Humps and hollows in wave resistance
Contents
| Section | Topic |
|---|---|
| 1 | Wave systems |
| 2 | Superposition |
| 3 | Humps and hollows |
| 4 | Speeds and Froude number |
1. Wave systems
Conditions:
- deep water
- transverse waves only: \(\theta = 0\)
- the bow and the stern each generate a wave system of the same amplitude \(a\)
- the bow system starts with a crest at the bow
- the stern system starts with a trough at the stern
Transverse waves stationary relative to the ship (Wavelength of surface waves for a translating body, with \(\theta = 0\)):
\[ \lambda = \frac{2\pi u^2}{g} \tag{1} \]
\[ k = \frac{2\pi}{\lambda} = \frac{2\pi g}{2\pi u^2} = \frac{g}{u^2} \tag{2} \]
\(x\) measured aft from the bow; stern at \(x = L\).
Bow system, crest at \(x = 0\):
\[ \eta_b = a\cos(kx), \qquad x \ge 0 \]
Stern system, trough at \(x = L\):
\[ \eta_s = -a\cos\big(k(x - L)\big), \qquad x \ge L \]
2. Superposition
Behind the ship both systems are present:
\[ \eta = \eta_b + \eta_s = a\cos(kx) - a\cos\big(k(x - L)\big) \]
\(\cos P - \cos Q = -2\sin\left(\dfrac{P + Q}{2}\right)\sin\left(\dfrac{P - Q}{2}\right)\), with \(P = kx\) and \(Q = k(x - L)\):
\[ \frac{P + Q}{2} = kx - \frac{kL}{2}, \qquad \frac{P - Q}{2} = \frac{kL}{2} \]
\[ \eta = -2a\sin\left(\frac{kL}{2}\right)\sin\left(kx - \frac{kL}{2}\right) \tag{3} \]
Amplitude of the wave left behind the ship:
\[ A = 2a\left|\sin\left(\frac{kL}{2}\right)\right| \tag{4} \]
Wave-making resistance, deep water (Wave energy and wave-making resistance: \(R = \frac{1}{4}\rho gA^2\)):
\[ R = \frac{1}{4}\rho g\cdot 4a^2\sin^2\left(\frac{kL}{2}\right) = \rho ga^2\sin^2\left(\frac{kL}{2}\right) \tag{5} \]
3. Humps and hollows
Hollows: \(\sin\left(\dfrac{kL}{2}\right) = 0\) \(\rightarrow\) the two systems cancel behind the ship
\[ \frac{kL}{2} = n\pi \;\rightarrow\; kL = 2\pi n \;\rightarrow\; L = n\lambda, \qquad n = 1, 2, 3, \dots \tag{6} \]
Humps: \(\left|\sin\left(\dfrac{kL}{2}\right)\right| = 1\) \(\rightarrow\) the two systems reinforce behind the ship
\[ \frac{kL}{2} = \frac{(2n - 1)\pi}{2} \;\rightarrow\; kL = (2n - 1)\pi \;\rightarrow\; L = \left(n - \frac{1}{2}\right)\lambda, \qquad n = 1, 2, 3, \dots \tag{7} \]
4. Speeds and Froude number
Speeds. From (2), \(kL = \dfrac{gL}{u^2}\):
Hollows, from (6):
\[ \frac{gL}{u^2} = 2\pi n \;\rightarrow\; u = \sqrt{\frac{gL}{2\pi n}} \tag{8} \]
Humps, from (7):
\[ \frac{gL}{u^2} = (2n - 1)\pi \;\rightarrow\; u = \sqrt{\frac{gL}{(2n - 1)\pi}} \tag{9} \]
Froude number: \(Fr = \dfrac{u}{\sqrt{gL}}\)
\[ \frac{gL}{u^2} = \frac{1}{Fr^2} \]
\[ \text{Hollows: } Fr = \frac{1}{\sqrt{2\pi n}}, \qquad \text{Humps: } Fr = \frac{1}{\sqrt{(2n - 1)\pi}} \tag{10} \]
| \(n\) | Hump \(Fr\) | Hump \(u\) | Hollow \(Fr\) | Hollow \(u\) |
|---|---|---|---|---|
| 1 | 0.564 | \(1.77\sqrt{L}\) | 0.399 | \(1.25\sqrt{L}\) |
| 2 | 0.326 | \(1.02\sqrt{L}\) | 0.282 | \(0.88\sqrt{L}\) |
| 3 | 0.252 | \(0.79\sqrt{L}\) | 0.230 | \(0.72\sqrt{L}\) |
| 4 | 0.213 | \(0.67\sqrt{L}\) | 0.199 | \(0.63\sqrt{L}\) |
(\(u\) in m/s with \(L\) in m, \(g = 9.81\) m/s\(^2\).)
- First hollow, \(n = 1\): \(L = \lambda\), \(\;u = \sqrt{\dfrac{gL}{2\pi}} = 1.25\sqrt{L}\) \(\rightarrow\) the limiting hull speed.
- First hump, \(n = 1\): \(L = \dfrac{\lambda}{2}\), \(\;Fr = 0.564\) \(\rightarrow\) above the hull speed the two systems reinforce again; the last hump.
- At lower speeds the humps and hollows crowd together, since \(\dfrac{1}{Fr^2}\) changes rapidly.