as Eq. (2) of Chapter 3, with the pressure term set to zero, with $\rho$ taken as constant, and with $\nu = \rho^{-1}\left(\frac{4}{3}\mu + \mu_{B}\right)$. Bateman then proceeded to find a particular solution of this equation by a method similar to that discussed further in Section 5.3, and showed that his solution did indeed yield a discontinuity in the limit of vanishing $\nu$. It appears, however, that no further attention was given to this partial dilferential equation in subsequent literature.
为了举一个相对简单的例子，说明一个方程与流体力学方程有某些相似之处，以及零粘度极限会导致不连续性，Bateman 写下了偏微分方程
$$ \begin{equation} \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \nu\frac{\partial^{2}u}{\partial x^{2}}.\tag{192} \end{equation} $$
这可以看作是 Navier—Stokes 方程的一维版本，即第 3 章中的公式 (2)，压力项设为零，$\rho$ 为常数，$\nu = \rho^{-1}\left(\frac{4}{3}\mu + \mu_{B}\right)$。Bateman 接着用一种类似于在第 5.3 节中进一步讨论的方法找到了这个方程的一个特定解，并证明了他的解确实在 $\nu$ 消失的极限中产生了不连续性。然而，在随后的文献中，人们似乎没有进一步关注这个偏微分方程。
The contribution by Burgers to this history begins witha series of papers concerned with turbulence, which were published in the Netherlands during the early years of World War II, and of which the most frequently cited is Burgers (1940). Following the end of the war, von Kàrmàn and von Mises established a set of review volumes to report advances in applied mechanics; for the inaugural volume of this series, Burgers was invited to write an expository paper (Burgers, 1948), intended to fully describe and extend the theoretical ideas that he had developed in the earlier papers.
Burgers 对这段历史的贡献始于二战初期在荷兰发表的一系列有关湍流的论文，其中最常被引用的是 Burgers (1940)。战争结束后，von Kàrmàn 和 von Mises 创办了一套综述集，以报道应用力学的进展；Burgers 应邀为这套评论集的首卷撰写了一篇阐述性论文（Burgers, 1948），旨在全面描述和扩展他在前几篇论文中提出的理论观点。
Burgers’s 1948 article was apparently widely noticed and continues to have an influence on turbulence research to the present day. (The Term “Burgerlence” to describe phenomena governed by Burgers’s mathematical model has become part of the vocabulary of workers in turbulence.) At the outset of the paper, Burgers set down two coupled equations (without much of a motivational preamble), which he then proceeded to discuss in considerable detail with the objective of demonstrating that their solutions had many features analogous to what is commonly associated with turbulence. The second of these equations was
$$ \begin{equation} \frac{\partial v}{\partial t} = \frac{U}{b}v + \nu\frac{\partial^{2}v}{\partial y^{2}} - 2v\frac{\partial v}{\partial y},\tag{193}\label{eq:193} \end{equation} $$
where $v$ and $U$ were his two dependent variables, and $b$ and $\nu$ were constants, it being understood that $U$ depended only on $t$, while $v$ depended on both $t$ and $y$.
Burgers 1948 年的文章显然受到了广泛关注，并对湍流研究产生了持续至今的影响。( “Burgerlence” 一词以描述 Burgers 数学模型支配下的现象，已成为湍流研究工作者的词汇之一)。在论文的开头，Burgers 列出了两个耦合方程（没有过多的动机性前言），然后他对这两个方程进行了相当详细的讨论，目的是证明它们的解具有许多与通常的湍流类似的特征。其中第二个方程是
$$ \begin{equation} \frac{\partial v}{\partial t} = \frac{U}{b}v + \nu\frac{\partial^{2}v}{\partial y^{2}} - 2v\frac{\partial v}{\partial y},\tag{193} \end{equation} $$
其中，$v$ 和 $U$ 是他的两个因变量，$b$ 和 $\nu$ 是常数，不言而喻，$U$ 只取决于 $t$，而 $v$ 则取决于 $t$ 和 $y$。
Burgers’s discussion is somewhat murky on the relation of the variables that appear here to actual physical variables: he states tha $U$ is the analog of “mean motion in the case of a liquid flowing through a channel,” while $v$, when different from zero, corresponds to “turbulence in the channel,” $b$ corresponds to the width of the channel, and $\nu$ is a coefficient associated with “frictional effects.”
Burgers 在讨论中对此处出现的变量与实际物理变量的关系有些模糊：他说 $U$ 类似于 “液体流经通道时的平均运动”，而 $v$ 非零时，对应于 “通道中的湍流”，$b$ 对应于通道的宽度，而 $\nu$ 是与 “摩擦效应 “相关的系数。
Farther on in the paper, he considers the case in which $U$ is a constant and gives arguments to the effect that, even though $U$ may be nonzero, there may be some domains in $y$ where it may be an appropriate approximation to neglect the term $(U/b)v$ in the above equation, in which case the considered partial differential equation becomes
$$ \begin{equation} \frac{\partial v}{\partial t} + 2v\frac{\partial v}{\partial y} - \nu\frac{\partial^{2}v}{\partial y^{2}} = 0.\tag{194}\label{eq:194} \end{equation} $$
Still farther on in the paper, Burgers develops an explicit solution of this partial differential equation that corresponds to a discontinuity in $v$ in the limit as $\nu\rightarrow 0$. The solution is similar to that given by Bateman (1915) and to what is presented further below in Section 5.3. Burgers gives no reference to Bateman (1915), but in all fairness there is no reason to have expected him to have done so.
在论文的更深层处，他考虑了 $U$ 是常数的情况，并给出了一些论据，大意是：即使 $U$ 可能不为零，但在 $y$ 的某些域中，忽略上述方程中的 $(U/b)v$ 项可能是一个合适的近似值，在这种情况下，所考虑的偏微分方程变为
$$ \begin{equation} \frac{\partial v}{\partial t} + 2v\frac{\partial v}{\partial y} - \nu\frac{\partial^{2}v}{\partial y^{2}} = 0.\tag{194} \end{equation} $$
在论文的更远处，Burgers 提出了这个偏微分方程的显式解，它对应于 $v$ 在 $\nu\rightarrow 0$ 时的极限不连续。该解与 Bateman（1915 年）给出的解以及下文第 5.3 节进一步介绍的解相似。Burgers 没有提到 Bateman (1915)，但平心而论，我们没有理由期望如此 (1915)。
In the concluding section of his paper, Burgers(1948) states that “the group of terms
$$ \begin{equation} \frac{\partial v}{\partial t} + 2v\frac{\partial v}{\partial y} - \nu\frac{\partial^{2}v}{\partial y^{2}}\tag{195}\label{eq:195} \end{equation} $$
will find its closest analogy [in fluid-dynamic contexts other than turbulence] in the terms
$$ \begin{equation} \left(\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x}\right) - \nu\frac{\partial^{2}u}{\partial x^{2}}\tag{196}\label{eq:196} \end{equation} $$
which are decisive in determining the appearance of shock waves in the supersonic motion of a gas.” This is the only place in the paper where there is any hint that there might he an analogous partial differential equation of the same general form that would govern nonlinear sound propagation.
Burgers（1948 年）在其论文的结论部分指出：“这一组项
$$ \begin{equation} \frac{\partial v}{\partial t} + 2v\frac{\partial v}{\partial y} - \nu\frac{\partial^{2}v}{\partial y^{2}}\tag{195} \end{equation} $$
在流体动力学背景下（除了湍流之外）将在以下项中找到最接近的类比
$$ \begin{equation} \left(\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x}\right) - \nu\frac{\partial^{2}u}{\partial x^{2}}\tag{196} \end{equation} $$
这些项决定了气体超音速运动中激波的出现” 这是论文中唯一一处暗示可能存在一个类似的偏微分方程，其形式与非线性声音传播有关。
The first appearance of an equation that is of the form of Eq. $\eqref{eq:191}$ and that also had a tangible connection with nonlinear acoustics can arguably be attributed to Julian Cole. In 1949, Lagerstrom, Cole, and Trilling (LCT) issued a report, which they stated was intended to be in the nature of a progress report rather than a report on finished research. The report was almost entirely concerned with the linearized equations of viscous compressible flow, but Appendix B dealt with “nonlinear longitudinal waves.” A principal result, derived in the context of an ideal gas. in that appendix was their Eq. (B24), which had the form
$$ \begin{equation} \phi_{t} + \frac{1}{4}(\gamma + 1)(\phi_{x}^{2} - w_{-\infty}^{2}) = \frac{2}{3}\nu^{*}\phi_{xx},\tag{197}\label{eq:197} \end{equation} $$
where the subscripts imply partial derivatives and $\phi$ is a velocity potential, $\nu^{*}$ is a kinematic viscosity, $\gamma = c_{p}/c_{v}$, is the specific heat ratio, and $w_{-\infty}$ is the value of $u = \partial\phi/\partial x$ in the limit of $x\rightarrow -\infty$.
$\eqref{eq:191}$ 形式的方程首次出现，并且与非线性声学有着切实的联系，可以说是 Julian Cole 的功劳。1949 年，Lagerstrom、Cole 和 Trilling（LCT）发表了一份报告，他们表示该报告的性质是进度报告，而非已完成研究的报告。报告几乎全部涉及粘性可压缩流的线性化方程，但附录 B 涉及了 “非线性纵波”。在该附录中，以理想气体为背景推导出的一个主要结果是公式 (B24)，其形式为
$$ \begin{equation} \phi_{t} + \frac{1}{4}(\gamma + 1)(\phi_{x}^{2} - w_{-\infty}^{2}) = \frac{2}{3}\nu^{*}\phi_{xx},\tag{197} \end{equation} $$
其中，下标表示偏导数，$\phi$ 是速度势，$\nu^{*}$ 是运动粘度，$\gamma = c_{p}/c_{v}$，是比热比，$w_{-\infty}$ 是 $u = \partial\phi/\partial x$ 在 $x\rightarrow -\infty$ 时的值。
The velocity $u$ is the flow velocity relative to a reference sound speed, here denoted by $c^{*}$ (and denoted by $c_{0}$ in the following discussion). If one takes the $x$ derivative of the above equation and reexpresses it in the notation of the present text, the result takes the form
$$ \begin{equation} u_{t} + \beta u u_{x} = \frac{\delta}{2}u_{xx},\tag{198}\label{eq:198} \end{equation} $$
where $\beta = \frac{1}{2}(\gamma + 1)$ is the coefficient of nonlinearity for an ideal gas and $\delta$ is the diffusivity of sound, given by Eq. $\eqref{eq:190}$, in the limit where the bulk viscosity $\mu_{B}$ and the thermal conductivity $\kappa$ are both set to zero. Although Eq. $\eqref{eq:198}$ does not appear explicitly in the report, it was set down (without any derivation but with a citation to the 1949 report) in a paper submitted in April 1950 by Cole (1951).
速度 $u$ 是相对于参考声速的流速，这里用 $c^{*}$ 表示（在下面的讨论中用 $c_{0}$ 表示）。如果对上式进行 $x$ 导数运算，并用本文的符号重新表达，结果为
$$ \begin{equation} u_{t} + \beta u u_{x} = \frac{\delta}{2}u_{xx},\tag{198} \end{equation} $$
其中，$\beta = \frac{1}{2}(\gamma + 1)$ 是理想气体的非线性系数，$\delta$ 是声音的耗散性，由公式 $\eqref{eq:190}$ 给出，其中在体积粘度 $\mu_{B}$ 和热导率 $\kappa$ 都设为零极限。尽管公式 $\eqref{eq:198}$ 在报告中没有明确出现，但它是由 Cole（1951）在 1950 年 4 月提交的一篇论文中写下的（没有任何推导，但引用了 1949 年的报告）。
Because Eq. $\eqref{eq:198}$ is not invariant under a Galilean transformation, it is incomplete unless it is accompanied by a specification of the coordinate system in which it applies. The authors were concerned at that time with transonic flow, with the flow velocity taken as $c_{0} + u(x,t)$. If one instead adopts a coordinate system with its origin moving to the right with speed $c_{0}$ relative to the LCT coordinate origin, and denotes the $x$, $t$, and $u$ that appear in the above two equations by $x_{\text{LCT}}$, $t_{\text{LCT}}$, and $u_{\text{LCT}}$, then the transformed position and time coordinates are given by
$$ \begin{equation} x_{\text{trans}} = x_{\text{LCT}} - c_{0}t_{\text{LCT}},\quad t_{\text{trans}} = t_{\text{LCT}},\tag{199}\label{eq:199} \end{equation} $$
and the total flow velocity in the transformed coordinate system, relative to a flow at rest, is simply
$$ \begin{equation} u_{\text{trans}}(x_{\text{trans}},t_{\text{trans}}) = u_{\text{LCT}}(x_{\text{LCT}},t_{\text{LCT}})\tag{200}\label{eq:200} \end{equation} $$
since the original $u_{\text{LCT}}$ was flow velocity relative to the sound speed. It also follows from the above relations that
$$ \begin{equation} \frac{\partial u_{\text{LCT}}}{\partial t_{\text{LCT}}} = \frac{\partial u_{\text{trans}}}{\partial t_{\text{trans}}} + c_{0}\frac{\partial u_{\text{trans}}}{\partial x_{\text{trans}}},\quad \frac{\partial u_{\text{LCT}}}{\partial t_{\text{LCT}}} = \frac{\partial u_{\text{trans}}}{\partial t_{\text{trans}}}.\tag{201}\label{eq:201} \end{equation} $$
Consequently, Eq. $\eqref{eq:198}$ transforms to
$$ \begin{equation} u_{t} + (c_{0} + \beta u)u_{x} = \frac{\delta}{2}u_{xx}.\tag{202}\label{eq:202} \end{equation} $$
Here, for brevity, the subscript “trans” has been omitted on $x$, $t$, and $u$, but it should be understood that these denote quantities that are different from what appears in Eq. $\eqref{eq:198}$.
由于公式 $\eqref{eq:198}$ 在伽利略变换下并非守恒，因此它是不完整的，除非同时说明它适用的坐标系。作者当时关注的是超音速流动，流速取为 $c_{0}+u(x,t)$。如果采用一个坐标系，其原点相对于 LCT 坐标原点以 $c_{0}$ 的速度向右移动，并用 $x_{\text{LCT}}$、$t_{\text{LCT}}$ 和 $u_{\text{LCT}}$ 表示上述两个方程中出现的 $x$、$t$ 和 $u$，那么变换后的位置坐标和时间坐标为
$$ \begin{equation} x_{\text{trans}} = x_{\text{LCT}} - c_{0}t_{\text{LCT}},\quad t_{\text{trans}} = t_{\text{LCT}},\tag{199} \end{equation} $$
相对于静止的流体，变换坐标系中的总流速为
$$ \begin{equation} u_{\text{trans}}(x_{\text{trans}},t_{\text{trans}}) = u_{\text{LCT}}(x_{\text{LCT}},t_{\text{LCT}})\tag{200} \end{equation} $$
因为最初的 $u_{\text{LCT}}$ 是相对于声速的流速。根据上述关系还可以得出
$$ \begin{equation} \frac{\partial u_{\text{LCT}}}{\partial t_{\text{LCT}}} = \frac{\partial u_{\text{trans}}}{\partial t_{\text{trans}}} + c_{0}\frac{\partial u_{\text{trans}}}{\partial x_{\text{trans}}},\quad \frac{\partial u_{\text{LCT}}}{\partial t_{\text{LCT}}} = \frac{\partial u_{\text{trans}}}{\partial t_{\text{trans}}}.\tag{201} \end{equation} $$
因此，公式 $\eqref{eq:198}$ 变式为
$$ \begin{equation} u_{t} + (c_{0} + \beta u)u_{x} = \frac{\delta}{2}u_{xx}.\tag{202} \end{equation} $$
为简洁起见，这里省略了 $x$、$t$ 和 $u$ 的下标 “$\text{trans}$"，但应该理解的是，这些表示的量与公式 $\eqref{eq:198}$ 中出现的量不同。
Apart from the restriction of its derivation to ideal gases without thermal conductivity. Eq. $\eqref{eq:202}$ applies to the same circumstances as Eq.$\eqref{eq:189}$, which is what is referred to as the Burgers equation in this book. To show that the two equations are actually equivalent to the order of approximations with which either was derived, one first divides through by $c_{0}+\beta u$ and, since $\beta|u|\ll c_{0}$, approximates the reciprocal by $1/c_{0}-(\beta/c_{0}^{2})u$, with the intermediate result
$$ \begin{equation} u_{x} + \frac{1}{c_{0}}u_{t} - \frac{\beta}{c_{0}^{2}}uu_{t} = \frac{1}{c_{0}}\frac{\delta}{2}u_{xx} - \frac{\delta\beta}{2c_{0}^{2}} uu_{xx}. \tag{203}\label{eq:203} \end{equation} $$
除了其推导仅限于无热传导的理想气体之外，公式 $\eqref{eq:202}$ 与公式 $\eqref{eq:189}$ 适用于相同的情况, 也就是本书中所说的 Burgers 方程。为了证明这两个方程实际上等价于其中任何一个方程的近似阶数，我们首先通过 $c_{0}+\beta u$ 除以 $c_{0}$，由于 $\beta|u|\ll c_{0}$，所以用 $1/c_{0}-(\beta/c_{0}^{2})u$ 近似倒数，得出中间结果
$$ \begin{equation} u_{x} + \frac{1}{c_{0}}u_{t} - \frac{\beta}{c_{0}^{2}}uu_{t} = \frac{1}{c_{0}}\frac{\delta}{2}u_{xx} - \frac{\delta\beta}{2c_{0}^{2}} uu_{xx}. \tag{203} \end{equation} $$
The two dominant terms are the first two on the left, although the others can have major accumulative effects over large propagation distances or large propagation times. Both $\delta$ and $u$ are regarded as small in a relative sense, so the last term on the right can be discarded. Furthermore, in the first term on the right, it is consistent, given the approximate balancing out of the first two terms on the left, to make the substitution $\partial/\partial x\rightarrow -c_{0}^{-1}\partial/\partial t$. This then yields
$$ u_{x} + \frac{1}{c_{0}}u_{t} - \frac{\beta}{c_{0}^{2}}uu_{t} = \frac{\delta}{2c_{0}^{3}}u_{tt}. \tag{204}\label{eq:204} $$
Finally, if one introduces the retarded time $\tau = t - x/c_{0}$, Eq. $\eqref{eq:204}$ reduces to
$$ u_{x} - \frac{\beta}{c_{0}^{2}}uu_{\tau} = \frac{\delta}{2c_{0}^{3}}u_{\tau\tau}. \tag{205}\label{eq:205} $$
The substitution of the plane-wave relation $u = p/\rho_{0}c_{0}$ then yields Eq. $\eqref{eq:189}$.
两个主要项是左边的前两个项，尽管其他项在大的传播距离或大的传播时间内会产生重大的累积效应。$\delta$ 和 $u$ 都被认为是相对意义上的小项，因此右边的最后一项可以舍弃。此外，在右边第一个项中，考虑到左边前两个项的近似平衡，将 $\partial/\partial x\rightarrow -c_{0}^{-1}\partial/\partial t$ 替换为 $\partial/\partial t$ 是一致的。这样就得出
$$ \begin{equation} u_{x} + \frac{1}{c_{0}}u_{t} - \frac{\beta}{c_{0}^{2}}uu_{t} = \frac{\delta}{2c_{0}^{3}}u_{tt}. \tag{204} \end{equation} $$
最后，如果引入延迟时间 $\tau = t - x/c_{0}$，公式 $\eqref{eq:204}$ 化简为
$$ \begin{equation} u_{x} - \frac{\beta}{c_{0}^{2}}uu_{\tau} = \frac{\delta}{2c_{0}^{3}}u_{\tau\tau}. \tag{205} \end{equation} $$
然后，将平面波关系 $u = p/\rho_{0}c_{0}$ 代入，得到公式 $\eqref{eq:189}$。
Although the above shows that the Cole equation, appearing here as Eq. $\eqref{eq:198}$, is trivially related to an equation governing nonlinear sound propagation through a medium nominally at rest, this was not explicitly pointed out in either the 1949 report or the closely associated journal article by Cole(1951). The first author to derive a partial differential equation specifically for plane-wave sound propagation was apparently Mendousse (1953). Mendousse refers to Cole (1951) only tangentially and does not refer to the Lagerstrom, Cole, and Trilling (1949) report at all; he apparently did not perceive the connection of Eq. $\eqref{eq:198}$ with sound propagation in the sense discussed above. His derivation begins with a Lagrangian description of one-dimensional flow in an ideal gas with viscosity. Rewritten in terms of the symbols used in Chapter 3, Section 2, his starting point was a “Navier—Stokes equation,” which would ordinarily have the correct form
$$ \begin{equation} \rho_{0}\frac{\partial^{2}\xi}{\partial t^{2}} - \frac{4}{3}\mu\frac{\partial}{\partial a}\left[\left(1 + \frac{\partial\xi}{\partial a}\right)\frac{\partial^{2}\xi}{\partial a\partial t}\right] + \frac{\partial P}{\partial a} = 0.\tag{206}\label{eq:206} \end{equation} $$
where $\xi$ is particle displacement relative to a references position of $a$, in which case the actual $x$ coordinate of the considered fluid particle is
$$ \begin{equation} x = a + \xi(a,t).\tag{207}\label{eq:207} \end{equation} $$
尽管上述内容表明，Cole 方程（在此以公式 $\eqref{eq:198}$ 出现）与表征在名义上静止的介质中非线性声传播的方程有着微妙的联系，但无论是在 1949 年的报告中，还是在 Cole（1951 年）发表的密切相关的期刊文章中，都没有明确指出这一点。第一个推导出专门用于平面波声传播的偏微分方程的作者显然是 Mendousse（1953 年）。Mendousse 只是略微提到了 Cole (1951)，而根本没有提到 Lagerstrom、Cole 和 Trilling（1949）的报告；他显然没有意识到方程 $\eqref{eq:198}$ 与上述意义上的声传播的联系。他的推导始于对具有粘度的理想气体中一维流动的拉格朗日量描述。根据第 3 章第 2 节中使用的符号重写，他的出发点是 “Navier—Stokes 方程”，其正确形式通常为
$$ \begin{equation} \rho_{0}\frac{\partial^{2}\xi}{\partial t^{2}} - \frac{4}{3}\mu\frac{\partial}{\partial a}\left[\left(1 + \frac{\partial\xi}{\partial a}\right)\frac{\partial^{2}\xi}{\partial a\partial t}\right] + \frac{\partial P}{\partial a} = 0.\tag{206} \end{equation} $$
其中，$\xi$ 是相对于参考位置 $a$ 的粒子位移，此时所考虑的流体粒子的实际 $x$ 坐标为
$$ \begin{equation} x = a + \xi(a,t).\tag{207} \end{equation} $$
Mendousse omitted the factor $1 + (\partial\xi/\partial a)$ in the viscous term, but including it would have been of no consequence, given the further approximations discussed below. Also, with the neglect of the thermal conductivity and witn the implicit neglect of second-order terms multiplied by the viscosity, it is consistent [see Eq. (3) of Chapter 3] to take the pressure as a function of only density, and independent of entropy. Moreover, conservation of mass requires that
$$ \begin{equation} \rho_{0} = \left(1 + \frac{\partial\xi}{\partial a}\right)\rho.\tag{208}\label{eq:208} \end{equation} $$
and therefore expansion of the pressure to second order in $\partial\xi/\partial a$ yields
$$ \begin{equation} P = P_{0} - \rho_{0}c_{0}\left[\frac{\partial\xi}{\partial a} - \beta\left(\frac{\partial\xi}{\partial a}\right)^{2}\right],\tag{209}\label{eq:209} \end{equation} $$
where $\beta$ is the coefficient of nonlinearity. Thus, with the approximations as described, Eq. $\eqref{eq:206}$ takes the form
$$ \rho_{0}\frac{\partial^{2}\xi}{\partial t^{2}} - \rho_{0}\delta\frac{\partial^{3}}{\partial a^{2}\partial t} - \rho_{0}c_{0}^{2}\frac{\partial^{2}\xi}{\partial a^{2}} + 2\rho_{0}c_{0}\beta\frac{\partial\xi}{\partial a}\frac{\partial^{2}\xi}{\partial a^{2}} = 0,\tag{210}\label{eq:210} $$
which is comparable in form to, but not quite the same as, the one-dimensional version of the Westervelt equation, given by Eq. (47) of Chapter 3. [Mendousse does not number hisequations, but the above equation appears, albeit in a different notation, at the end of page 53 in Mendousse (1953).]
Mendousse 在粘性项中省略了系数 $1 + (\partial\xi/\partial a)$ ，但考虑到下面讨论的进一步近似，加入这个系数并没有什么影响。此外，由于忽略了热导率，并隐含地忽略了乘以粘度的二阶项，把压力看作仅是密度的函数而与熵无关是一致的（见第 3 章公式 (3)）。此外，质量守恒要求
$$ \begin{equation} \rho_{0} = \left(1 + \frac{\partial\xi}{\partial a}\right)\rho.\tag{208} \end{equation} $$
因此，将压力在 $\partial\xi/\partial a$ 的二阶展开为
$$ \begin{equation} P = P_{0} - \rho_{0}c_{0}\left[\frac{\partial\xi}{\partial a} - \beta\left(\frac{\partial\xi}{\partial a}\right)^{2}\right],\tag{209} \end{equation} $$
其中，$\beta$ 是非线性系数。因此，根据上述近似，公式 $\eqref{eq:206}$ 变为
$$ \begin{equation} \rho_{0}\frac{\partial^{2}\xi}{\partial t^{2}} - \rho_{0}\delta\frac{\partial^{3}}{\partial a^{2}\partial t} - \rho_{0}c_{0}^{2}\frac{\partial^{2}\xi}{\partial a^{2}} + 2\rho_{0}c_{0}\beta\frac{\partial\xi}{\partial a}\frac{\partial^{2}\xi}{\partial a^{2}} = 0,\tag{210} \end{equation} $$
这与 Westervelt 方程的一维形式相似，但并不完全相同，Westervelt 方程的一维版本由第 3 章公式 (47) 给出。[Mendousse 没有对他的方程进行编号，但上述方程出现在 Mendousse (1953) 的第 53 页末尾，尽管使用了不同的符号。]
Mendousse subsequently considered Iwo transformations of Eq. $\eqref{eq:210}$ above. The first resulted from the change of coordinates to $\chi = a - c_{0}t$ and $t$, in terms of which $\partial/\partial a\rightarrow \partial/\partial\xi$ and $\partial/\partial t\rightarrow \partial/\partial t - c_{0}\partial/\partial\chi$. In the accompanying discussion, Mendousse gives quantitative arguments for expecting that, for a wave propagating in the $+x$ direction, derivatives with respect to $t$ should be much smaller than $c_{0}$ times derivatives with respect to $\chi$, and he thus neglected the terms $\rho_{0}\partial^{2}\xi/\partial t^{2}$ and $-\rho_{0}\delta\partial^{3}\xi/\partial\chi^2\partial t$ in the resulting transformed expression. The remaining terms yielded the result
$$ \begin{equation} -2\rho_{0}c_{0}\frac{\partial\theta}{\partial t} + \rho_{0}c_{0}\delta\frac{\partial^{2}\theta}{\partial\chi^{2}} + 2\rho_{0}c_{0}^{2}\beta\theta\frac{\partial\theta}{\partial\chi} = 0,\tag{211}\label{eq:211} \end{equation} $$
where $\theta$ here abbreviates $\partial\xi/\partial\chi$. [Mendousse gives this ina dimensionless form in the middle of the first column of p.54 as $\theta^{\prime\prime} = \dot{\theta} - 2\theta\theta^{\prime}$.]
Mendousse 随后考虑了上述公式 $\eqref{eq:210}$ 的两种变换。第一种是坐标变为 $\chi = a - c_{0}t$ 和 $t$，其中 $\partial/\partial a\rightarrow \partial/\partial/xi$ 和 $\partial/\partial t\rightarrow \partial/\partial t - c_{0}\partial/\partial\chi$。在随附的讨论中，Mendousse 给出了定量论据，说明对于沿 $+x$ 方向传播的波，与 $t$ 有关的导数应该比与 $\chi$ 有关的导数的 $c_{0}$ 倍小得多、因此，他忽略了转换表达式中的 $\rho_{0}\partial^{2}\xi/\partial t^{2}$ 和 $-\rho_{0}\delta\partial^{3}\xi/\partial\chi^2\partial t$ 项。其余项的结果是
$$ \begin{equation} -2\rho_{0}c_{0}\frac{\partial\theta}{\partial t} + \rho_{0}c_{0}\delta\frac{\partial^{2}\theta}{\partial\chi^{2}} + 2\rho_{0}c_{0}^{2}\beta\theta\frac{\partial\theta}{\partial\chi} = 0,\tag{211} \end{equation} $$
其中 $\theta$ 在这里为 $\partial\xi/\partial\chi$ 的缩写. [Mendousse 在第 54 页的第一列中间给出了无量纲形式，即 $\theta^{\prime\prime} = \dot{\theta} - 2\theta\theta^{\prime}$]
The second considered transformation resulted from the change of coordinates to $a$ and $\tau = t - a/c_{0}$, in terms of which $\partial/\partial a\rightarrow \partial/\partial a + c_{0}^{-1}\partial/\partial\tau$ and $\partial/\partial t\rightarrow\partial/\partial\tau$. Although Mendousse omits the details, it appears that his intent was that one neglect derivatives with respect to $a$ when compared to $c_{0}^{-1}$ times derivatives with respect to $\tau$. Doing sowould have yielded
$$ \begin{equation} -2\rho_{0}c_{0}\frac{\partial\phi}{\partial a} + \rho_{0}c_{0}^{-2}\delta\frac{\partial^{2}\phi}{\partial\tau^{2}} + 2\rho_{0}c_{0}^{-1}\beta\phi\frac{\partial\phi}{\partial\tau} = 0,\tag{212}\label{eq:212} \end{equation} $$
where $\phi$ here abbreviated $\partial\xi/\partial\tau$. [Mendousse gives this in a dimensionless form in the middle of the first column of p. 54 as $\ddot{\theta} = \theta^{\prime} + 2\theta\theta^{\prime}$, his $\theta$ corresponding toa negative constant times the $\phi$ that appears above.]
第二个考虑的变换是坐标变为 $a$ 和 $\tau = t - a/c_{0}$，其中 $\partial/\partial a\rightarrow \partial/\partial a + c_{0}^{-1}\partial/\partial\tau$ 和 $\partial/\partial t\rightarrow\partial/\partial\tau$ 。虽然 Mendousse 省略了细节，但他的意图似乎是，与 $c_{0}^{-1}$ 乘以与 $\tau$ 相关的导数相比，忽略与$a$相关的导数。这样做的结果是
$$ \begin{equation} -2\rho_{0}c_{0}\frac{\partial\phi}{\partial a} + \rho_{0}c_{0}^{-2}\delta\frac{\partial^{2}\phi}{\partial\tau^{2}} + 2\rho_{0}c_{0}^{-1}\beta\phi\frac{\partial\phi}{\partial\tau} = 0,\tag{212} \end{equation} $$
其中 $\phi$ 在这里缩写为 $\partial\xi/\partial\tau$.[Mendousse 在第 54 页第一列中间给出了无量纲形式，即 $\ddot{\theta} = \theta^{\prime} + 2\theta\theta^{\prime}$，他的 $\theta$ 对应于一个负常数乘以上面出现的 $\phi$]
In regard to the eguivalence of Mendousse’s two equations to other equations written in terms of Eulerian variables. one should realize that, in accordance with the statements following Eq. (53) in Ch apter 3, all three terms in either of the two equations are of the same order. Consequently, the order of approximation of the two equations will be unchanged if $a$ is replaced by $x$, so that $\tau$ can be regarded as $t - x/c_{0}$ and $\chi$ can be regarded as $x - c_{0}t$. Also, the linear acoustic relations connecting field variables in a plane wave propagating in the $+x$ direction can be used in choosing replacements for either $\theta$ or $\phi$ without changing the extent of the validity of either equation. Thus, one can set
$$ \begin{align} \frac{\partial\xi}{\partial\chi} &= \frac{\partial\xi}{\partial a}\simeq\frac{\rho_{0} - \rho}{\rho_{0}}\simeq -\frac{p}{\rho_{0}c_{0}^{2}}\simeq -\frac{u}{c_{0}},\tag{213}\label{eq:213}\\ \frac{\partial\xi}{\partial\tau} &= \frac{\partial\xi}{\partial t} = u,\tag{214}\label{eq:214} \end{align} $$
in which case Eqs. $\eqref{eq:211}$ and $\eqref{eq:212}$ reduce to
$$ \begin{align} \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial\chi} &= \frac{\delta}{2}\frac{\partial^{2}u}{\partial\chi^{2}},\tag{215}\label{eq:215}\\ \frac{\partial u}{\partial x} - \frac{\beta}{c_{0}^{2}}u\frac{\partial u}{\partial\tau} &= \frac{\delta}{2c_{0}^{3}}\frac{\partial^{2}u}{\partial\tau^{2}},\tag{216}\label{eq:216} \end{align} $$
Equation $\eqref{eq:215}$ is the same as Cole’s result. which appears above as Eq. $\eqref{eq:198}$. provided one interprets Cole’s $x$ as the $\chi$ that appears here. Equation $\eqref{eq:216}$ is the same as Eq. $\eqref{eq:205}$, which in turn is equivalent to Eq. $\eqref{eq:189}$.
关于 Mendousse 的两个方程与其他用 Euler 变量写成的方程的等价性，我们应该意识到，根据第 3 章公式 (53) 后面的陈述，这两个方程中的所有三个项都是同阶的。因此，如果将 $a$ 替换为 $x$，两个方程的近似阶数将保持不变，因此 $\tau$ 可视为 $t - x/c_{0}$，而 $\chi$ 可视为 $x - c_{0}t$。另外，在 $+x$ 方向传播的平面波中，联系各场变量的线性声学关系也可以用来选择 $\theta$ 或 $\phi$ 的替代变量，且不改变两个方程的有效性范围。因此，我们可以设置
$$ \begin{align} \frac{\partial\xi}{\partial\chi} &= \frac{\partial\xi}{\partial a}\simeq\frac{\rho_{0} - \rho}{\rho_{0}}\simeq -\frac{p}{\rho_{0}c_{0}^{2}}\simeq -\frac{u}{c_{0}},\tag{213}\\ \frac{\partial\xi}{\partial\tau} &= \frac{\partial\xi}{\partial t} = u,\tag{214} \end{align} $$
在这种情况下，公式 $\eqref{eq:211}$ 和 $\eqref{eq:212}$ 简化为
$$ \begin{align} \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial\chi} &= \frac{\delta}{2}\frac{\partial^{2}u}{\partial\chi^{2}},\tag{215}\\ \frac{\partial u}{\partial x} - \frac{\beta}{c_{0}^{2}}u\frac{\partial u}{\partial\tau} &= \frac{\delta}{2c_{0}^{3}}\frac{\partial^{2}u}{\partial\tau^{2}},\tag{216} \end{align} $$
等式 $\eqref{eq:215}$ 与 Cole 的结果相同，即上面的等式 $\eqref{eq:198}$，前提是把 Cole 的 $x$ 解释为这里出现的 $\chi$。等式 $\eqref{eq:216}$ 与等式 $\eqref{eq:205}$ 相同，而后者又等同于等式 $\eqref{eq:189}$。
The most influential article in this history was undoubtedly one published by Lighthill it (1956) in a special volume commemorating the seventieth birthday of G. I. Taylor. This was a long article(152 pages), partly review, but conveying some new results and adding considerable fresh insight to the subject of dissipative propagation of nonlinear waves. Early in the article, Lighthill gives the disclaimer that although “the paper seeks to be a self-contained account of its subject,… no attempt has been made to compile a bibliography, or even to give full references, especially when describing the classical parts of the paper.”
在这段历史中，最有影响力的文章无疑是 Lighthill（1956 年）在纪念 G. I. Taylor 七十寿辰的特辑中发表的一篇文章。这是一篇很长的文章（152 页），部分内容是评论，但传达了一些新的结果，并为非线性波的耗散传播这一主题增添了相当多的新见解。在文章的开头，Lighthill 给出了这样的免责声明：虽然 “本文力图自成一体地阐述其主题，但……没有试图编制参考书目，甚至没有给出完整的参考文献，尤其是在描述本文的经典部分时”。
The cited references include Burgers(1948), Cole (1951), and Taylor (1910), but not Bateman (1915), Lagerstrom, Cole, and Trilling (1949), or Mendousse (1953). Section 7.1 of Lighthill’s article has the title “Equation of progressive sound waves with convection and diffusion allowed for to first approximation (‘Burgers’s equation’).” The principal result in that section is Lighthill’s Eq. (121), which is the same as Eq. $\eqref{eq:215}$ above.
引用的参考文献包括 Burgers（1948 年）、Cole（1951 年）和 Taylor（1910 年），但不包括 Bateman（1915 年）、Lagerstrom、Cole 和 Trilling（1949 年）或 Mendousse（1953 年）。Lighthill 文章第 7.1 节的标题是 “允许对流和扩散的一阶近似渐进声波方程（‘Burgers 方程’）"。该节的主要结果是 Lighthill 的公式 (121)，与上文的公式 $\eqref{eq:215}$ 相同。
Lighthill’s $\delta$, however, given by his Eq. (19) earlier in the article, includes bulk viscosity and thermal conduction [asin Eq. $\eqref{eq:190}$ above], which were not included in the considerations of either Cole (Lagerstrom, Cole, and Trilling, 1949) or Mendousse (1953). The derivation is in the context of an ideal gas. Lighthill appreciated that this equation might have been derived earlier, and made the subsequent stalement, in relation to the papers giving a general solution of that equation by Hopf(1950) and Cole(1951), that “Cole and co-workers have followed up the implications of the solution for sound waves of finite amplitude, although they do not appear to have given an explicit derivation of [Eq. $\eqref{eq:215}$ above], and regard the equation more as an analogy than as an approximation of a definite order in a successive approximation scheme.”
然而，Lighthill 在文章前面的公式 (19) 中给出的 $\delta$ 包括了体积粘度和热传导 [如上面的公式 $\eqref{eq:190}$]，而 Cole（Lagerstrom, Cole, and Trilling, 1949）或 Mendousse（1953）在考虑这些因素时均未将其包括在内。该推导以理想气体为背景。Lighthill 意识到这个方程可能在更早的时候就已经被推导出来了，并在随后发表的关于 Hopf（1950）和 Cole（1951）给出该方程一般解的论文中声明：“科尔及其合作者已经跟进了有限振幅声波解的含义，尽管他们似乎并没有给出 [Eq.$\eqref{eq:215}$] 的明确推导，并更多地将该方程视为一种类比，而非连续逼近方法中某一确定阶次的近似”。
It is probably because of Lighthill’s article that any one-dimensional partial differential equation governing nonlinear unidirectional sound propagation in a dissipative medium, such as those discussed in the present section, is now ubiquitously referred to as the Burgers equation. Lighthill may not have intended that the appellation be permanent, as he placed the term “Burgers’s equation” within quotation marks. He did, however, perceive a stronger analogy between nonlinear sound propagation and the physical phenomena that Burgers (1948) was seeking to portray with his model than might have been apparent to a casual reader of Burgers’s work, as he refers to Burgers’s proffering of this equation as “the simplest equation embodying together the convective and diffusive effects whose conflict is the subject of [Lighthill’s] article. [Burgers] has used it principally to throw light on turbulence, where also these two effects are fundamental, although the balance struck between them there is more complicated, being both three-dimensional and statistically random.”
可能正是因为 Lighthill 的这篇文章，现在人们普遍将在耗散介质中控制非线性单向声传播的任何一维偏微分方程（如本节讨论的那些方程）称为 Burgers 方程。Lighthill 可能无意将这一称谓永久化，因为他将 “Burgers 方程” 一词加了引号。不过，他确实认为非线性声传播与 Burgers（1948 年）试图用他的模型描绘的物理现象之间的类比关系要比 Burgers 作品的普通读者所看到的更强，因为他提到 Burgers 提出的这个方程是 “最简单的方程，它同时体现了对流效应和扩散效应，而这两种效应之间的冲突正是 [Lighthill] 文章的主题”。[Burgers] 主要用它来揭示湍流，在湍流中这两种效应也是基本的，尽管它们之间的平衡更为复杂，既是三维的，又是统计随机的”。
This short history concludes with the article, also of considerable subsequent influence and also a part of a survey volume, by Hayes (1958) in which the Burgers equation was derived, apparently for the first time, for a fluid with an arbitrary equation of state, rather than specifically for an ideal gas. His derivation also included thermal conductivity, as well as shear and bulk viscosity. Hayes cites Burgers (1948), Hopf(1950). and Lighthill (1956), but not Bateman (1915), Lagerstrom, Cole. and Trilllng (1949), or Mendousse (1953). Apart from some minor differences in notation, Hayes’s Eq. (5-42) is identical in form to Cole’s result, which is written above, also with some minor changes in notation, as Eq. $\eqref{eq:198}$. Hayes (1958) uses the symbol $\Gamma$ for the coefficient of nonlinearity $\beta$ and $\nu^{\prime\prime}$ for the diffusivity of sound $\delta$. Like Cole’s, Hayes’s coordinate system is one in which the flow appears to be slightly transonic. His $u$ is the total fluid velocity as seen in such a coordinate system, so the quantity $u-c_{0}$ is regarded as small.
这段简短的历史以 Hayes（1958 年）的一篇文章而告终，这篇文章也具有相当大的后续影响，同时也是调查卷的一部分，在这篇文章中，Burgers 方程显然是第一次针对具有任意状态方程的流体，而不是专门针对理想气体进行推导。他的推导还包括热导率以及剪切粘度和体积粘度。Hayes 引用了 Burgers (1948)、Hopf(1950) 和 Lighthill (1956)，但没有引用 Bateman (1915)、Lagerstrom、Cole 和 Trilllng (1949) 或 Mendousse (1953)。除了符号上的一些细微差别之外，Hayes 的公式 (5-42) 与 Cole 的结果在形式上完全相同，后者在符号上也做了一些细微的改动，写成了公式 $\eqref{eq:198}$。Hayes（1958）使用符号 $\Gamma$ 表示非线性系数 $\beta$，$\nu^{\prime\prime}$ 表示声扩散系数 $\delta$。与 Cole 的坐标系一样，Hayes 的坐标系中的流动似乎略带跨音速。他的 $u$ 是在这样的坐标系中看到的流体总速度，因此 $u-c_{0}$ 被认为是很小的量。
Alternatively, Hayes’s result can be regarded as being the same as Eq. $\eqref{eq:215}$, with his $x$ being the $\chi = x - c_{0}t$ and his $u-c_{0}$ being the $u$ that appear in Eq. $\eqref{eq:215}$. That the labeling of this partial differential equation in the context of nonlinear acoustics as the Burgers equation had taken hold is evident in Hayes’s remarks: “lt should be evident that [Hayes’s] Eq. (5-42) is essentially the same as the equation extensively treated by Burgers [( 1948)]. Burgers’ equation was first obtained for the case of a perfect gas (with viscosity and heat conduction) by Lighthill [(1956)].”
或者说, Hayes 的结果可以被视为与公式 $\eqref{eq:215}$ 相同，他的 $x$ 是 $\chi = x - c_{0}t$，他的 $u-c_{0}$ 是公式 $\eqref{eq:215}$ 中出现的 $u$。在非线性声学的背景下，将这个偏微分方程标记为 Burgers 方程的做法在 Hayes 的评论中得到了证实：“显而易见，[Hayes 的]公式 (5-42) 与 Burgers[（1948 年）]广泛处理的方程本质上是相同的。Burgers 方程是 Lighthill[（1956 年）]在完全气体（具有粘性和热传导）情况下首次得到的”。
General Solution
One of the most appealing features of the Burgers equation is that, when the coefficients are constants, it has an exact solution expressible in terms of definite integrals. An abbreviated form of the derivation of this solution appeared in Appendix B of the Lagerstrom, Cole, and Trilling (1949) report, and more comprehensive accounts were subsequently published by Hopf (1950) and Cole (1951). These two papers were apparently completely independent, as neither references any of the other’s work. Also, Hopf (1950) states in a footnote that “[the solution] was known to me since the end of 1946. However, it was not until 1949 that [Hopf] became sufficiently acquainted with the recent development of fluid dynamics to be convinced that a theory of [the Burgers equation] could serve as an instructive introduction into some of the mathematical problems involved.” (Hopf. incidentally, is the Hopf whose 1931 paper published with Norbert Wiener introduced the well-known Wiener—Hopf technique.)
Burgers 方程最吸引人的特点之一是，当系数为常数时，它有一个可以用定积分表示的精确解。Lagerstrom、Cole 和 Trilling（1949 年）的报告附录 B 中出现了这一解的简略推导形式，随后 Hopf（1950 年）和 Cole（1951 年）发表了更全面的说明。这两篇论文显然是完全独立的，因为它们都没有引用对方的任何研究成果。此外，Hopf（1950 年）在脚注中指出：“我早在 1946 年底就知道[解]了。然而，直到 1949 年，[Hopf] 才对流体动力学的最新发展有了足够的了解，从而确信 [Burgers 方程] 的理论可以作为对某些相关数学问题的启发性介绍”。(顺便一提，Hopf 就是 1931 年与 Norbert Wiener 共同发表论文，提出著名的 Wiener-Hopf 方法的 Hopf)。
The discussion here is in the context of the version of the Burgers equation given by Eq.$\eqref{eq:189}$. Let us suppose that one has an initial-value problem in which $p$ is given at $x = 0$ as a function $F(t)$, this funetion being denoted by
$$ \begin{equation} p(0,t) = p_{0}F(t),\tag{217}\label{eq:217} \end{equation} $$
where $p_{0}$ is a pressure amplitude, and one seeks to determine $p$ as a function of $t$ for any value of $x$ that is greater than or equal to zero. It is evident that $p$ should vanish as $t\rightarrow -\infty$ for any positive value of $x$, given that $F(t)$ has this same property, and we expect the raie of vanishing to be sufficiently fast that it is meaningful to deal with the integral over $\tau$ of $p(x, \tau)$ from $-\infty$ up to any given value of $\tau$. If this is so, then another function that should exist for any positive $x$ is
$$ \begin{equation} \zeta(x,\tau) = \text{exp}\left\{\frac{1}{\rho_{0}c_{0}L}\int_{-\infty}^{\tau}p(x,\tau^{\prime})\mathrm{d}\tau\right\},\tag{218}\label{eq:218} \end{equation} $$
where $L$ is any positive constant having the units of length, this definition being such that
$$ \begin{equation} p(x,\tau) = \rho_{0}c_{0}L\frac{\partial}{\partial\tau}\ln{\zeta}. \tag{219}\label{eq:219} \end{equation} $$
这里的讨论是以 Burgers 方程的公式 $\eqref{eq:189}$ 为背景的。让我们假设有一个初值问题，其中 $p$ 是在 $x = 0$ 时作为函数 $F(t)$ 给出的，这个函数用以下公式表示
$$ \begin{equation} p(0,t) = p_{0}F(t),\tag{217} \end{equation} $$
其中，$p_{0}$ 是(声)压力振幅，而我们要确定的是，对于大于或等于零的任何 $x$ 值，$p$ 作为 $t$ 的函数。很明显，考虑到 $F(t)$ 具有同样的性质，对于任何正值的 $x$ 而言，$p$ 应该随着 $t\rightarrow -\infty$ 的消失而消失，我们希望消失的速度足够快，以至于处理 $p(x, \tau)$ 从 $-\infty$ 到任何给定的 $\tau$ 值对 $\tau$ 的积分是有意义的。如果是这样的话，那么对于任何正的 $x$ 都应该存在的另一个函数是
$$ \begin{equation} \zeta(x,\tau) = \text{exp}\left\{\frac{1}{\rho_{0}c_{0}L}\int_{-\infty}^{\tau}p(x,\tau^{\prime})\mathrm{d}\tau\right\},\tag{218} \end{equation} $$
其中，$L$ 是任何具有长度单位的正常数，其定义为
$$ \begin{equation} p(x,\tau) = \rho_{0}c_{0}L\frac{\partial}{\partial\tau}\ln{\zeta}. \tag{219} \end{equation} $$
The method of solution is to determine some function $\zeta(x,\tau)$ and some choice of length $L$ that will guarantee that the $p$ determined subsequently from Eq. $\eqref{eq:219}$ does indeed satisfy the original partial differential equation (the Burgers equation) and the specified initial condition at $x = 0$. To discover what $\zeta$ and $L$ should be, one inserts Eq. $\eqref{eq:219}$ into Eq. $\eqref{eq:189}$, with the result
$$ \begin{equation} \zeta^{2}(\zeta_{\tau x}-\frac{1}{2}\delta c_{0}^{-3}\zeta_{\tau\tau\tau}) - \zeta\zeta_{\tau}[\zeta_{x} - (\frac{3}{2}\delta c_{0}^{-3} - \beta Lc_{0}^{-2})\zeta\zeta_{\tau\tau}]+(\zeta_{\tau})^{3}(\beta Lc_{0}^{-2} - \delta c_{0}^{-3}) = 0.\tag{220}\label{eq:220} \end{equation} $$
Given the latitude one has in choosing $L$, the most propitious choice is $L = \delta/\beta c_{0}$, which yields
$$ \begin{equation} p = \frac{\rho_{0}\delta}{\beta}\frac{\partial}{\partial\tau}\ln{\zeta} = \frac{\rho_{0}\delta}{\beta}\frac{\zeta_{\tau}}{\zeta}.\tag{221}\label{eq:221} \end{equation} $$
This choice of $L$ has the effect of reducing Eq. $\eqref{eq:220}$ to
$$ \begin{equation} \frac{\partial}{\partial\tau}\left\{\frac{1}{\zeta}\left(\zeta_{x} - \frac{\delta}{2c_{0}^{3}}\zeta_{\tau\tau}\right)\right\} = 0,\tag{222}\label{eq:222} \end{equation} $$
which inturn will certainly be satisfied if one takes $\zeta$ to satisfy
$$ \begin{equation} \frac{\partial\zeta}{\partial x} - \frac{\delta}{2c_{0}^{3}}\frac{\partial^{2}\zeta}{\partial\tau^{2}} = K(x)\zeta,\tag{223}\label{eq:223} \end{equation} $$
where $K(x)$ is an arbitrary function of $x$.
求解的方法是确定某个函数 $\zeta(x,\tau)$ 和某个长度 $L$ 的选择，以保证随后根据公式 $\eqref{eq:219}$ 确定的 $p$ 确实满足原始偏微分方程（Burgers 方程）和指定的初始条件 $x = 0$。要想知道 $\zeta$ 和 $L$ 应该是多少，可以把公式 $\eqref{eq:219}$ 插入公式 $\eqref{eq:189}$，结果是
$$ \begin{equation} \zeta^{2}(\zeta_{\tau x}-\frac{1}{2}\delta c_{0}^{-3}\zeta_{\tau\tau\tau}) - \zeta\zeta_{\tau}[\zeta_{x} - (\frac{3}{2}\delta c_{0}^{-3} - \beta Lc_{0}^{-2})\zeta\zeta_{\tau\tau}]+(\zeta_{\tau})^{3}(\beta Lc_{0}^{-2} - \delta c_{0}^{-3}) = 0.\tag{220} \end{equation} $$
考虑到选择 $L$ 的纬度，最有利的选择是 $L = \delta/\beta c_{0}$，由此得出
$$ \begin{equation} p = \frac{\rho_{0}\delta}{\beta}\frac{\partial}{\partial\tau}\ln{\zeta} = \frac{\rho_{0}\delta}{\beta}\frac{\zeta_{\tau}}{\zeta}.\tag{221} \end{equation} $$
所选的 $L$ 可以将公式 $\eqref{eq:220}$ 简化为
$$ \begin{equation} \frac{\partial}{\partial\tau}\left\{\frac{1}{\zeta}\left(\zeta_{x} - \frac{\delta}{2c_{0}^{3}}\zeta_{\tau\tau}\right)\right\} = 0,\tag{222} \end{equation} $$
如果我们认为 $\zeta$ 满足以下关系, 上式必定满足:
$$ \begin{equation} \frac{\partial\zeta}{\partial x} - \frac{\delta}{2c_{0}^{3}}\frac{\partial^{2}\zeta}{\partial\tau^{2}} = K(x)\zeta,\tag{223} \end{equation} $$
其中，$K(x)$ 是$x$ 的任意函数。
The arbitrariness of $\zeta$ associated with the arbitrariness of $K(x)$ is illusory because one can always make the substitution
$$ \begin{equation} \zeta(x,\tau) = \widetilde{\zeta}(x,\tau)\text{exp}\left[\int_{0}^{x}K(x^{\prime})\mathrm{d}x^{\prime}\right] \tag{224}\label{eq:224} \end{equation} $$
and find that $\widetilde{\zeta}(x,\tau)$ satisfies the equation
$$ \begin{equation} \frac{\partial\widetilde{\zeta}}{\partial x} - \frac{\delta}{2c_{0}^{3}}\frac{\partial^{2}\widetilde{\zeta}}{\partial\tau^{2}} = 0,\tag{225}\label{eq:225} \end{equation} $$
which is of the same form as Eq. $\eqref{eq:223}$, only with $K(x)$ set to zero. Also, were Eq. $\eqref{eq:224}$ to be inserted into Eq. $\eqref{eq:221}$, the result would be independent of $K(x)$. Moreover, the initial $(x = 0)$ values of $\zeta$ and $\widetilde{\zeta}$ would be identical. Consequently, it is sufficient to proceed as if satisfied Eq. $\eqref{eq:225}$ or, equivalently, Eq. $\eqref{eq:223}$ with $K(x)$ set identically to zero. With a relabeling of the variables. Eq. $\eqref{eq:225}$ is recognized as the linear diffusion equation. which governs one-dimensional unsteady heat conduction and a variety of other phenomena that are studied in mathematical physics. The transformation Eq. $\eqref{eq:221}$ that leads from the (nonlinear) Burgers equation to the linenr diffusion equation is often referred to as the Hopf-Cole transformation.
与 $K(x)$ 的任意性相关联的 $\zeta$ 的任意性是虚幻的，因为我们总是可以做如下替换
$$ \begin{equation} \zeta(x,\tau) = \widetilde{\zeta}(x,\tau)\text{exp}\left[\int_{0}^{x}K(x^{\prime})\mathrm{d}x^{\prime}\right] \tag{224} \end{equation} $$
并发现 $\widetilde{\zeta}(x,\tau)$ 满足方程
$$ \begin{equation} \frac{\partial\widetilde{\zeta}}{\partial x} - \frac{\delta}{2c_{0}^{3}}\frac{\partial^{2}\widetilde{\zeta}}{\partial\tau^{2}} = 0,\tag{225} \end{equation} $$
与公式 $\eqref{eq:223}$的形式相同，只是 $K(x)$ 设为零。另外，如果把公式 $\eqref{eq:224}$ 插入公式 $\eqref{eq:221}$ 中，结果将与 $K(x)$ 无关。此外，$\zeta$ 和 $\widetilde{\zeta}$ 的初始 $(x = 0)$ 值将是相同的。因此，我们只需像满足公式 $\eqref{eq:225}$ 或等价于公式 $\eqref{eq:223}$ 一样，将 $K(x)$ 设为零即可。重新标注变量后, 公式 $\eqref{eq:225}$ 被认为是线性扩散方程，它支配着一维非稳态热传导和数学物理中研究的其他各种现象。$\eqref{eq:221}$ 从（非线性） Burgers 方程到线性扩散方程的变换通常被称为 Hopf-Cole 变换。
Solution of the initial-value problem for the linear diffusion equation, Eq. $\eqref{eq:225}$, proceeds with the writing of the solution as a convolution integral over the initial values in the form
$$ \begin{equation} \zeta(x,\tau) = \int_{-\infty}^{+\infty}\zeta(0,\tau^{\prime})G(x,\tau-\tau^{\prime})\mathrm{d}\tau^{\prime},\tag{226}\label{eq:226} \end{equation} $$
where $G(x,\tau-\tau^{\prime})$ isa “Green’s function” that satisfies the homogeneous partial differential equation
$$ \begin{equation} \frac{\partial G}{\partial x} - \frac{\delta}{2c_{0}^{3}}\frac{\partial^{2}G}{\partial\tau^{2}} = 0,\tag{227}\label{eq:227} \end{equation} $$
and that reduces to the Dirac delta function $\delta(\tau-\tau^{\prime})$ in the limit as $x\rightarrow 0$, so that
$$ \begin{align} \lim_{x\rightarrow 0}G(x,\tau-\tau^{\prime}) = 0,\quad \tau-\tau^{\prime}\neq 0,\tag{228}\label{eq:228}\\ \lim_{x\rightarrow 0}\int_{-\infty}^{+\infty}G(x,\tau-\tau^{\prime})\mathrm{d}\tau^{\prime} = 1.\tag{229}\label{eq:229} \end{align} $$
在求解线性扩散方程 $\eqref{eq:225}$ 的初值问题时，可以把解写成对初值的卷积积分，其形式为
$$ \begin{equation} \zeta(x,\tau) = \int_{-\infty}^{+\infty}\zeta(0,\tau^{\prime})G(x,\tau-\tau^{\prime})\mathrm{d}\tau^{\prime},\tag{226} \end{equation} $$
其中，$G(x,\tau-\tau^{\prime})$ 是一个 “Green 函数”，它满足均质偏微分方程
$$ \begin{equation} \frac{\partial G}{\partial x} - \frac{\delta}{2c_{0}^{3}}\frac{\partial^{2}G}{\partial\tau^{2}} = 0,\tag{227} \end{equation} $$
并在 $x\rightarrow 0$ 的极限值上还原为 Dirac-$\delta$ 函数 $\delta(\tau-\tau^{\prime})$ ，因此
$$ \begin{align} \lim_{x\rightarrow 0}G(x,\tau-\tau^{\prime}) = 0,\quad \tau-\tau^{\prime}\neq 0,\tag{228}\\ \lim_{x\rightarrow 0}\int_{-\infty}^{+\infty}G(x,\tau-\tau^{\prime})\mathrm{d}\tau^{\prime} = 1.\tag{229} \end{align} $$
This Green’s function is derived in many texts on mathematical physics and heat transfer, so the derivation is omitted here; the reader can independently verify that the equations above are satisfied by
$$ \begin{equation} G(x,\tau-\tau^{\prime}) = \sqrt{\frac{c_{0}^{3}}{2\pi x\delta}}e^{-E_{G}},\quad E_{G} = \frac{c_{0}^{3}(\tau-\tau^{\prime})^{2}}{2x\delta}. \tag{230}\label{eq:230} \end{equation} $$
As a consequence, Eq. $\eqref{eq:226}$ yields
$$ \begin{equation} \zeta(x,\tau) = \sqrt{\frac{c_{0}^{3}}{2\pi x\delta}}\int_{-\infty}^{+\infty}\zeta(0,\tau^{\prime})e^{-E_{G}}\mathrm{d}\tau^{\prime} \tag{231}\label{eq:231} \end{equation} $$
as the general solution of the initial-value problem for the auxiliary function $\zeta$.
这个 Green 函数在许多数学物理和传热学著作中都有推导，因此此处省略推导过程；读者可以通过以下方法独立验证上述方程是否满足要求
$$ \begin{equation} G(x,\tau-\tau^{\prime}) = \sqrt{\frac{c_{0}^{3}}{2\pi x\delta}}e^{-E_{G}},\quad E_{G} = \frac{c_{0}^{3}(\tau-\tau^{\prime})^{2}}{2x\delta}. \tag{230} \end{equation} $$
因此，$\eqref{eq:226}$ 公式得出
$$ \begin{equation} \zeta(x,\tau) = \sqrt{\frac{c_{0}^{3}}{2\pi x\delta}}\int_{-\infty}^{+\infty}\zeta(0,\tau^{\prime})e^{-E_{G}}\mathrm{d}\tau^{\prime} \tag{231} \end{equation} $$
作为辅助函数 $\zeta$ 初值问题的通解。
The solution to the Burgers equation now results from substitution of Eq. $\eqref{eq:231}$ into Eq. $\eqref{eq:221}$ to obtain
$$ \begin{equation} p = \frac{\rho_{0}c_{0}^{3}}{\beta x}\frac{\int_{-\infty}^{+\infty}\zeta(0,\tau^{\prime})(\tau-\tau^{\prime})e^{-E_{G}}\mathrm{d}\tau^{\prime}}{\int_{-\infty}^{+\infty}\zeta(0,\tau^{\prime})e^{-E_{G}}\mathrm{d}\tau^{\prime}},\tag{232}\label{eq:232} \end{equation} $$
where
$$ \begin{equation} \zeta(0,\tau^{\prime}) = e^{E_{\zeta}},\quad E_{\zeta}(\tau^{\prime}) = \frac{\beta p_{0}}{\rho_{0}\delta}\int_{-\infty}^{\tau^{\prime}}F(\tau^{\prime\prime})\mathrm{d}\tau^{\prime\prime}. \tag{233}\label{eq:233} \end{equation} $$
将公式 $\eqref{eq:231}$ 代入公式 $\eqref{eq:221}$ 即可得到 Burgers 方程的解:
$$ \begin{equation} p = \frac{\rho_{0}c_{0}^{3}}{\beta x}\frac{\int_{-\infty}^{+\infty}\zeta(0,\tau^{\prime})(\tau-\tau^{\prime})e^{-E_{G}}\mathrm{d}\tau^{\prime}}{\int_{-\infty}^{+\infty}\zeta(0,\tau^{\prime})e^{-E_{G}}\mathrm{d}\tau^{\prime}},\tag{232} \end{equation} $$
其中
$$ \begin{equation} \zeta(0,\tau^{\prime}) = e^{E_{\zeta}},\quad E_{\zeta}(\tau^{\prime}) = \frac{\beta p_{0}}{\rho_{0}\delta}\int_{-\infty}^{\tau^{\prime}}F(\tau^{\prime\prime})\mathrm{d}\tau^{\prime\prime}. \tag{233} \end{equation} $$
An alternative expression for the solution results after an integration by parts in the integral in the numerator:
$$ \begin{equation} p = p_{0}\frac{\int_{-\infty}^{+\infty}F(\tau^{\prime})e^{E_{\zeta}}e^{-E_{G}}\mathrm{d}\tau^{\prime}}{\int_{-\infty}^{+\infty}e^{E_{\zeta}}e^{-E_{G}}\mathrm{d}\tau^{\prime}}. \tag{234}\label{eq:234} \end{equation} $$
This form makes it manifestly evident that the solution so obtained reduces to the initial condition in the limit as $x\rightarrow 0$. Note also that the intrinsic nonlinear nature of the Burgers equation is exemplified by the appearance of the integral over the time history of $F$ in the exponent $E_{\zeta}$. If $\beta$ is set to zero, but $\delta$ remains finite, the result reduces to the linear-acoustics prediction for the attenuation and distortion of a transient pulse, each of whose frequency components is being attenuated as $\text{exp}(-\omega^{2}x\delta/2c_{0}^{3})$ after propagation over a distance $x$. That such is so is confirmed by the computation
$$ \begin{equation} \frac{\int_{-\infty}^{+\infty}\sin{(\omega\tau^{\prime})e^{-E_{G}}\mathrm{d}\tau^{\prime}}}{\int_{-\infty}^{+\infty}e^{-E_{G}}\mathrm{d}\tau^{\prime}} = e^{-\omega^{2}x\delta/2c_{0}^{3}}\sin{(\omega\tau)}. \tag{235}\label{eq:235} \end{equation} $$
在对分子中的积分进行分式积分后，会得出另一种解法：
$$ \begin{equation} p = p_{0}\frac{\int_{-\infty}^{+\infty}F(\tau^{\prime})e^{E_{\zeta}}e^{-E_{G}}\mathrm{d}\tau^{\prime}}{\int_{-\infty}^{+\infty}e^{E_{\zeta}}e^{-E_{G}}\mathrm{d}\tau^{\prime}}. \tag{234} \end{equation} $$
这种形式明显表明，如此求得的解在极限值 $x\rightarrow 0$ 时会还原为初始条件。还要注意的是，Burgers 方程的内在非线性体现在指数 $E_{\zeta}$ 中出现了对 $F$ 的时间历史的积分。如果将 $\beta$ 设为零，但 $\delta$ 保持有限，结果就会还原为线性声学对瞬态脉冲衰减和失真的预测，即在传播距离为 $x$ 后，每个频率成分都会衰减为 $\text{exp}(-\omega^{2}x\delta/2c_{0}^{3})$。计算结果证实了这一点:
$$ \begin{equation} \frac{\int_{-\infty}^{+\infty}\sin{(\omega\tau^{\prime})e^{-E_{G}}\mathrm{d}\tau^{\prime}}}{\int_{-\infty}^{+\infty}e^{-E_{G}}\mathrm{d}\tau^{\prime}} = e^{-\omega^{2}x\delta/2c_{0}^{3}}\sin{(\omega\tau)}. \tag{235} \end{equation} $$
Rise Time And Thickness Of Weak Shocks
The weak shock model discussed in Section 4 leads to abrupt discontinuities, but when the model incorporates dissipation processes, such discontinuities become instead transition regions over which the pressure and fluid velocity change rapidly. Insight into the nature of the transition results from consideration of the idealized model of a wave that moves without change of form in the $x$ direction with speed $U$, and that is distinguished by the fact that the most rapid variations of the fluid-dynamic variables occur in the vicinity of points and times at which $x-Ut$ is relatively small. (The quantity $U$ will subsequently be identified as the shock speed $U_{\text{sh}}$.) For $x\gg Ut$, $p$ and $u$ should be zero, while for $x\ll Ut$, $p$ and $\rho_{0}c_{0}u$ approach the shock overpressure $\Delta P = \Delta p = \rho_{0}c_{0}\Delta u$.
第 4 节讨论的弱冲击模型会导致突然的不连续性，但当模型包含耗散过程时，这种不连续性会变成压力和流体速度快速变化的过渡区域。流体动力学变量的最快速变化发生在 $x-Ut$ 相对较小的点和时间附近（后续我们将 $U$ 量确定为冲击速度 $U_{\text{sh}}$）。对于 $x\gg Ut$，$p$ 和 $u$ 应该为零，而对于 $x\ll Ut$，$p$ 和 $\rho_{0}c_{0}u$ 接近冲击超压 $\Delta P = \Delta p = \rho_{0}c_{0}\Delta u$。
The discussion here is based on the retarded-time version of the Burgers equation, as given by Eq. $\eqref{eq:189}$. In accordance with the notion of a frozen profile, we begin with the assumption of a profile moving with some speed $U$, but without change of form, that is,
$$ \begin{equation} p = p(t - x/U) = \rho_{0}c_{0}u(t^{\prime}),\tag{236}\label{eq:236} \end{equation} $$
where
$$ \begin{equation} t^{\prime} = t - x/U = \tau + (c_{0}^{-1} - U^{-1})x. \tag{237}\label{eq:237} \end{equation} $$
This, when inserted into Eq. $\eqref{eq:189}$, yields the ordinary differential equation
$$ \begin{equation} c_{0}^{2}(U-c_{0})u_{t^{\prime}} - \frac{\delta}{2}Uu_{t^{\prime}t^{\prime}} = \beta c_{0}uUu_{t^{\prime}}. \tag{238}\label{eq:238} \end{equation} $$
which integrates, with the (causality) boundary condition $u\rightarrow 0$ as $t^{\prime}\rightarrow -\infty$, to
$$ \begin{equation} c_{0}^{2}(U-c_{0})u - \frac{\delta}{2}Uu_{t^{\prime}} = \frac{1}{2}\beta c_{0}u^{2}U. \tag{239}\label{eq:239} \end{equation} $$
这里的讨论基于 Burgers 方程的延迟时间版本，如公式 $\eqref{eq:189}$ 所示。根据凝固剖面的概念，我们首先假设剖面以某种速度 $U$ 运动，但不改变波形，即：
$$ \begin{equation} p = p(t - x/U) = \rho_{0}c_{0}u(t^{\prime}),\tag{236} \end{equation} $$
其中
$$ \begin{equation} t^{\prime} = t - x/U = \tau + (c_{0}^{-1} - U^{-1})x. \tag{237} \end{equation} $$
将其插入公式 $\eqref{eq:189}$，可以得到常微分方程
$$ \begin{equation} c_{0}^{2}(U-c_{0})u_{t^{\prime}} - \frac{\delta}{2}Uu_{t^{\prime}t^{\prime}} = \beta c_{0}uUu_{t^{\prime}}. \tag{238} \end{equation} $$
当 $t^{\prime}\rightarrow -\infty$ 时，在（因果关系）边界条件 $u\rightarrow 0$ 的作用下，积分为
$$ \begin{equation} c_{0}^{2}(U-c_{0})u - \frac{\delta}{2}Uu_{t^{\prime}} = \frac{1}{2}\beta c_{0}u^{2}U. \tag{239} \end{equation} $$
The second boundary condition, that $u\rightarrow\Delta u$ as $t^{\prime}\rightarrow \infty$, requires that $u_{t^{\prime}}\rightarrow 0$ as $u\rightarrow \Delta u$, which yields
$$ \begin{equation} c_{0}(c_{0} - U) = -\frac{1}{2}\beta U\Delta u,\tag{240}\label{eq:240} \end{equation} $$
or
$$ \begin{equation} U = \frac{c_{0}^{2}}{c_{0} - \frac{1}{2}\beta\Delta u}. \tag{241}\label{eq:241} \end{equation} $$
Since $\Delta u\ll c_{0}$, this can be replaced, to the same order of approximation, by
$$ \begin{equation} U = c_{0} + \frac{1}{2}\beta\Delta u. \tag{242}\label{eq:242} \end{equation} $$
One should note that Eq. $\eqref{eq:242}$ is consistent with the approximate Rankine-Hugoniot relation
$$ \begin{equation} U_{\text{sh}} = c_{\text{av}} + u_{\text{av}} \tag{243}\label{eq:243} \end{equation} $$
in Eq. (123). Ahead of the shock, one has $c_{a} = c_{0}$ and $u_{a} = 0$, while far hehind the shock, one has $c_{b} = c_{0} + (\beta - 1)\Delta u$ and $u_{b} = \Delta u$. The identification of $U_{\text{sh}} = U$, $c_{\text{av}} = c_{0} + \frac{1}{2}(\beta - 1)\Delta u$ and $u_{\text{av}} = \frac{1}{2}\Delta u$ results in Eq. $\eqref{eq:242}$.
