Motion Transfer Functions
By Ivar J. Fylling, MARINTEK
1. Introduction
Motion transfer functions have proved to give an efficient description of the motion characteristics of floating vessels. The transfer functions are calculated for the 6 motion degrees of freedom for a specified reference point on the vessel, either from potential theory or from model test results. In the interpretation of motion characteristics, we normally focus on the amplitudes and often neglect the phase angles.
In practical use one will often have to calculate motions of other points on the vessel, motions relative to other vessels, or motions referring to other coordinate systems. This will require a combination of two or more degrees of freedom, and the result will depend on both amplitudes and phase angles.
A lot of trouble has been caused by erroneous interpretation of transfer functions and erroneous transformation. In many cases a simple checking of asymptotic phase angles can be useful in verifying the actual interpretation. The present note is an effort to describe the asymptotic phase angles, the different coordinate systems that are often used, and to show a recipe for transformation of phase angles.
2. Terminology and Conventions

Phase angle  If nothing else is said this means forward phase shift. Backwards phase shift, delay, may be denoted
phase lag
. 
Coordinate system  All coordinate systems are righthanded Cartesian systems.

Transfer function  Physical relation between harmonic wave and linear response \(\mathrm {X(t)=R_x\zeta_a\sin(\omega t+\phi _x)}\). The transfer function consists of an amplitude ratio, \(\mathrm {R_x=\displaystyle \frac{X_a}{\zeta_a}}\), and a phase angle, \(\mathrm {\phi _x}\).
Vectorial components 
\(\mathrm {x_1}\) 
\(\mathrm {x_2}\) 
\(\mathrm {x_3}\) 
\(\mathrm {x_4}\) 
\(\mathrm {x_5}\) 
\(\mathrm {x_6}\) 
Names of axes 
X 
Y 
Z 

Wave particle motions 
\(\mathrm {\xi }\) 
\(\mathrm {\eta }\) 
\(\mathrm {\zeta}\) 
\(\mathrm {\gamma _1}\) 
\(\mathrm {\gamma _2}\) 
\(\mathrm {\gamma _3}\) 
Names of motions 
Surge 
Sway 
Heave 
Roll 
Pitch 
Yaw 
Observe that the naming of motions refers to the coordinate system,
regardless of whether the Xaxis is pointing forwards
or
backwards
in a vessel.
3. Wave Potential
The velocity potential for a regular wave according to Airy’s theory can be expressed as follows:
where

\(\mathrm {\zeta_a\qquad}\) is the wave amplitude

\(\mathrm {g\qquad\:\:}\) is the acceleration due to gravity

\(\mathrm {k\qquad\:\:}\) is the wave number

\(\mathrm {\beta \qquad\:}\) is the direction of wave propagation (\(\mathrm {\beta =0}\) corresponds to wave propagation along the positive xaxis)

\(\mathrm {\phi _\zeta\qquad}\) is the wave component phase angle .
\(\mathrm {C_1}\) is given by
where \(\mathrm {d}\) is the water depth.
In deep water, \(\mathrm {C_1}\) can be approximated by
We then obtain the following relations for the particle velocities and accelerations in the undisturbed wave field:
where
Using the deep water approximation we have
Taking into account finite water depth we have
The surface elevation is given by
It is important to notice these definitions.
Similarly the linearized dynamic pressure is given by
4. Phase Angles of Wave Particle Motions
4.1. Particle motions
Particle motions are obtained by integrating the velocity functions, Equation (4)Equation (5).
\(\mathrm {C_2}\) and \(\mathrm {C_3}\) are depthdependent functions, according to Equation (7).
The integration constants \(\mathrm {x_0}\), \(\mathrm {y_0}\) and \(\mathrm {z_0}\) have been selected so that the average value is zero.
Thus, the particle motions are expressed by
4.2. Phase angles of particle translations
In nearly all contexts, the surface elevation is selected as the reference process when describing waves and waveinduced responses. According to the previous section, the particle translations can be written
where
Differentiation with respect to time gives an increase of phase angle of \(\mathrm {\displaystyle \frac{\pi }{2}}\).
4.3. Transfer functions of wave angular motions
The wave angular motions of a wave particle rod
that at rest is
parallel with the xaxis (right handed rotations positive) are defined
by
This gives phase angles
These will be the asymptotic phase angles of roll, pitch, and yaw motion of a slender ship when the wave length becomes large compared with the ship and the ship is oriented parallel with the xaxis.
4.3.1. Other motions
All other motions or other responses, \(\mathrm {r}\), that are linearly dependent on the waves are expressed in the following way:
where

\(\mathrm {r_a\quad }\) = harmonic amplitude

\(\mathrm {\phi _p\quad }\) = phase angle relative to the surface elevation.
4.4. Summary of phase angles of wave particle motions
This overview can be used to check asymptotic behaviour of floating structures (long wavelength) and can serve as a basis for converting transfer functions from one reference system to another.
4.4.1. Coordinate system
The wave profile is at \(\mathrm {t=0}\) for \(\mathrm {\phi _\zeta=0}\).
4.4.2. Surface elevation
where

\(\mathrm {\phi _p\quad }\) = \(\mathrm {kx\cos(\beta )ky\sin(\beta )}\)

\(\mathrm {\phi _\zeta\quad }\) = defines the wave state at the origin at time \(\mathrm {t=0}\)

\(\mathrm {\zeta_a\quad }\) = wave amplitude at the surface

\(\mathrm {C_3\quad }\) = is a depth and frequencydependent function
Motion  Displacement  Velocity  Acceleration  

Surge 
\(\mathrm {\xi }\) 
\(\mathrm {\displaystyle \frac{\pi }{2}}\) 
\(\mathrm {0}\) 
\(\mathrm {+\displaystyle \frac{\pi }{2}}\) 
Sway 
\(\mathrm {\eta }\) 
\(\mathrm {\displaystyle \frac{\pi }{2}}\) 
\(\mathrm {0}\) 
\(\mathrm {+\displaystyle \frac{\pi }{2}}\) 
Heave 
\(\mathrm {\zeta}\) 
\(\mathrm {0}\) 
\(\mathrm {+\displaystyle \frac{\pi }{2}}\) 
\(\mathrm {+\pi }\) 
Roll 
\(\mathrm {\gamma _1}\) 
\(\mathrm {\displaystyle \frac{\pi }{2}}\) 
\(\mathrm {0}\) 
\(\mathrm {+\displaystyle \frac{\pi }{2}}\) 
Pitch 
\(\mathrm {\gamma _2}\) 
\(\mathrm {+\displaystyle \frac{\pi }{2}}\) 
\(\mathrm {+\pi }\) 
\(\mathrm {+\displaystyle \frac{3\pi }{2}}\) 
Yaw 
\(\mathrm {\gamma _3}\) 
\(\mathrm {+\pi }\) 
\(\mathrm {+\displaystyle \frac{3\pi }{2}}\) 
\(\mathrm {0}\) 
Motion  Displacement  Velocity  Acceleration  

Surge 
\(\mathrm {\xi }\) 
\(\mathrm {\cos(\beta )C_2\zeta_a}\) 
\(\mathrm {\omega \cos(\beta )C_2\zeta_a}\) 
\(\mathrm {\omega ^2\cos(\beta )C_2\zeta_a}\) 
Sway 
\(\mathrm {\eta }\) 
\(\mathrm {\sin(\beta )C_2\zeta_a}\) 
\(\mathrm {\omega \sin(\beta )C_2\zeta_a}\) 
\(\mathrm {\omega ^2\sin(\beta )C_2\zeta_a}\) 
Heave 
\(\mathrm {\zeta}\) 
\(\mathrm {C_3\zeta_a}\) 
\(\mathrm {\omega C_3\zeta_a}\) 
\(\mathrm {\omega ^2C_3\zeta_a}\) 
Roll 
\(\mathrm {\gamma _1}\) 
\(\mathrm {k\sin(\beta )C_3\zeta_a}\) 
\(\mathrm {\omega k\sin(\beta )C_3\zeta_a}\) 
\(\mathrm {\omega ^2k\sin(\beta )C_3\zeta_a}\) 
Pitch 
\(\mathrm {\gamma _2}\) 
\(\mathrm {k\cos(\beta )C_3\zeta_a}\) 
\(\mathrm {\omega k\cos(\beta )C_3\zeta_a}\) 
\(\mathrm {\omega ^2k\cos(\beta )C_3\zeta_a}\) 
Yaw 
\(\mathrm {\gamma _3}\) 
\(\mathrm {k\sin(\beta )}\) \(\mathrm {\cdot \cos(\beta )C_2\zeta_a}\) 
\(\mathrm {\omega k\sin(\beta )}\) \(\mathrm {\cdot \cos(\beta )C_2\zeta_a}\) 
\(\mathrm {\omega ^2k\sin(\beta )}\) \(\mathrm {\cdot \cos(\beta )C_2\zeta_a}\) 
\(\mathrm {C_2}\) is a depth and frequencydependent function.
Observe that \(\mathrm {\sin(\beta )}\) and \(\mathrm {\cos(\beta )}\) enter the expression for amplitudes and include a sign.
When changing coordinate system this can be taken care of in the phase angles if \(\mathrm {\beta }\) goes out of the first quadrant and one wishes to have positive amplitude expressions.
5. Complex Notation of Harmonic Wave Field, Transfer Functions and Responses
We define a complex harmonic wave component by:
Thus, the surface elevation is
All other responses, \(\mathrm {r}\), are related to the surface elevation by complex transfer functions, \(\mathrm {H_r}\), and can be derived from a complex harmonic function, \(\mathrm {R}\):
6. Change of Coordinate Systems and Sign Conventions
6.1. Basis
A righthanded Cartesian coordinate system is used with the Zaxis pointing upwards. The sea surface elevation is:
where

\(\mathrm {\beta \quad \:\:}\) is the propagation direction

\(\mathrm {\phi _\zeta\quad }\) is the phase angle, interpreted as forward phase shift
The transfer function of any response \(\mathrm {j}\) for a given frequency is defined by an amplitude ratio, \(\mathrm {\bigH_j(\beta )\big}\) and a phase angle, \(\mathrm {\phi _j(\beta )}\).

Denoting the base case as case no. 1, the following alternatives are discussed: # 180 deg. rotation about the Zaxis. This is often used in order to obtain a head wave condition with \(\mathrm {\beta =0}\).

180 deg. rotation about Xaxis. This may be done in order to get positive rotation with the bow turning starboard, which is a nautical convention. This requires that the Zaxis points downwards in a righthanded system. # Change of direction convention of the waves, so that \(\mathrm {\beta }\) is the direction from which the wave originates (
comingfrom
direction) instead of the propagation direction. This is also according to nautical practice. 
Change of phase angle convention so that \(\mathrm {\phi }\) denotes phase lag instead of phase addition. This is purely a matter of taste and habit.

Utilization of XZ symmetry by mirroring the transfer functions about the XZ plane.
Figure 3 gives an overview of these alternatives. For the coordinate selections, cases 1, 2, 3, directions and phase angles are written fully. For the alternative sign conventions, cases 4, 5, 6, only the changes are given. These can apply to any one of the cases 1, 2, 3.
Notes:

A change of \(\mathrm {+\pi }\) is equivalent to a change of \(\mathrm {\pi }\).

All cases except 6 represent the same physical system and responses.

The cases 4, 5, 6 can be superimposed on any of the coordinate system alternatives 1, 2, 3.

If relative motions are to be calculated, the same modifications must be carried out both for vessel transfer functions and for wave particle motion transfer function.
7. Examples
7.1. Example 1
The NSRDC program has been run (case 2, 5). The resulting transfer functions are to be used as MOSSI input (case 3, 4).
According to Figure 3:
MOSSI (M), (Case 3, 4) 
NSRDC (N), (Case 2, 5) 
MOSSI with \(\mathrm {\phi _N}\) as basis 
\(\mathrm {\beta _M=\beta +\pi }\) 
\(\mathrm {\beta _N=\beta +\pi }\) 
\(\mathrm {\beta _M=\beta _N}\) 
\(\mathrm {\Phi_1}\) 
\(\mathrm {\Phi_2}\) 
\(\mathrm {\Phi_3}\) 
where \(\mathrm {\Phi_1=\phi _M=\begin{cases}\phi _1\\\phi _2+\pi \\\phi _3+\pi \\\phi _4\\\phi _5+\pi \\\phi _6+\pi \end{cases}}\), \(\mathrm {\qquad\Phi_2=\phi _N=\begin{cases}\phi _1\pi \\\phi _2\pi \\\phi _3\\\phi _4\pi \\\phi _5\pi \\\phi _6\end{cases}}\), \(\mathrm {\qquad\Phi_3=\phi _M=\begin{cases}\phi _{N1}\pi \\\phi _{N2}\\\phi _{N3}\pi \\\phi _{N4}\pi \\\phi _{N5}\\\phi _6\pi \end{cases}}\).
In order to use the result for the same wave directions (nominal values), the result is mirrored and \(\mathrm {\pi }\) is added to all phase angles, giving:
\(\mathrm {\begin{array}{l}\beta _M\:\:=\beta _N\\\\\phi _{M1}=\phi _{N1}\\\\\phi _{M2}=\phi _{N2}\\\\\phi _{M3}=\phi _{N3}\\\\\phi _{M4}=\phi _{N4}\pi \\\\\phi _{M5}=\phi _{N5}\pi \\\\\phi _{M6}=\phi _{N6}+\pi \end{array}}\)
The addition of \(\mathrm {\pi }\) is done in order to refer to wave downwards, see note 4 to Figure 3.
7.2. Example 2
Results from the NSRDC program (case 2, 5) are to be compared to those of WADIF (case 2). The difference is simply a sign change of all phases, so the comparable quantities are:
\(\mathrm {\begin{array}{l}\beta _N\quad \:=\beta _W\\\\\phi _{N1}=\phi _{W1}\\\\\phi _{N2}=\phi _{W2}\\\\\phi _{N3}=\phi _{W3}\\\\\phi _{N4}=\phi _{W4}\\\\\phi _{N5}=\phi _{W5}\\\\\phi _{N6}=\phi _{W6}\end{array}}\)
7.3. Example 3
WAMOF (case 1) is to be run to create input to MOSSI (case 3, 4) for wave directions \(\mathrm {\beta _M=0^\circ,30^\circ,60^\circ,90^\circ,120^\circ,150^\circ,180^\circ}\).
WAMOF  MOSSI 

\(\mathrm {\beta _W=\beta }\) 
\(\mathrm {\beta _M=\beta \pi \rightarrow\beta =\beta _M\pi }\) 
\(\mathrm {\Phi_4}\) 
\(\mathrm {\Phi_5}\) 
where \(\mathrm {\Phi_4=\phi _W=\begin{cases}\phi _1\\\phi _2\\\phi _3\\\phi _4\\\phi _5\\\phi _6\end{cases}}\), \(\mathrm {\qquad\Phi_5=\phi _M=\begin{cases}\phi _1\\\phi _2+\pi \\\phi _3+\pi \\\phi _4\\\phi _5+\pi \\\phi _6+\pi \end{cases}}\).
MOSSI 
\(\mathrm {\beta _M}\) 
\(\mathrm {0^\circ}\) 
\(\mathrm {30^\circ}\) 
\(\mathrm {60^\circ}\) 
\(\mathrm {90^\circ}\) 
\(\mathrm {120^\circ}\) 
\(\mathrm {150^\circ}\) 
\(\mathrm {180^\circ}\) 
WAMOF 
\(\mathrm {\beta _W}\) 
\(\mathrm {180^\circ}\) 
\(\mathrm {150^\circ}\) 
\(\mathrm {120^\circ}\) 
\(\mathrm {90^\circ}\) 
\(\mathrm {60^\circ}\) 
\(\mathrm {30^\circ}\) 
\(\mathrm {0^\circ}\) 
Conversion of phase angles carried out by WAMOF when results are written to a MOSSI file:  For relative motion problems, the \(\mathrm {180^\circ}\) phase shift of the wave from case 1 to case 3 has to be included: 

\(\mathrm {\phi _{M1}=\phi _{W1}}\) 
\(\mathrm {\phi _{M1}=\phi _{W1}+\pi }\) 
\(\mathrm {\phi _{M2}=\phi _{W2}+\pi }\) 
\(\mathrm {\phi _{M2}=\phi _{W2}}\) 
\(\mathrm {\phi _{M3}=\phi _{W3}+\pi }\) 
\(\mathrm {\phi _{M3}=\phi _{W3}}\) 
\(\mathrm {\phi _{M4}=\phi _{W4}}\) 
\(\mathrm {\phi _{M4}=\phi _{W4}+\pi }\) 
\(\mathrm {\phi _{M5}=\phi _{W5}+\pi }\) 
\(\mathrm {\phi _{M5}=\phi _{W5}}\) 
\(\mathrm {\phi _{M6}=\phi _{W6}+\pi }\) 
\(\mathrm {\phi _{M6}=\phi _{W6}}\) 
For problems involving only absolute motions these two sets of phase angles are equivalent.
7.4. Sample Transfer Functions
7.4.1. Sample 1: A ship
Coordinate system according to case 1, Figure 3.