7 Forces and Moments on a Floating Object in Waves
Learning Objectives
7.1 Motivation
The previous chapter discussed about the radiation problem, where body moving in calm water results in the outward propagating radiated wave. The forces and moments on the body due to this radiated wave can be expressed in terms of components proportional to acceleration and velocity of the body. This gives rise to the concept of added mass and radiation damping. However, when a structure is floating in the open ocean, it experiences forces and moments due to the pre-existing waves in the ocean. This chapter discusses the forces and moments acting on a floating body due to the waves present in the ocean.
7.2 Incident and Diffraction Waves
Unlike the surface of a small lake that maybe calm on some days, the surface of the ocean always has waves. These waves on the ocean surface are generated due to the wind blowing over the vast areas of the ocean. The ocean surface over which the wind blows and transfers energy is known as fetch. When wind acts on a fetch for a long time, the small ripples generated at the interface interact with each other and grow into larger waves. In general, the waves in the ocean will be random. However, using Fourier transform we can consider the irregular wave as a superposition of infinitely many regular waves.
These wind generated waves are incident on any structure floating in the ocean and are known as incident waves. Due to the presence of the floating body, the incident waves get reflected from the body surface. This process is known as diffraction or scattering. The wave scattered off from the body surface is known as the scattered wave. Both the incident and scattered waves will exert forces and moments on the floating body and these components are known as incident and scattered wave forces respectively.
7.3 Incident Wave Forces
Consider a regular incident wave with amplitude \(A\), frequency \(\omega\), phase \(\varepsilon\) and direction \(\beta\). Note that the direction \(\beta\) is defined as the angle between the x-axis of the GCS and the direction of wave propagation. For a ship with BCS initially oriented along GCS in calm condition, \(\beta = 0^{\circ}\) corresponds to following wave, \(\beta = 180^{\circ}\) corresponds to head wave, \(\beta = 90^{\circ}\) corresponds to beam wave propagating towards the port side. According to the Airy’s wave theory, the first order wave elevation is given by:
\[\begin{align} \eta^{(1)} &= A \cos(-k(x\cos(\beta) + y\sin(\beta)) + \omega t + \varepsilon) \\ &= \operatorname{Re} \left\{A e^{-ik(x\cos(\beta) + y\sin(\beta)) + i\omega t + i\varepsilon}\right\} \end{align}\]
The corresponding velocity potential upto first order due to the regular wave is given by:
\[\begin{align} \varphi_I^{(1)} &= -\frac{gA}{\omega} \frac{\cosh(k(z+h))}{\cosh(kh)} \sin\left(-k(x\cos(\beta) + y\sin(\beta)) + \omega t + \varepsilon\right) \\ &= \operatorname{Im} \left\{-\frac{gA}{\omega} \frac{\cosh(k(z+h))}{\cosh(kh)} e^{-ik(x\cos(\beta) + y\sin(\beta)) + i\omega t + i\varepsilon}\right\} \\ &= \operatorname{Re} \left\{\frac{igA}{\omega} \frac{\cosh(k(z+h))}{\cosh(kh)} e^{-ik(x\cos(\beta) + y\sin(\beta)) + i\omega t + i\varepsilon}\right\} \end{align}\]
From here on we will use the complex notation of the wave elevation and the velocity potential with the understanding that the real part is physical solution. If deep waters are considered, \(h \rightarrow \infty\) and the expression for the potential reduces to:
\[\begin{align} \varphi_I^{(1)} = \frac{igA}{\omega} e^{kz} e^{-ik(x\cos(\beta) + y\sin(\beta)) + i\omega t + i\varepsilon} \end{align}\]
For a ship moving steadily with forward speed \(U\), the observed frequency of waves changes due to the Doppler effect. The observed frequency of the waves is known as encounter frequency \(\omega_e\) and relates to the actual frequency \(\omega\) through:
\[\begin{align} \omega_e = \omega - k U \cos(\beta) \end{align}\]
where \(k = 2\pi/\lambda\) is the wave number and is related to the wave frequency \(\omega\) through the dispersion relationship \(\omega^2 = g k \tanh(kh)\). When considering deep waters, the dispersion relationship simplifies to \(\omega^2 = gk\) and the encounter frequency \(\omega_e\) is given by:
\[\begin{align} \omega_e = \omega - \frac{\omega^2U}{g}\cos(\beta) \end{align}\]
Defining a new GCS system that moves forward with constant design speed \(U\), the expressions for wave elevation in this coordinate frame is given by:
\[\begin{align} \eta^{(1)} = A e^{-ik(x\cos(\beta) + y\sin(\beta)) + i\omega_e t + i\varepsilon} \end{align}\]
and the velocity potential is given by:
\[\begin{align} \varphi_I^{(1)} = \frac{igA}{\omega_e} e^{kz} e^{-ik(x\cos(\beta) + y\sin(\beta)) + i\omega_e t + i\varepsilon} \end{align}\]
The dynamic pressure due to the wave is obtained from the Bernoulli’s principle and the wave forces and moments acting on the structure upto the first order can be calculated as (previously derived in Chapter 6):
\[\begin{align} \vec{F}_I^{(1)} &= - \rho\iint_{S_{B0}} \left(\frac{\partial }{\partial t} - U \frac{\partial }{\partial x}\right)\varphi_I^{(1)} \hat{n}^{(0)} dS \\ \vec{M}_I^{(1)} &= - \rho \iint_{S_{B0}}\left(\frac{\partial }{\partial t} - U \frac{\partial }{\partial x}\right)\varphi_I^{(1)} (\vec{r}_P^{(0)} \times \hat{n}^{(0)}) dS \end{align}\]
These forces and moments due to the incident waves is known as the Fourde-Krylov forces and moments. Taking a Fourier transform yields:
\[\begin{align} \left\{\hat{\tau}_I(\omega_e)\right\} = \begin{bmatrix} \hat{\vec{F}}_I^{(1)} (\omega_e) \\ \hat{\vec{M}}_I^{(1)} (\omega_e) \end{bmatrix} = -\rho \begin{bmatrix} \iint_{S_{B0}} \left(i\omega_e - U \frac{\partial }{\partial x}\right)\hat{\varphi}_I^{(1)} \hat{n}^{(0)} dS \\ \iint_{S_{B0}} \left(i\omega_e - U \frac{\partial }{\partial x}\right)\hat{\varphi}_I^{(1)} (\vec{r}_P^{(0)} \times \hat{n}^{(0)}) dS \end{bmatrix} \label{eq-incident-force-ft} \end{align}\]
where
\[\begin{align} \hat{\varphi}_I^{(1)}(\omega_e) = \frac{igA}{\omega_e} e^{kz} e^{-ik(x\cos(\beta) + y\sin(\beta)) + i\varepsilon} \end{align}\]
7.3.1 Example: Froude–Krylov loads on a box barge
Consider a box barge freely floating in deep water with length \(L\), breadth \(B\) and draft \(T\). Place the BCS origin at midship, on the centreline, in the plane of the calm waterline, so that the mean wetted hull occupies
\[\begin{align} -\frac{L}{2} \le x \le \frac{L}{2} \qquad -\frac{B}{2} \le y \le \frac{B}{2} \qquad -T \le z \le 0 \end{align}\]
The barge has no forward speed, \(U = 0\). Then \(\omega_e = \omega\) and the incident-wave force and moment amplitudes in \(\eqref{eq-incident-force-ft}\) reduce to
\[\begin{align} \hat{\vec{F}}_I^{(1)} (\omega) &= -\rho \iint_{S_{B0}} i\omega\,\hat{\varphi}_I^{(1)}\,\hat{n}^{(0)}\,dS \\ \hat{\vec{M}}_I^{(1)} (\omega) &= -\rho \iint_{S_{B0}} i\omega\,\hat{\varphi}_I^{(1)}\,(\vec{r}_P^{(0)}\times\hat{n}^{(0)})\,dS \end{align}\]
with the deep-water incident potential
\[\begin{align} \hat{\varphi}_I^{(1)} = \frac{igA}{\omega}\,e^{kz}\,e^{-ik(x\cos\beta + y\sin\beta)+i\varepsilon} \end{align}\]
The factor multiplying the normal is therefore
\[\begin{align} i\omega\,\hat{\varphi}_I^{(1)} &= i\omega\cdot\frac{igA}{\omega}\,e^{kz}\,e^{-ik(x\cos\beta + y\sin\beta)+i\varepsilon} = -gA\,e^{kz}\,e^{-ik(x\cos\beta + y\sin\beta)+i\varepsilon} \end{align}\]
Substituting into the force integral gives
\[\begin{align} \hat{\vec{F}}_I^{(1)} &= -\rho\iint_{S_{B0}}\bigl(-gA\,e^{kz}\,e^{-ik(x\cos\beta + y\sin\beta)+i\varepsilon}\bigr)\hat{n}^{(0)}\,dS \nonumber\\ &= \rho g A e^{i\varepsilon} \iint_{S_{B0}} e^{kz}\,e^{-ik(x\cos\beta + y\sin\beta)}\,\hat{n}^{(0)}\,dS \end{align}\]
Likewise,
\[\begin{align} \hat{\vec{M}}_I^{(1)} &= \rho g A e^{i\varepsilon} \iint_{S_{B0}} e^{kz}\,e^{-ik(x\cos\beta + y\sin\beta)}\,(\vec{r}_P^{(0)}\times\hat{n}^{(0)})\,dS \end{align}\]
These are the Froude–Krylov force and moment: the first-order dynamic pressure of the undisturbed incident wave, integrated over the mean wetted surface. To evaluate the surface integrals, introduce the shorthand
\[\begin{align} \Psi(x,y,z) &\equiv e^{kz}\,e^{-ik(x\cos\beta + y\sin\beta)} \end{align}\]
so that
\[\begin{align} \hat{F}_{Ij}^{(1)} &= \rho g A e^{i\varepsilon}\iint_{S_{B0}}\Psi\,n_j\,dS \\ \hat{M}_{Ij}^{(1)} &= \rho g A e^{i\varepsilon}\iint_{S_{B0}}\Psi\,(\vec{r}_P^{(0)}\times\hat{n}^{(0)})_j\,dS \end{align}\]
Consistent with the convention used in Chapter 6 (force written as \(\iint p\,\hat{n}\,dS\)), the unit normal \(\hat{n}^{(0)}\) points out of the fluid and into the body. On the five faces of the mean wetted barge this gives:
\[\begin{align} \begin{array}{c|c|c} \text{face} & \text{location} & \hat{n}^{(0)} \\ \hline \text{bottom} & z=-T & (0,0,1) \\ \text{port} & y=+B/2 & (0,-1,0) \\ \text{starboard} & y=-B/2 & (0,1,0) \\ \text{bow} & x=+L/2 & (-1,0,0) \\ \text{stern} & x=-L/2 & (1,0,0) \end{array} \end{align}\]
The following elementary integrals will appear repeatedly. Define
\[\begin{align} u &\equiv \frac{kL\cos\beta}{2} \qquad v &\equiv \frac{kB\sin\beta}{2} \end{align}\]
and
\[\begin{align} J_x &\equiv \int_{-L/2}^{L/2} e^{-ik x\cos\beta}\,dx = \begin{cases} \dfrac{2\sin u}{k\cos\beta} = L\dfrac{\sin u}{u} & \cos\beta\neq 0\\[1em] L & \cos\beta = 0 \end{cases} \\[1em] J_y &\equiv \int_{-B/2}^{B/2} e^{-ik y\sin\beta}\,dy = \begin{cases} \dfrac{2\sin v}{k\sin\beta} = B\dfrac{\sin v}{v} & \sin\beta\neq 0\\[1em] B & \sin\beta = 0 \end{cases} \\[1em] J_z &\equiv \int_{-T}^{0} e^{kz}\,dz = \frac{1-e^{-kT}}{k} \\[1em] K_z &\equiv \int_{-T}^{0} z\,e^{kz}\,dz = \frac{T e^{-kT}}{k}-\frac{1-e^{-kT}}{k^2} \\[1em] L_x &\equiv \int_{-L/2}^{L/2} x\,e^{-ik x\cos\beta}\,dx = \begin{cases} \dfrac{2i}{(k\cos\beta)^2}\bigl(u\cos u-\sin u\bigr) & \cos\beta\neq 0\\[1em] 0 & \cos\beta = 0 \end{cases} \\[1em] L_y &\equiv \int_{-B/2}^{B/2} y\,e^{-ik y\sin\beta}\,dy = \begin{cases} \dfrac{2i}{(k\sin\beta)^2}\bigl(v\cos v-\sin v\bigr) & \sin\beta\neq 0\\[1em] 0 & \sin\beta = 0 \end{cases} \end{align}\]
(The special cases \(\cos\beta=0\) and \(\sin\beta=0\) are the continuous limits of the corresponding formulae.)
7.3.1.1 Surge force
Only the bow and stern contribute to \(n_1\). On the stern, \(x=-L/2\) and \(n_1=1\); on the bow, \(x=+L/2\) and \(n_1=-1\). Therefore
\[\begin{align} \hat{F}_{I1}^{(1)} &= \rho g A e^{i\varepsilon}\Biggl[ \int_{-T}^{0}\int_{-B/2}^{B/2} \Psi\!\left(-\tfrac{L}{2},y,z\right)\,dy\,dz - \int_{-T}^{0}\int_{-B/2}^{B/2} \Psi\!\left(\tfrac{L}{2},y,z\right)\,dy\,dz \Biggr] \end{align}\]
Factoring the \(x\)-dependence out of \(\Psi\),
\[\begin{align} \Psi\!\left(-\tfrac{L}{2},y,z\right) &= e^{kz}\,e^{-ik y\sin\beta}\,e^{ik(L/2)\cos\beta} = e^{kz}\,e^{-ik y\sin\beta}\,e^{iu} \\ \Psi\!\left(\tfrac{L}{2},y,z\right) &= e^{kz}\,e^{-ik y\sin\beta}\,e^{-ik(L/2)\cos\beta} = e^{kz}\,e^{-ik y\sin\beta}\,e^{-iu} \end{align}\]
The double integrals therefore separate:
\[\begin{align} \hat{F}_{I1}^{(1)} &= \rho g A e^{i\varepsilon}\bigl(e^{iu}-e^{-iu}\bigr) \Biggl(\int_{-T}^{0}e^{kz}\,dz\Biggr) \Biggl(\int_{-B/2}^{B/2}e^{-ik y\sin\beta}\,dy\Biggr) \nonumber\\ &= \rho g A e^{i\varepsilon}\cdot(2i\sin u)\cdot J_z\cdot J_y \nonumber\\ &= 2i\,\rho g A e^{i\varepsilon}\,J_z\,J_y\sin u \end{align}\]
Writing \(J_y\) and \(J_z\) explicitly,
\[\begin{align} \hat{F}_{I1}^{(1)} &= 2i\,\rho g A e^{i\varepsilon}\,\frac{1-e^{-kT}}{k}\, B\frac{\sin v}{v}\,\sin u \end{align}\]
7.3.1.2 Sway force
Only the port and starboard faces contribute to \(n_2\). On the starboard face, \(y=-B/2\) and \(n_2=1\); on the port face, \(y=+B/2\) and \(n_2=-1\). Therefore
\[\begin{align} \hat{F}_{I2}^{(1)} &= \rho g A e^{i\varepsilon}\Biggl[ \int_{-T}^{0}\int_{-L/2}^{L/2} \Psi\!\left(x,-\tfrac{B}{2},z\right)\,dx\,dz - \int_{-T}^{0}\int_{-L/2}^{L/2} \Psi\!\left(x,\tfrac{B}{2},z\right)\,dx\,dz \Biggr] \end{align}\]
Now
\[\begin{align} \Psi\!\left(x,-\tfrac{B}{2},z\right) &= e^{kz}\,e^{-ik x\cos\beta}\,e^{iv} \qquad \Psi\!\left(x,\tfrac{B}{2},z\right) &= e^{kz}\,e^{-ik x\cos\beta}\,e^{-iv} \end{align}\]
so
\[\begin{align} \hat{F}_{I2}^{(1)} &= \rho g A e^{i\varepsilon}\bigl(e^{iv}-e^{-iv}\bigr)\,J_z\,J_x = 2i\,\rho g A e^{i\varepsilon}\,J_z\,J_x\sin v \nonumber\\ &= 2i\,\rho g A e^{i\varepsilon}\,\frac{1-e^{-kT}}{k}\, L\frac{\sin u}{u}\,\sin v \end{align}\]
7.3.1.3 Heave force
Only the bottom contributes to \(n_3\). On \(z=-T\), \(n_3=1\), and
\[\begin{align} \hat{F}_{I3}^{(1)} &= \rho g A e^{i\varepsilon}\int_{-B/2}^{B/2}\int_{-L/2}^{L/2} \Psi(x,y,-T)\,dx\,dy \nonumber\\ &= \rho g A e^{i\varepsilon}\,e^{-kT} \Biggl(\int_{-L/2}^{L/2}e^{-ik x\cos\beta}\,dx\Biggr) \Biggl(\int_{-B/2}^{B/2}e^{-ik y\sin\beta}\,dy\Biggr) \nonumber\\ &= \rho g A e^{i\varepsilon}\,e^{-kT}\,J_x\,J_y \nonumber\\ &= \rho g A e^{i\varepsilon}\,e^{-kT}\, L\frac{\sin u}{u}\,B\frac{\sin v}{v} \end{align}\]
7.3.1.4 Moments: the lever arm \(\vec{r}_P^{(0)}\times\hat{n}^{(0)}\)
With \(\vec{r}_P^{(0)}=(x,y,z)\),
\[\begin{align} \vec{r}_P^{(0)}\times\hat{n}^{(0)} = \begin{pmatrix} y n_3 - z n_2 \\ z n_1 - x n_3 \\ x n_2 - y n_1 \end{pmatrix} \end{align}\]
7.3.1.5 Roll moment
The roll component is \(y n_3 - z n_2\). The bow and stern give \(n_2=n_3=0\) and do not contribute. The remaining three faces give
\[\begin{align} \hat{M}_{I1}^{(1)} &= \rho g A e^{i\varepsilon}\Biggl[ \underbrace{\iint_{\text{bottom}} y\,\Psi\,dx\,dy}_{n_3=1} + \underbrace{\iint_{\text{port}} z\,\Psi\,dx\,dz}_{n_2=-1\;\Rightarrow\; -z n_2 = z} + \underbrace{\iint_{\text{starboard}} (-z)\,\Psi\,dx\,dz}_{n_2=+1\;\Rightarrow\; -z n_2 = -z} \Biggr] \end{align}\]
Bottom contribution:
\[\begin{align} \iint_{\text{bottom}} y\,\Psi\,dx\,dy &= e^{-kT}\Biggl(\int_{-L/2}^{L/2}e^{-ik x\cos\beta}\,dx\Biggr) \Biggl(\int_{-B/2}^{B/2} y\,e^{-ik y\sin\beta}\,dy\Biggr) \nonumber\\ &= e^{-kT}\,J_x\,L_y \end{align}\]
Port and starboard contributions:
\[\begin{align} &\iint_{\text{port}} z\,\Psi\,dx\,dz = e^{-iv}\,K_z\,J_x \\ &\iint_{\text{starboard}} (-z)\,\Psi\,dx\,dz = -e^{iv}\,K_z\,J_x \end{align}\]
and their sum is
\[\begin{align} \bigl(e^{-iv}-e^{iv}\bigr)K_z J_x = -2i\sin v\,K_z J_x \end{align}\]
Collecting terms,
\[\begin{align} \hat{M}_{I1}^{(1)} &= \rho g A e^{i\varepsilon}\bigl(e^{-kT}\,J_x\,L_y - 2i\sin v\,K_z J_x\bigr) = \rho g A e^{i\varepsilon}\,J_x\bigl(e^{-kT}\,L_y - 2i K_z\sin v\bigr) \end{align}\]
7.3.1.6 Pitch moment
The pitch component is \(z n_1 - x n_3\). Port and starboard do not contribute. The remaining faces give
\[\begin{align} \hat{M}_{I2}^{(1)} &= \rho g A e^{i\varepsilon}\Biggl[ \underbrace{\iint_{\text{bottom}} (-x)\,\Psi\,dx\,dy}_{n_3=1} + \underbrace{\iint_{\text{stern}} z\,\Psi\,dy\,dz}_{n_1=1} + \underbrace{\iint_{\text{bow}} (-z)\,\Psi\,dy\,dz}_{n_1=-1} \Biggr] \end{align}\]
Bottom contribution:
\[\begin{align} \iint_{\text{bottom}} (-x)\,\Psi\,dx\,dy &= -e^{-kT}\,L_x\,J_y \end{align}\]
Stern and bow contributions:
\[\begin{align} \iint_{\text{stern}} z\,\Psi\,dy\,dz &= e^{iu}\,K_z\,J_y \qquad \iint_{\text{bow}} (-z)\,\Psi\,dy\,dz &= -e^{-iu}\,K_z\,J_y \end{align}\]
and their sum is
\[\begin{align} \bigl(e^{iu}-e^{-iu}\bigr)K_z J_y = 2i\sin u\,K_z J_y \end{align}\]
Collecting terms,
\[\begin{align} \hat{M}_{I2}^{(1)} &= \rho g A e^{i\varepsilon}\bigl(-e^{-kT}\,L_x\,J_y + 2i\sin u\,K_z J_y\bigr) = \rho g A e^{i\varepsilon}\,J_y\bigl(-e^{-kT}\,L_x + 2i K_z\sin u\bigr) \end{align}\]
7.3.1.7 Yaw moment
The yaw component is \(x n_2 - y n_1\). The bottom does not contribute. The four vertical faces give
\[\begin{align} \hat{M}_{I3}^{(1)} &= \rho g A e^{i\varepsilon}\Biggl[ \iint_{\text{port}} (-x)\,\Psi\,dx\,dz + \iint_{\text{starboard}} x\,\Psi\,dx\,dz \nonumber\\ &\quad + \iint_{\text{stern}} (-y)\,\Psi\,dy\,dz + \iint_{\text{bow}} y\,\Psi\,dy\,dz \Biggr] \end{align}\]
Evaluating each face,
\[\begin{align} \iint_{\text{port}} (-x)\,\Psi\,dx\,dz &= -e^{-iv}\,L_x\,J_z \\ \iint_{\text{starboard}} x\,\Psi\,dx\,dz &= e^{iv}\,L_x\,J_z \\ \iint_{\text{stern}} (-y)\,\Psi\,dy\,dz &= -e^{iu}\,L_y\,J_z \\ \iint_{\text{bow}} y\,\Psi\,dy\,dz &= e^{-iu}\,L_y\,J_z \end{align}\]
Adding these contributions,
\[\begin{align} \hat{M}_{I3}^{(1)} &= \rho g A e^{i\varepsilon}\Bigl[ \bigl(e^{iv}-e^{-iv}\bigr)L_x J_z + \bigl(e^{-iu}-e^{iu}\bigr)L_y J_z \Bigr] \nonumber\\ &= \rho g A e^{i\varepsilon}\,J_z\bigl(2i\sin v\,L_x - 2i\sin u\,L_y\bigr) \nonumber\\ &= 2i\,\rho g A e^{i\varepsilon}\,J_z\bigl(L_x\sin v - L_y\sin u\bigr) \end{align}\]
7.3.1.8 Summary
The six Froude–Krylov amplitudes on the box barge (deep water, \(U=0\)) are
\[\begin{align} \hat{F}_{I1}^{(1)} &= 2i\,\rho g A e^{i\varepsilon}\,J_z\,J_y\sin u \\ \hat{F}_{I2}^{(1)} &= 2i\,\rho g A e^{i\varepsilon}\,J_z\,J_x\sin v \\ \hat{F}_{I3}^{(1)} &= \rho g A e^{i\varepsilon}\,e^{-kT}\,J_x\,J_y \\ \hat{M}_{I1}^{(1)} &= \rho g A e^{i\varepsilon}\,J_x\bigl(e^{-kT}\,L_y - 2i K_z\sin v\bigr) \\ \hat{M}_{I2}^{(1)} &= \rho g A e^{i\varepsilon}\,J_y\bigl(-e^{-kT}\,L_x + 2i K_z\sin u\bigr) \\ \hat{M}_{I3}^{(1)} &= 2i\,\rho g A e^{i\varepsilon}\,J_z\bigl(L_x\sin v - L_y\sin u\bigr) \end{align}\]
with \(u\), \(v\), \(J_x\), \(J_y\), \(J_z\), \(K_z\), \(L_x\), \(L_y\) as defined above. The physical time-domain loads are recovered by taking the real part after restoring the factor \(e^{i\omega t}\),
\[\begin{align} \vec{F}_I^{(1)}(t) &= \operatorname{Re}\bigl\{\hat{\vec{F}}_I^{(1)} e^{i\omega t}\bigr\} \\ \vec{M}_I^{(1)}(t) &= \operatorname{Re}\bigl\{\hat{\vec{M}}_I^{(1)} e^{i\omega t}\bigr\} \end{align}\]
Two useful special cases follow at once. In head seas (\(\beta=\pi\), so \(u=-kL/2\), \(v=0\), \(J_y=B\), \(L_y=0\), \(\sin v=0\))
\[\begin{align} \hat{F}_{I2}^{(1)}=\hat{M}_{I1}^{(1)}=\hat{M}_{I3}^{(1)}=0 \end{align}\]
and only surge, heave and pitch survive. In beam seas (\(\beta=\pi/2\), so \(u=0\), \(v=kB/2\), \(J_x=L\), \(L_x=0\), \(\sin u=0\))
\[\begin{align} \hat{F}_{I1}^{(1)}=\hat{M}_{I2}^{(1)}=\hat{M}_{I3}^{(1)}=0 \end{align}\]
and only sway, heave and roll survive. In both cases the yaw Froude–Krylov moment vanishes by symmetry; a non-zero \(\hat{M}_{I3}^{(1)}\) appears only for oblique heading.
Interactive Froude–Krylov RAO
The expressions above are the Froude–Krylov load RAOs: for each mode \(j\), the complex amplitude \(\hat{\tau}_{Ij}/A\) as a function of wave frequency (with the wave phase \(\varepsilon\) set to zero at the BCS origin). Use the tool below to plot their magnitude and phase for the box barge. Choose a mode from the dropdown; the heading \(\beta\) and the barge dimensions \(L\), \(B\), \(T\) can be varied with the sliders. The marker tracks the frequency selected by the \(\omega\) slider.
7.4 Scattering Wave Forces
The consideration of the incident wave forces, disregards the scattered wave generated by the reflection of the incident wave off of the body boundary. The velocity potential associated to this scattered wave field is denoted as \(\varphi_S\). With the application of perturbation approach the total scattered potential can be separated into various orders as:
\[\begin{align} \varphi_S = \epsilon\varphi_S^{(1)} + \epsilon^2 \varphi_S^{(2)} + ~ ... \end{align}\]
The boundary value problem corresponding to the first order scattering potential \(\varphi_S^{(1)}\) is described by the governing equation:
\[\begin{align} \nabla^2 \varphi_S^{(1)} = 0 \end{align}\]
with free surface boundary conditions:
\[\begin{align} \frac{\partial \varphi_S^{(1)}}{\partial t} - U \frac{\partial \varphi_S^{(1)}}{\partial x} + g \eta = 0 \text{ over } z = 0 \end{align}\]
\[\begin{align} \frac{\partial \eta^{(1)}}{\partial t} - U\frac{\partial \eta^{(1)}}{\partial x} - \frac{\partial \varphi_S^{(1)}}{\partial z} = 0 \text{ over } z = 0 \end{align}\]
bottom boundary condition:
\[\begin{align} \frac{\partial \varphi_S^{(1)}}{\partial z} = 0 \text{ over } S_Z = z + h = 0 \end{align}\]
Sommerfield radiation boundary condition:
\[\begin{align} \lim_{r\rightarrow\infty} \sqrt{r}\left[\frac{\partial \varphi_S^{(1)}}{\partial r} - i k \varphi_S^{(1)}\right] = 0 \text{ over } S_{\infty} \end{align}\]
and body boundary condition:
\[\begin{align} \frac{\partial \varphi_S^{(1)}}{\partial n} = -\frac{\partial \varphi_I^{(1)}}{\partial n} \text{ over } S_{B0} \end{align}\]
where \(S_{B0}\) is the surface of the hull in its equilibrium position \((\{\xi\} = 0)\). The body boundary condition for the scattering potential enforces that the combined wave field due to the incident wave and the scattered wave result in no normal velocity on the hull surface. Taking a Fourier transform of the boundary value problem leads to the boundary value problem for \(\hat{\varphi}_S^{(1)} (\omega_e) = \mathcal{F} \left(\varphi_S^{(1)}\right)\) with governing equation:
\[\begin{align} \nabla^2 \hat{\varphi}_S^{(1)} = 0 \end{align}\]
with combined free surface boundary condition:
\[\begin{align} \left(i\omega_e - U \frac{\partial }{\partial x}\right)^2 \hat{\varphi}_S^{(1)} + g \frac{\partial \hat{\varphi}_S^{(1)}}{\partial z} = 0 \text{ over } z = 0 \end{align}\]
bottom boundary condition:
\[\begin{align} \frac{\partial \hat{\varphi}_S^{(1)}}{\partial z} = 0 \text{ over } S_Z = z + h = 0 \end{align}\]
Sommerfield radiation boundary condition:
\[\begin{align} \lim_{r\rightarrow\infty} \sqrt{r}\left[\frac{\partial \hat{\varphi}_S^{(1)}}{\partial r} - i k \hat{\varphi}_S^{(1)}\right] = 0 \text{ over } S_{\infty} \end{align}\]
and body boundary condition:
\[\begin{align} \frac{\partial \hat{\varphi}_S^{(1)}}{\partial n} = -\frac{\partial \hat{\varphi}_I^{(1)}}{\partial n} \text{ over } S_{B0} \end{align}\]
Since \(\hat{\varphi}_I^{(1)}\) is already known analytically, the boundary value problem can be solved in a similar manner (using boundary element methods) as the radiation potential boundary value problem discussed in the previous chapter. Most softwares such as WAMIT, ANSYS AQWA or HydRA that solve the radiation potential problem also solve the scattering potential problem.
7.4.1 Calculating Scattering Forces and Moments
Once the linearized boundary value problem has been solved, the potential \(\hat{\varphi}_S^{(1)}\) is known. The Fourier transform of the first order generalized force acting on the vessel due to the scattering waves can be evaluated as:
\[\begin{align} \left\{\hat{\tau}_S(\omega_e)\right\} = \begin{bmatrix} \hat{\vec{F}}_S^{(1)} (\omega_e) \\ \hat{\vec{M}}_S^{(1)} (\omega_e) \end{bmatrix} = -\rho \begin{bmatrix} \iint_{S_{B0}} \left(i\omega_e - U \frac{\partial }{\partial x}\right)\hat{\varphi}_S^{(1)} \hat{n}^{(0)} dS \\ \iint_{S_{B0}} \left(i\omega_e - U \frac{\partial }{\partial x}\right)\hat{\varphi}_S^{(1)} (\vec{r}_P^{(0)} \times \hat{n}^{(0)}) dS \end{bmatrix} \label{eq-scattering-force-ft} \end{align}\]
The combined forcing due to incident and scattering waves is known as diffraction force and is given by:
\[\begin{align} \left\{\hat{\tau}_D(\omega_e)\right\} = \begin{bmatrix} \hat{\vec{F}}_D^{(1)} (\omega_e) \\ \hat{\vec{M}}_D^{(1)} (\omega_e) \end{bmatrix} = -\rho \begin{bmatrix} \iint_{S_{B0}} \left(i\omega_e - U \frac{\partial }{\partial x}\right)\hat{\varphi}_D^{(1)} \hat{n}^{(0)} dS \\ \iint_{S_{B0}} \left(i\omega_e - U \frac{\partial }{\partial x}\right)\hat{\varphi}_D^{(1)} (\vec{r}_P^{(0)} \times \hat{n}^{(0)}) dS \end{bmatrix} \label{eq-diffraction-force-ft} \end{align}\]
where \(\hat{\varphi}_D^{(1)} = \hat{\varphi}_I^{(1)} + \hat{\varphi}_S^{(1)}\).
Interactive: Wave Loads on the KCS Container Ship
The box barge above is the only hull for which the Froude–Krylov integrals close in a few lines. For a real ship both the incident-wave integral in \(\eqref{eq-incident-force-ft}\) and the scattering problem of \(\eqref{eq-scattering-force-ft}\) are evaluated numerically over a panel mesh. The tool below shows the result of such a computation for the KCS, the MOERI container ship that serves as a benchmark hull for the ship hydrodynamics community.
The loads are from a frequency-domain panel-method (HydRA) run on the KCS at full scale — \(L_{pp} = 230\) m, \(B = 32.2\) m, \(T = 10.8\) m, \(\nabla = 52\,030\) m\(^3\), \(C_B = 0.651\), see the SIMMAN 2008 hull description — in deep water, for 50 wave frequencies \(\omega = 0.03, 0.06, \ldots, 1.50\) rad/s, eight headings in \(45^{\circ}\) steps, and three forward speeds: \(U = 0\), \(6.17\) m/s (12 kn, \(Fn = 0.13\)) and \(12.35\) m/s (24 kn, the service speed, \(Fn = 0.26\)). The Froude–Krylov and scattering parts are output separately, which is what makes the split of this chapter visible in numbers. All loads are per unit wave amplitude \(A\), with the wave phase \(\varepsilon = 0\) at the BCS origin (midship, on the calm waterline), so the phase of each load is measured relative to the incident-wave crest passing the origin. The Scale button switches between dimensional loads (N/m, N·m/m) and the non-dimensional forms \(\hat{F}/(\rho g A L^2)\) for the forces and \(\hat{M}/(\rho g A L^3)\) for the moments, with \(L = L_{pp} = 230\) m.
The three speeds are always drawn together, and the Axis button re-plots the same data against the encounter frequency \(\omega_e = \omega - (\omega^2/g)U\cos\beta\) instead of the wave frequency \(\omega\). Choose a mode and a heading, and move the frequency marker with the slider or by dragging on any plot.
Four things are worth chasing:
The Froude–Krylov load does not know the ship is moving. It is the pressure of the undisturbed incident wave integrated over the same mean wetted surface, and neither depends on \(U\). On the \(\omega\) axis the three FK curves lie exactly on top of one another. On the \(\omega_e\) axis they separate — not because the load changed, but because the same wave is met at a different frequency. This is the whole content of the Doppler shift for the incident-wave problem.
The scattering load does. Its boundary value problem carries the operator \((i\omega_e - U\,\partial/\partial x)\) in both the free-surface condition and the pressure, so its curves differ genuinely between speeds, not merely by a relabelling of the axis. Compare pitch in head seas (\(\beta = 180^{\circ}\)): the scattering moment changes by tens of percent between \(0\) and \(24\) kn while the Froude–Krylov moment does not move at all.
Following seas fold the encounter axis. With \(\cos\beta > 0\) the encounter frequency rises with \(\omega\), peaks at \(\omega = g/(2U\cos\beta)\) and then falls through zero at \(\omega = g/(U\cos\beta)\), beyond which the ship overtakes the waves. On the \(\omega_e\) axis the curves for \(U > 0\) double back on themselves: two very different wavelengths are met at the same \(\omega_e\). Try \(\beta = 0^{\circ}\) or \(45^{\circ}\) and watch the 24 kn curve.
Long waves and short waves. As \(\omega \rightarrow 0\) the heave FK force tends to \(\rho g A_{WP}\) per unit amplitude — the ship is simply lifted by a slowly rising sea — while at high frequency the hull spans many wavelengths, the pressure cancels along the length and every load decays. In between, the pitch moment in head seas peaks near \(\lambda \approx 1.5\,L_{pp}\): long enough for the bow and stern to sit on opposite slopes of the wave, short enough for the wave slope itself to be appreciable.