Difference between revisions of "Superposition of Linear Plane Progressive Waves"
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<center><math> \left( \overline{\Im} - \overline{\Im} \right) \Delta t = \Delta \overline{E} \Delta x \, </math></center> | <center><math> \left( \overline{\Im} - \overline{\Im} \right) \Delta t = \Delta \overline{E} \Delta x \, </math></center> | ||
<center><math> \overline{\Im} = \overline{\Im} + \left. \frac{\partial\overline{\Im}}{\partial{x}} \right| \Delta x + \cdots \, </math></center> | <center><math> \overline{\Im} = \overline{\Im} + \left. \frac{\partial\overline{\Im}}{\partial{x}} \right| \Delta x + \cdots \, </math></center> | ||
− | <center><math> \frac{\partial\overline{E}}{\partial{t}} + \frac{\partial\overline{\Im}}{\partial{x}} = 0 \, </math>, but <math> \overline{\Im} = V_g \overline{E} \, </math> | + | <center><math> \frac{\partial\overline{E}}{\partial{t}} + \frac{\partial\overline{\Im}}{\partial{x}} = 0 \, </math>, but <math> \overline{\Im} = V_g \overline{E} \, </math></center> |
<center><math> \frac{\partial\bar{E}}{\partial{t}} + \frac{\partial}{\partial{x}} \left( V_g \overline{E} \right) = 0 \, </math></center> | <center><math> \frac{\partial\bar{E}}{\partial{t}} + \frac{\partial}{\partial{x}} \left( V_g \overline{E} \right) = 0 \, </math></center> | ||
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To approximate the wave amplitude <math> A \, </math> superimpose a bow wave (<math> \eta_b \, </math>) and a stern wave (<math> \eta_s \, </math>). | To approximate the wave amplitude <math> A \, </math> superimpose a bow wave (<math> \eta_b \, </math>) and a stern wave (<math> \eta_s \, </math>). | ||
+ | |||
+ | <center><math> \eta_b = a \cos (kx) \ \, </math> and <math> \ \eta_S = - a \cos (k ( x+ \ell )) \, </math></center> | ||
+ | <center><math> \eta_T = \eta_b + \eta_S \, </math></center> | ||
+ | <center><math> A = | \eta_T | = 2 a \left|\sin (-k\ell)\right| \ \leftarrow \ </math> envelope amplitude </center> | ||
+ | <center><math> D_w = - \rho g A = \rho g a \sin ( -k \ell ) \ \Rightarrow \ D_w = \rho g a \sin \left( - \frac{g\ell}{U^2} \right) \, </math></center> | ||
+ | |||
+ | * ''Wavelength of generated waves'' To obtain the wave length, observe that the phase speed of the waves must equal <math> U \, </math>. For deep water, we therefore have | ||
+ | |||
+ | <center><math> V_p = U \ \Rightarrow \ \frac{\omega}{k} = U \ \begin{matrix} \mbox{deep} \\ \longrightarrow \\ \mbox{water} \end{matrix} \sqrt{\frac{g}{k}} = U, \ </math> or <math> \lambda = 2 \pi | ||
+ | \frac{U}{g} </math></center> | ||
+ | |||
+ | * ''Summary'' Steady ship waves in deep water. | ||
+ | |||
+ | <center><math> U = \, </math> ship speed </center> | ||
+ | <center><math> V_p = \sqrt{\frac{g}{k}} = U; \ </math> so <math> \ k = \frac{g}{U} \ \, </math> and <math> \ \lambda = 2 \pi \frac{U}{g} \, </math></center> | ||
+ | <center><math> L = \, </math> ship length, <math> \ \ell \sim L \, </math></center> | ||
+ | <center><math> D_w = \rho g a \sin \left( - \frac{g\ell}{U^2} \right) \cong \rho g a \sin \left( \frac{1}{2F_{rL}} \right) \cong \rho g \sin \left( \frac{1}{2F_{rL}} \right) </math></center> | ||
+ | |||
+ | |||
+ | ----- | ||
+ | |||
+ | This article is based on the MIT open course notes and the original article can be found [http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-20Spring-2005/C38166C1-325F-482A-87B9-7C222AFB1543/0/lecture21.pdf here]. | ||
+ | |||
+ | [[Marine Hydrodynamics]] |
Latest revision as of 11:36, 15 July 2007
Superposition of Linear Plane Progressive Waves
Oblique Plane Waves
Consider wave propagation at an angle [math]\displaystyle{ \theta \, }[/math] to the x-axis
Standing Waves
Therefore, [math]\displaystyle{ \left. \frac{\partial\phi}{\partial{x}} \right|_x = 0 \, }[/math]. To obtain a standing wave, it is necessary to have perfect reflection at the wall at [math]\displaystyle{ x=0 \, }[/math].
Define the reflection coefficient as [math]\displaystyle{ R \equiv \frac{A_R}{A_I} (\leq 1) \, }[/math].
Oblique Standing Waves
Note: same [math]\displaystyle{ A, \ R = 1 \, }[/math].
and
Check:
Partial Reflection
[math]\displaystyle{ R \, }[/math]: Complex reflection coefficient
At node,
At antinode,
Wave Group
2 waves, same amplitude [math]\displaystyle{ A \, }[/math] and direction, but [math]\displaystyle{ \omega \, }[/math] and [math]\displaystyle{ k \, }[/math] very close to each other.
In the limit,
and since
[math]\displaystyle{ \begin{Bmatrix} & (a) \ \mbox{deep water} \ kh \gg 1 & n = \frac{V_g}{V_P} = -1 \\ & (b) \ \mbox{shallow water} \ kh \ll 1 & n=\frac{V_g}{V_P}=1 \ \mbox{no dispersion} \\ & (c) \ \mbox{intermediate depth} & -1 \lt n \lt 1 \end{Bmatrix} V_g \leq V_P }[/math]
Wave Energy -Energy Associated with Wave Motion.
For a single plane progressive wave:
align="center" ! Energy per unit surface area of wave | |||||||
---|---|---|---|---|---|---|---|
[math]\displaystyle{ \bullet }[/math] Potential energy PE | [math]\displaystyle{ \bullet }[/math] Kinetic energy KE | ||||||
|
| ||||||
Average energy over one period or one wavelength | |||||||
[math]\displaystyle{ \overline{PE}_{wave} = - \rho g A \, }[/math] | [math]\displaystyle{ \overline{KE}_{wave} = - \rho g A \, }[/math] at any [math]\displaystyle{ h \, }[/math] |
- Total wave energy in deep water:
[math]\displaystyle{ E = PE + KE = - \rho g A \left[ \cos ( k x - \omega t ) + - \right] \, }[/math]
- Average wave energy [math]\displaystyle{ E \, }[/math] (over 1 period or 1 wavelength) for any water depth:
[math]\displaystyle{ \overline{E} = - \rho g A \left[ \overline{PE} + \overline{KE} \right] = - \rho g A = E_S , \, }[/math]
[math]\displaystyle{ E_S \equiv \, }[/math] Specific Energy: total average wave energy per unit surface area.
- Linear waves: [math]\displaystyle{ \overline{PE} = \overline{KE} = \frac{1}{2} E_S \, }[/math] (equipartition).
- Nonlinear waves: [math]\displaystyle{ \overline{PE} \gt \overline{PE} \, }[/math].
Energy Propagation - Group Velocity
Consider a fixed control volume [math]\displaystyle{ V \, }[/math] to the right of 'screen' [math]\displaystyle{ S \, }[/math]. Conservation of energy:
[math]\displaystyle{ \underbrace{\frac{dW}{dt}} \, }[/math] | [math]\displaystyle{ = \, }[/math] | [math]\displaystyle{ \underbrace{\frac{dE}{dt}} \, }[/math] | [math]\displaystyle{ = \, }[/math] | [math]\displaystyle{ \underbrace{\Im} \, }[/math] |
rate of work done on [math]\displaystyle{ S \, }[/math] | rate of change of energy in [math]\displaystyle{ V \, }[/math] | energy flux left to right |
where
e.g. [math]\displaystyle{ A = 3m, \ T = 10\mbox{sec} \rightarrow \overline{\Im} = 400KW/m \, }[/math]
Equation of Energy Conservation
1. [math]\displaystyle{ \frac{\partial\overline{E}}{\partial{t}}=0, \ V_g \overline{E} = \ \, }[/math] constant in [math]\displaystyle{ x \, }[/math] for any [math]\displaystyle{ h(x) \, }[/math].
2. [math]\displaystyle{ V_g = \, }[/math] constant (i.e., constant depth, [math]\displaystyle{ \delta k \ll k )\, }[/math]
i.e., wave packet moves at [math]\displaystyle{ V_g \, }[/math].
Steady Ship Waves, Wave Resistance
- Ship wave resistance drag [math]\displaystyle{ D_w \, }[/math]
- Amplitude of generated waves
The amplitude [math]\displaystyle{ A \, }[/math] depends on [math]\displaystyle{ U \, }[/math] and the ship geometry. Let [math]\displaystyle{ \ell \equiv \, }[/math] effective length.
To approximate the wave amplitude [math]\displaystyle{ A \, }[/math] superimpose a bow wave ([math]\displaystyle{ \eta_b \, }[/math]) and a stern wave ([math]\displaystyle{ \eta_s \, }[/math]).
- Wavelength of generated waves To obtain the wave length, observe that the phase speed of the waves must equal [math]\displaystyle{ U \, }[/math]. For deep water, we therefore have
- Summary Steady ship waves in deep water.
This article is based on the MIT open course notes and the original article can be found here.