Pressure to the surface to a stream
notation
\(\varpi\): -> r ///////p439 (22)
\(\sigma\): -> \(\omega\)
\(c\): -> \(c_p\) //////p457 (15)
\(U\): ->
Assumptions 📄P398 Art. 242
consider only steady motion, dissipative forces but small, irrotational flow
Core
Prerequisite notes
\(X, Y, Z\) the components of the extraneous forces per unit mass, at the point \((x, y, z)\) at the time \(t\).
Framework
\[ X=-\mu(u-c), \quad Y=-g-\mu v, \quad Z=-\mu w \]
where \(\mu\) is small
the canonical form of the solution is: \[ g \rho y=\kappa C \cdot \frac{(k-\kappa) \cos k x-\mu_1 \sin k x}{(k-\kappa)^2+\mu_1^2} \] where \(\kappa\) for \(g / c^2\) and \(\mu/c=\mu_1\)
📄P401 eqn(17) Art. 243
put \(c=\kappa+i \mu_1\) and \(\zeta=k+i m\)
Denominator:
\[ (k-\kappa)^2+\mu_1^2=0 \] \(\Rightarrow\)
\(c=\kappa+i \mu_1\), and \(k\), \(\kappa\) are real
\[ \begin{aligned} (k-\kappa)^2+\mu_1^2 & =\left|k-\left(\kappa+i \mu_1\right)\right|^2 \\ & =\left|(k-\kappa)+i \mu_1\right|^2 \\ & =(k-c)^2 \end{aligned} \] and rlumerator
\[ \begin{aligned} & (k-\kappa) \cos kx-\mu_1 \sin kx \\ &= \left(k-\kappa-i \mu_1\right) e^{i k x} \\ &= (k-c) e^{i k x} \end{aligned} \] extend \(k\) to the complex plane \((k,m)\), eqn (16) become \[ \int \frac{e^{i x \xi}}{\xi-c} d \xi \]
why \(-2 \pi i\)
－留数定理：若在这个＂环形＂区域里函数只有一个奇点 \(c\) ，则把外边界与内边界合起来 （构成一个完整的闭合曲线，称之为 \(C_{\text {total }}\) ，逆时针绕之，满足 \[ \oint_{C_{\text {total }}} f(\zeta) \mathrm{d} \zeta=2 \pi i(\text { 留数 } \text { at } \zeta=c) \text {. } \]
－但是内边界小圆实际走向是顺时针（与大边界方向相反）。如果我们单独看内小圆的贡献（称之为 \(\int_{C_{\text {inner }}}\) ），那么它的正向（逆时针）取值应为 \(+2 \pi i\)（留数），可是它在这张图里却是顺时针，所以要加一个负号：
\[ \int_{\text {内小圆(顺时针) }} f(\zeta) \mathrm{d} \zeta=-\int_{\text {内小圆(屰时针) }} f(\zeta) \mathrm{d} \zeta=-[2 \pi i(\text { 留数 })] . \]
因此，小圆那部分产生的贡献是 \({ }^{* *}-2 \pi i \cdot \ldots{ }^{* *}\) 。
📄P401-402 Art. 243
The integral along a closed path can be decomposed into： \[ \oint \frac{e^{i k x}}{k-c} d k=\int_{-\infty}^{\infty} \frac{e^{i k x}}{k-c} d k+\int_{C_1} \frac{e^{i k x}}{k-c} d k . \]
Contribution of semicircular arc integral by Jordan's Lemma, the integral over the semicicular arc in the upper half-plane vanishes when \(x>0\) \[ \int_{C_2} \frac{e^{i k x}}{k-c} d k \rightarrow 0 \text {. } \]
Contour integral: by Cauchy's integral theorem, the integral vanishes if the closed contour does not enclose any singularities. \[ \oint \frac{e^{i k x}}{k-c} d k=0 . \] when \(m<0\) \[ \int_0^{\infty} \frac{e^{i k x}}{k-\left(\kappa+i \mu_1\right)} d k=\int_0^{\infty} \frac{e^{-i k x}}{k+\left(\kappa+i \mu_1\right)} d k \] there is no poles in a closed path (\(\kappa\) and \(\mu_1\) are both positive.)
The formula (16) is equivalent, for \(x\) positive, to
\[ \begin{aligned} \frac{\pi g \rho}{\kappa P} . y & =-2 \pi e^{-\mu_1 x} \sin \kappa x+\int_0^{\infty} \frac{(k+\kappa) \cos k x-\mu_1 \sin k x}{(k+\kappa)^2+\mu_1{ }^2} d k \\ & =-2 \pi e^{-\mu_1 x} \sin \kappa x+\int_0^{\infty} \frac{\left(m-\mu_1\right) e^{-m x} d m}{\left(m-\mu_1\right)^2+\kappa^2}, \ldots \ldots \ldots \ldots \end{aligned} \tag{25} \]
and, for \(x\) negative, to
\[ \frac{\pi g \rho}{\kappa P} \cdot y=\int_0^{\infty} \frac{\left(m+\mu_1\right) e^{m x} d m}{\left(m+\mu_1\right)^2+\kappa^2} \tag{26} \]
The interpretation of these results is simple. The first term of (25) represents a train of simple-harmonic waves, on the down-stream side of the origin, of wave-length \(2 \pi c^2 / g\), with amplitudes gradually diminishing according to the law \(e^{-\mu_1 x}\). The remaining part of the deformation of the free surface, expressed by the definite integrals in (25) and (26), though very great for small values of \(x\), diminishes very rapidly as \(x\) increases in absolute value, however small the value of the frictional coefficient \(\mu_1\).
When \(\mu_1\) is infinitesimal, our results take the simpler forms
\[ \begin{aligned} \frac{\pi g \rho}{\kappa P} \cdot y & =-2 \pi \sin \kappa x+\int_0^{\infty} \frac{\cos k x}{k+\kappa} d k \\ & =-2 \pi \sin \kappa x+\int_0^{\infty} \frac{m e^{-m x}}{m^2+\kappa^2} d m \end{aligned} \]
📄P430 eqn(6)
in the case of simple-harmonic motion,
\[ \sigma^2 \phi=g \frac{\partial \phi}{\partial z} \]
if the time-factor be \(e^{i(\sigma t+\epsilon)}\).
\(\epsilon\) is the initial phase of the wave
📄P430 eqn(12-13)
we have normalization condition \[ \int^{\infty} f(\alpha) J_{0}(k \alpha) \alpha d \alpha=1 \]
If \(f(\alpha)\) is chosen to be the two-dimensional Dirac delta function in polar coordinates
\[ f(\alpha)=\frac{\delta(\alpha)}{2 \pi \alpha} \]
Then, it satisfies the normalization condiation
\[ \int^{\infty} f(\alpha) 2 \pi \alpha d \alpha=\int^{\infty} \frac{\delta(\alpha)}{2 \pi \alpha} \cdot 2 \pi \alpha d \alpha=\int^{\infty} \delta(\alpha) d \alpha=1 \]
Substituting \(f(\alpha)\) into the original equationo \[ \int^{\infty} f(\alpha) J_{0}(k \alpha) \alpha d \alpha=\int^{\infty} \frac{\delta(\alpha)}{2 \pi \alpha} J_0(k \alpha) \alpha d \alpha=\frac{J_{0}(0)}{2 \pi}=\frac{1}{2 \pi} \]
📄P467 eqn(20)
\[ p^{\prime}=\frac{P}{\pi} \frac{b}{b^2+x^2} \]
P469 eqn(3)
有一族直线，它们的方程可以写成
\[ x \cos \theta+y \sin \theta=p(\theta) \]
- \(\theta\) 可以看作这条直线（或其与原点连线的垂线）相对于 \(x\) 轴的方向角；
- \(p(\theta)\) 则是＂从原点 \(O\) 到这条直线的垂距＂（又称＂极线到极点的距离＂），是 \(\theta\) 的函数。
当 \(\theta\) 变化时，我们就得到一族（无穷多）的直线。如果这族直线在平面上包络成一条＂包络曲线＂，则这条曲线在每一个 \(\theta\) 对应的那条直线上有且只有一个公切点（即曲线与那一条直线相切于该点），那么这条曲线就可以写成参数方程 \(x(\theta), y(\theta)\) 。为什么这些 \((x(\theta), y(\theta))\) 正好满足
\[ \left\{\begin{array}{l} x(\theta)=p(\theta) \cos \theta-p^{\prime}(\theta) \sin \theta \\ y(\theta)=p(\theta) \sin \theta+p^{\prime}(\theta) \cos \theta \end{array}\right. \]
这里 \(p^{\prime}(\theta)=\frac{d p}{d \theta}\) 。
包络曲线与消去法
2.1 直线族方程 \[ x \cos \theta+y \sin \theta=p(\theta) \]
在几何上，若某条曲线 \(C\) 恰好是这族直线的包络，那么对每个 \(\theta\) ，直线（1）应该与曲线 \(C\) 相切。换言之，在该 \(\theta\) 下，这条直线跟曲线 \(C\) 有且仅有一个公切点（重合点）。要想从＂直线族＂找到＂包络＂，常见的方法是：将（1）式和它对 \(\theta\) 的导数同时联立求解 \(x, y\) ，因为曲线在该 \(\theta\) 处的切点不光满足直线方程本身，还要满足＂对 \(\theta\) 微小改变时仍然落在同一点上＂的条件（这是＂相切而非相交＂在解析上的体现）。
2.2 对 \(\theta\) 求导并联立
因此，对（1）式做 \(\theta\) 求导（并令结果 \(=0\) ），可以得到包络的另一条方程。 对（1）式求导： \[ \frac{d}{d \theta}[x \cos \theta+y \sin \theta-p(\theta)]=0 \]
注意，这里 \(x\) 和 \(y\) 也可能是 \(\theta\) 的函数（因为我们最终要找包络上满足条件的 \((x, y)\) ）。因此作全导数时，应写
\[ \frac{d}{d \theta}[x(\theta) \cos \theta+y(\theta) \sin \theta]-\frac{d p(\theta)}{d \theta}=0 \] 也就是
\[ \left[x^{\prime}(\theta) \cos \theta-x(\theta) \sin \theta\right]+\left[y^{\prime}(\theta) \sin \theta+y(\theta) \cos \theta\right]-p^{\prime}(\theta)=0 \]
这里 \(x^{\prime}(\theta)=\frac{d x}{d \theta}, y^{\prime}(\theta)=\frac{d y}{d \theta}\) 。 包络曲线上的点 \((x(\theta), y(\theta))\) 同时满足
\[ \left\{\begin{array}{l} x(\theta) \cos \theta+y(\theta) \sin \theta=p(\theta) \\ x^{\prime}(\theta) \cos \theta-x(\theta) \sin \theta+y^{\prime}(\theta) \sin \theta+y(\theta) \cos \theta=p^{\prime}(\theta) \end{array}\right. \]
这是通常的＂消去＂或＂联立＂思路。
some parameters
group-velocity \(U\)
\(\sigma\) is similar to \(\omega\), \(d \sigma / d k=U=x / t\)
\[ \left\{\begin{array}{l} \eta_t+U \eta_x-\varphi_y=0 \\ \varphi_t+U \varphi_x+g \eta-\frac{T}{\rho} \nabla_{\perp} \eta=-f(x) e^{\epsilon t} \end{array} \text { on } z=0\right. \] and \[ \nabla^2 \varphi=0 \]
256 b. To examine the modification produced in the wave-pattern when the depth of the water has to be taken into account, the argument on p. 433 must be put in a more general form. If, as before, \(t\) is the time the pressure point has taken to travel from \(Q\) to \(O\), it may be shewn that the phase of the disturbance at \(P\), due to the impulse delivered at \(Q\), will differ only by a constant from \[ k(V t-\varpi) \]
where \(2 \pi / k\) is the predominant wave-length in the neighbourhood of \(P\), and \(V\) the corresponding wave-velocity*. This predominant wave-length is determined by the condition that the phase is stationary for variations of the wave-length only, i.e.
\[ \frac{\partial}{\partial k} \cdot k(V t-\varpi)=0, \text { or }\quad \omega=U t \]
where \(U=d(k V) / d k\), is the group-velocity (Art. 236).
For the effective part of the disturbance at \(P\) ，the phase（22）must further be stationary as regards variations in the position of \(Q\) ；hence， differentiating partially with respect to \(t\) ，we have
\[ \dot{\varpi}=V, \text { or } V=c \cos \theta \]
since \(\dot{\varpi}=c \cos \theta\). (\(c\) is the velocity of the pressure-point over the water) Now, referring to the figure on \(p .433\), we have
\[ p=c t \cos \theta-\varpi=V t-\varpi . \]
Hence for a given wave-ridge \(p\) will bear a constant ratio to the wavelength \(\lambda\), and in passing from one wave-ridge to the next this ratio will increase (or decrease) by unity. Since \(\lambda\) is determined as a function of \(\theta\) by (24), this gives the relation between \(p\) and \(\theta\).
Thus in the case of infinite depth, the formula (24) gives
\[ c^2 \cos ^2 \theta=V^2=\frac{g \lambda}{2 \pi} \]
and the required relation is of the form
\[ p=a \cos ^2 \theta \]