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Interpreting Water Waves and Water Wave Refraction
DOI: 10.12677/MP.2024.142006, PDF, HTML, XML, 下载: 101  浏览: 156  科研立项经费支持

Abstract: Based on the trajectory equation of water wave particle motion and the expression of water wave propagation speed, this paper deeply understands the propagation characteristics of water waves, points out the inaccuracy of the description of water wave refraction concept in high school physics textbooks, and clarifies the concept of water wave refraction. Analyzing the essential reasons for water wave refraction from the perspective of the change in propagation speed caused by the change in water potential during the movement of water waves from one water body with different depths to another, and explaining water wave refraction based on the Huygens principle.

1. 引言

2. 水波的深度认识

2.1. 水波的频散关系和波速

$\phi =-\frac{gA}{\omega }\cdot \frac{\mathrm{cosh}\left[k\left(h+z\right)\right]}{\mathrm{cosh}\left(kh\right)}\mathrm{cos}\left(k\text{ }x-\omega \text{ }\text{ }t\right)$ (1)

$\eta =A\mathrm{sin}\left(k\text{ }x-\omega \text{ }\text{ }t\right)$ (2)

$|\begin{array}{cc}{\omega }^{2}-gk& {\omega }^{2}+gk\\ {\text{e}}^{-kh}& {\text{e}}^{kh}\end{array}|=0$ (3)

${\omega }^{2}=gk\cdot \frac{{\text{e}}^{kh}-{\text{e}}^{-kh}}{{\text{e}}^{kh}+{\text{e}}^{-kh}}=gk\mathrm{tan}\left(kh\right)$ (4)

$\lambda =\frac{g{T}^{2}}{2\pi }\mathrm{tanh}\left(\frac{2\pi }{\lambda }h\right)$ (5)

$v=\frac{gT}{2\pi }\mathrm{tanh}\left(\frac{2\pi }{\lambda }h\right)$ (6)

2.2. 深水波和浅水波的特殊性

1) 深水波

$\mathrm{tanh}\left(kh\right)\approx 1\text{\hspace{0.17em}},\text{ }\mathrm{sinh}\left(kh\right)\approx \frac{1}{2}{\text{e}}^{kh}\text{\hspace{0.17em}},\text{ }\mathrm{cosh}\left(kh\right)\approx \frac{1}{2}{\text{e}}^{kh}$ (7)

${\omega }^{2}=gk$ (8)

$\lambda =\frac{g{T}^{2}}{2\pi }$ (9)

$v=\frac{gT}{2\pi }=\sqrt{\frac{g\lambda }{2\pi }}$ (10)

Figure 1. Parameter variation relationship of deep water waves

2) 浅水波

$\mathrm{tanh}\left(kh\right)\approx kh\text{\hspace{0.17em}},\text{ }\mathrm{sinh}\left(kh\right)\approx kh\text{\hspace{0.17em}},\text{ }\mathrm{cosh}\left(kh\right)\approx 1+\frac{1}{2}{\left(kh\right)}^{\text{\hspace{0.17em}}2}$ (11)

$\omega =-\frac{gA}{\omega }\cdot \left[1+\frac{{k}^{2}{\left(h+z\right)}^{\text{\hspace{0.17em}}2}}{2}\right]\mathrm{cos}\left(k\text{ }x-\omega \text{ }\text{ }t\right)$ (12)

${\omega }^{2}=g{k}^{2}h$ (13)

$\lambda =T\sqrt{gh}$ (14)

$v=\sqrt{gh}$ (15)

Figure 2. Parameter variation relationship of shallow water waves

2.3. 水质点运动的轨迹

$\frac{{\left(x-{x}_{0}\right)}^{2}}{{\left\{A\cdot \frac{\mathrm{cosh}\left[k\left(h+{z}_{0}\right)\right]}{\mathrm{sinh}\left(kh\right)}\right\}}^{2}}+\frac{{\left(z-{z}_{0}\right)}^{2}}{{\left\{A\cdot \frac{\mathrm{sinh}\left[k\left(h+{z}_{0}\right)\right]}{\mathrm{sinh}\left(kh\right)}\right\}}^{2}}=1$ (16)

${\left(x-{x}_{0}\right)}^{2}+{\left(z-{z}_{0}\right)}^{2}={\left(A{\text{e}}^{k\text{ }{x}_{0}}\right)}^{2}$ (17)

$\frac{{\left(x-{x}_{0}\right)}^{2}}{{\left(\frac{A}{kh}\right)}^{2}}+\frac{{\left(z-{z}_{0}\right)}^{2}}{{\left[A\cdot \left(1+\frac{{z}_{0}}{h}\right)\right]}^{2}}=1$ (18)

Figure 3. Movement trajectories of water quality points in different water area

2.4. 水质点运动与波速的关系

$u=A\omega \text{\hspace{0.17em}}{\text{e}}^{k\text{ }x}\mathrm{sin}\left(k\text{ }x-\omega \text{ }\text{ }t\right)$ (19)

$w=-A\omega \text{\hspace{0.17em}}{\text{e}}^{k\text{ }x}\mathrm{cos}\left(k\text{ }x-\omega \text{ }\text{ }t\right)$ (20)

Figure 4. Relationship between water quality point movement and wave velocity

2.5. 水波的本质

3. 水波折射的本质原因

3.1. 波高、波长与波速的变化

Figure 5. Water waves entering shallow water from deep water

3.2. 水波的折射

$n=\frac{\mathrm{sin}\alpha }{\mathrm{sin}\beta }=\frac{{v}_{1}}{{v}_{2}}$ (21)

Figure 6. Refraction of water waves

3.3. 水波折射的解释

$\mathrm{sin}\alpha =\frac{{A}^{\prime }{A}^{″}}{{B}^{\prime }{A}^{″}}=\frac{{v}_{1}\text{d}t}{{B}^{\prime }{A}^{″}}$ (22)

$\mathrm{sin}\beta =\frac{{B}^{\prime }{B}^{″}}{{B}^{\prime }{A}^{″}}=\frac{{v}_{2}\text{d}t}{{B}^{\prime }{A}^{″}}$ (23)

$n=\frac{\mathrm{sin}\alpha }{\mathrm{sin}\beta }=\frac{{v}_{1}}{{v}_{2}}$ (24)

Figure 7. Wheatstone principle

4. 结论

1) 水波折射过程中，深水区和浅水区的水结构以及杂质成分等没有任何变化。因此，将深水区和浅水区视为两种不同的介质的这种观点是完全错误的。

2) 只考虑水质点的自身重力作用而没有考虑水的表面张力等因素影响的水波常称为重力波。如果考虑水的表面张力影响，那么深水中的水波速度即(10)式可改写为

$v=\sqrt{\frac{g\lambda }{2\pi }+\frac{2\pi \alpha }{\rho \lambda }}$ (25)

3) 在深海中，若由于风力等作用先后产生波长不同的几列水波，且后波的波长大于前波的波长，如图8所示。根据(10)式，由于后波速度大于前波速度，即图中 ${v}_{3}>{v}_{2}>{v}_{1}$ ，因此几列波将可能会同时到达同一位置，因而叠加形成大浪或巨浪。

Figure 8. Several columns of water waves with different wavelengths in the deep sea

2022年教育部高等学校大学物理课程教学指导委员会大中物理教育衔接工作委员会教学研究课题“三新背景下县域普通高中物理实验教学策略研究”(立项编号：WX202242)部分研究成果。

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