Chapter 13

The Fortran utility F2T (Frequency-to-Time-domain) is a post-processor to transform frequency-domain WAMIT output to time-domain impulse-response functions (IRFs). This program is intended to provide a utility which can be used for general purposes, based on standard WAMIT outputs. This program accepts as input all of the first-order (linear) outputs from WAMIT, including any combinations of Options 1-6 (added mass/damping, Haskind exciting forces, Diffraction exciting forces, RAO’s, body pressures/velocities, field-point pressures/velocities). In principle there are no restrictions regarding the numbers of rigid-body modes, generalized modes, or bodies. The computed IRFs are saved in output files which are analogous to the input files for each option and use the same filename extensions.

The Fourier transforms from the frequency-domain to the time-domain are evaluated in F2T by Filon numerical integration. This method provides relatively accurate results for large values of the time variable t. A fundamental requirement is that the frequency-domain data must be evaluated by WAMIT for a large number N of uniformly-spaced frequencies ωn where n=1,2,3,...,N. Special attention is necessary to ensure that the increment Δω = ωn+1 - ωn is sufficiently small (to preserve the accuracy of the numerical integration) and that the highest finite frequency ωN is sufficiently large to span the physically-significant range of frequencies for the application (or from the mathematical standpoint, to ensure that the truncated Fourier integrals are reasonable approximations of the infinite integrals). In view of the need to include high frequencies in the WAMIT analysis it is usually advisable to use the irregular-frequency option (IRR=1), unless the body is submerged or its waterplane area is very small. The requirement of accuracy over a broad range of frequencies means that either a large number of low-order panels should be used, or alternatively that the higher-order method is used with appropriate control of the panel subdivision indices NU,NV or the global parameter PANEL_SIZE.

Section 6.6 of Reference [26] contains additional information including an outline of the numerical method and comparisons with the results from the time-domain panel program TiMIT.


The outputs from WAMIT and F2T are considered to be of either the radiation or diffraction type, depending on whether they are caused by forced motions in calm water or by incident waves, respectively. The simplest physical distinction between these two types is in terms of the incident wave amplitude: if the response is proportional to the wave amplitude it is of the diffraction type, and vice versa.

The added-mass and damping coefficients (Option 1) are of the radiation type, whereas the exciting forces and RAO’s (Options 2,3,4) are of the diffraction type. Except as noted in Section 13.5, the pressures and fluid velocities on the body (Option 5) and in the PICT PICT fluid (Option 6) are of the diffraction type, since these are defined in the WAMIT convention as total responses with the body free to respond (or fixed) in incident waves.

An important difference between the two types of outputs is in terms of their limits at infinite frequency or zero period. In this limit the radiation outputs are generally real and nonzero, corresponding to the added mass, pressure, and fluid velocity induced by forced motions of the body without wave effects on the free surface. Conversely, in the same limit there are no diffraction effects since the ‘incident waves’ have vanishingly small wavelength and cause no disturbance of either the body or the fluid.

The principal radiation IRFs correspond to the added-mass and damping coefficients, evaluated by WAMIT in Option 1. To displace the body in calm water in mode j with time-varying acceleration ξ
 j(t) the component i of the force required to oppose the hydrodynamic pressure is given [29] by
F (t) = A  (∞ )ξ (t) +  ∞ L  (τ)ξ (t - τ )dτ
  i       ij     j      0    ij    j

where Aij() is the infinite-frequency limit of the added mass. The radiation IRF Lij(t) is evaluated from either the Fourier cosine or sine transforms (13.5-6). These IRFs are output from F2T in the files _IR.1, _JR.1 and _KR.1 as explained below in Section 13.4.

The exciting forces and RAOs are of the diffraction type, evaluated by WAMIT in Options 2, 3 and 4 and defined relative to an incident wave of uniform amplitude A, propagating in the direction β. The corresponding IRFs are defined with respect to an impulsive incident wave, moving in the same direction, where the free-surface elevation η(x,y,t) is equal to a delta-function δ(t) at the origin x = y = 0. Further details are given in [26] and [30]. Here we define a general output Ui(t) to be either the exciting force or the RAO, with respect to the mode i. It follows that
Ui(t) =  - ∞ Ki(τ)η(0,0, t - τ) dτ

The IRFs Ki(t) are evaluated from the Fourier transform of the frequency-domain exciting force or RAO using (13.10). These IRFs are output from F2T in the files _IR.n and _JR.n where (n=2,3,4), as explained below in Section 13.4.


The frequency-domain input data for F2T is evaluated by WAMIT. The algorithms used to evaluate the Fourier transforms in F2T require that the input data is restricted to a uniformly-spaced set of frequencies ωn = nΔω, where (n=1,2,3,...,NPER). When radiation IRFs are evaluated it is necessary to also evaluate the corresponding frequency-domain outputs for ω0 = 0 and ωN+1 = , using the inputs PER<0.0 and PER=0.0 respectively (See Section 4.2). In the WAMIT run this is done most easily by setting the parameter IPERIN=2 in the configuration files (inputs are radian frequencies), and by using the option to write the data -NPER and ω1, Δω on the lines normally used to specify NPER and the array PER.

The input files used for the tests of the ISSC TLP can be regarded as an example. These files are listed in Appendix A14a. They are essentially the same as for TEST14, with the exception of IPERIN=2, and the specification of the input frequencies.

When the input files to F2T are read, the data is sorted so that the frequencies are listed in ascending order, regardless of their order in the WAMIT output files. Thus the order of the periods (-1.0, 0.0) is irrelevant, and it is possible to patch together two or more separate sets of output files from WAMIT, e.g. one with a coarse set of wave frequencies and the other with intermediate frequencies, to provide a finer set, without concern regarding their order.


The program F2T can be executed after the appropriate WAMIT output files are available. The user must specify the filenames of these files and a small number of input parameters, either interactively in response to appropriate runtime prompts or by preparing the special input file inputs.f2t. The following example of this special input file corresponds to the TEST14a test PICT PICT run TEST14a described in Appendix A:

header line for inputs.f2t control file, TLP example
0.2 100 (DT NT time step and number of time steps)
0 (IOUTFCFS, output both cosine and sine transforms)
43.125 9.80665 ULEN GRAV

These inputs are described for each line as follows:

Line 1 is an ASCII header dimensioned CHARACTER*72 as in most WAMIT input files. This line should be used to insert a brief description of the file.

Line 2 is a list of the filenames (not the extensions) of the primary and secondary WAMIT output files. F2T attempts to open all numeric output files with the same filenames, and includes all of these files in the analysis. Thus the determination of which options to be included depends on the available WAMIT output files. In this example where the TEST14a.FRC control file was used as in the standard WAMIT test runs, Options 1,2,3,4 are included in the F2T analysis. If all of the input data is included in the primary file it is not necessary to list other filenames. Additional secondary files can also be included, up to a maximum limit of 256 ASCII characters for the complete line. At least one blank space must be used to separate each filename.

Lines 3 and 4 contain the six control parameters identified by the comments in parenthesis. These parameters must have the same values as in the WAMIT runs. (No distinction is made between IRAD,IDIFF=0 or 1, and the only important value to specify correctly is -1. For any input values of IRAD,IDIFF other than -1 the results are the same as for 0 or 1.) NUMHDR, which is optional in WAMIT with the default value 0, must be specified here with the value 0 (no headers) or 1 (one line of headers) to indicate the presence or absence of a header line in the WAMIT numeric output files. INUMOPT5 and INUMOPT6 must be specified here with the value 0 (default) or the separate-components values 1. IPEROUT=1 or 2 must be specified, to distinguish wave periods and frequencies in the WAMIT output files

Line 5 contains the time step and number of time steps for the computation and tabulation of the time-domain response functions. The radiation IRF’S are computed and tabulated for t=0 and for NT positive times DT, 2DT, 3DT, ..., NT*DT. The diffraction IRFs are PICT PICT evaluated for both positive and negative times, starting with -NT*DT and ending with +NT*DT.

Line 6 contains the optional parameter IOUTFCFS, with the following options for its value:

IOUTFCFS=1: output only the cosine transform of radiation irf’s

IOUTFCFS=2: output only the sine transform of radiation irf’s

IOUTFCFS=0 (or any other integer except 1 or 2): output both cosine and sine transforms

These transforms are redundant, as explained below. The value IOUTFCFS=0 is recommended except in cases involving a large number of radiation modes, where the number of columns in the _JR1 output files may be excessive.

Line 7 contains the optional parameters ULEN and GRAV, which are the same characteristic length scale and gravitational acceleration parameters as input in the GDF file. These parameters are only required when Options 5 or 6 are included, and when the radiation outputs are specified, as explained in Section 13.5. In all other circumstances the parameters ULEN and GRAV can be omitted from inputs.f2t. When ULEN and GRAV are included in inputs.f2t it is essential to also include IOUTFCFS on line 6.

The use of the special file inputs.f2t is optional. If this file does not exist, or if the first five lines cannot be read with the appropriate data, the user is prompted to specify all of the above inputs interactively. The special file can also be used in a partial form with some but not all of the above lines, but the lines included must be in the same order as above. This permits the user to interactively input different values of the time step and number, simply by omitting Line 5 from the special file.

The numeric data in the special file is read with free format READ statements, separately for each line. Any additional text on the same lines is ignored, so that comments may be inserted as in the example above. The filenames on Line 2 are read as ASCII text of unknown length (maximum of 256 characters, all on one line) and no additional comments may be included on this line.

The program F2T has been updated to function with output files from WAMIT Version 7. To ensure compatibility, users should verify that (1) WAMIT Version 7.0 or later has been used to generate the WAMIT numeric output files, and (b) the file f2t.exe is dated after 31 December 2011. It is possible to mix earlier versions, subject to the following restrictions:


The output files from F2T are in two complementary formats with duplication of the output data in the two formats. The filename assigned to all of the output files is primary, with different extensions. The first set of output files have appended filenames including _IR followed by the same extensions as the WAMIT output files. The second set have the appended filenames including _JR. The first set follow the same format as the WAMIT numeric output files of the same number, except that the period is replaced by the time step and the WAMIT force coefficients are replaced by their Fourier cosine and sine transforms. Different modes and mode combinations are listed on separate lines with the identifying mode indices, just as in the numeric output files of WAMIT.

To facilitate plotting and separation of the different modes and wave angles (BETA), all of the Fourier cosine/sine transforms are listed on one line in the output files denoted by _JR, in the same order of mode combinations but without explicit mode indices. The cosine/sine transforms are listed as pairs, unless one or the other is ommitted by setting IOUTFCFS equal to 1 or 2 as explained in the following paragraph. Column one of the _JR file contains the value of time t.

Either the cosine transforms of the added mass or the sine transforms of the damping can be used to evaluate the radiation IRFs (cf. equations 13.3 and 13.4 below). These two sets of data can be checked to verify their accuracy and consistency, in much the same way that the Haskind and Diffraction exciting forces or cross-coupling coefficients are compared. Alternatively, to achieve more compact output files, one of these transforms can be omitted using the parameter IOUTFCFS.

One extra output file is produced with the extension _KR.1, containing the impulse-response functions Kij which are defined in Reference [30]. These alternative IRF’s are evaluated in F2T by numerical differentiation of the IRF’s Lij, and can be used with a similar convolution integral as in (13.1), but using the velocity ξ˙j instead of the acceleration ξ

The diffraction files _JR are different from the radiation files in two respects, to facilitate their use. First, the time steps begin with -NT*DT, and end with +NT*DT. Secondly, the cosine and sine transforms are combined (adding for t < 0 and subtracting for t > 0) to give the actual IRFs for the corresponding exciting forces and RAOs (cf. equation 13.2).

For practical purposes the .JRn files will be most useful, and the .IRn files may be useful only to clarify the identity of the different columns in the .JRn files. PICT PICT

Some experience and/or trial computations will be needed to determine appropriate values of the input frequencies and time steps. The dimensions of these parameters correspond to GRAV in the WAMIT run.

13.5 OPTIONS 5 AND 6

The F2T utility has been developed primarily for use with Options 1 to 4 (global forces and RAO’s). Local pressures, velocities, and wave elevations may be difficult to transform accurately, due to limited or non-convergence of the Fourier transforms at high frequencies. The outputs from F2T include transforms of the WAMIT outputs for Options 5 and 6, which may be useful for special purposes. A distinction is made between radiation and diffraction components. Radiation components are output from WAMIT separately if INUMOPT5=1 and/or INUMOPT6 = 1, and IRAD = 0 or 1. All other outputs are of the diffraction type, including both the separate diffraction components of the pressure and velocity, and the total superpositions which are output when INUMOPT5=0 and/or INUMOPT6=0.

The diffraction-type pressure and velocity are transformed in the same manner as the exciting forces and RAO’s. The IRFs Ki(t) are evaluated using (13.10), with Xi the frequency-domain output from WAMIT for the same pressure or velocity. The free-surface elevation is equivalent to the pressure on z = 0, as defined in Section 3.6.

The radiation components of the pressure are not output directly by F2T. Instead, the radiation potentials φj defined in Section 3.5 are transformed, using (13.5-6) with the real and imaginary parts of the potential substituted for the added mass and damping. The radiation components of the fluid velocity are defined as the nondimensional gradients of these potentials, as in Section 3.7.

If IDIFF=-1 is specified in the WAMIT run, the WAMIT outputs from Options 5 and 6 are the total responses from superposition of all specified radiation modes. If more than one mode is considered, the output is for nonzero finite frequencies only and is not suitable for transform to the time domain unless the configuration parameters INUMOPT5=1 and INUMOPT6=1 are used.


The fundamental relations between the time- and frequency-domain express the added-mass coefficient Aij and damping coefficient Bij in terms of Fourier transforms of the impulse-response function Lij(t):
Aij(ω ) - Aij(∞ ) = 0  Lij(t)cos ωtdt

            ∫ ∞
Bij(ω ) = ω     Lij(t)sin ωtdt

The inverse-transforms of (13.1-2) give complementary relations for the impulse-response function:
           ∫ ∞
L  (t) = 2-    [A   (ω ) - A  (∞  )]cosωt dω
  ij      π  0    ij       ij

L  (t) = 2-  ∞  Bij(ω)sin ωtdω
 ij     π  0     ω

Similar relations exist for the exciting forces and RAOs. Define one of these quantities by the complex function Xi(ω) The corresponding impulse-response function is real, denoted by Ki(t). The appropriate physical ranges are (0 ω < ) and (-∞ < t < ). Then the complex Fourier transform pairs are as follows:
         ∫ ∞
Xi (ω) =     Ki (t)e- iωtdt

           ∫ ∞
2πKi (t) =     Xi (ω)eiωtdω

Formally, since Ki is real, Xi(-ω) = Xi*(ω), and thus
          ∫ ∞ [       iωt     *    -iωt]
2πKi (t) =  0   Xi (ω )e   + X i (ω)e    dω

          ∫ ∞
K (t) = 1-    [Re(X  )cosωt -  Im (X )sinωt ] dω
  i     π  0        i               i

The Fourier transofrms (13.5), (13.6), and (13.10) are evaluated by truncating the infinite integrations at the largest value of the evaluated frequency, and using Filon quadratures to evaluate the resulting finite integrals. A truncation correction is applied to (13.5). Further details regarding this procedure are given in [26]. PICT PICT


In the WAMIT output files all of the output data is nondimensional, except for the wave period. Definitions of the nondimensional outputs are given in Chapter 3. The wave period, defined in seconds, has the same dimension as ∘----------------
 U LEN  ∕GRAV, where the parameters ULEN and GRAV are input in the GDF file. Since the hydrodynamic force coefficients and other outputs from WAMIT are nondimensional, and the frequency ω has the dimension of inverse time, it follows that the impulse response functions defined by equations 13.3, 13.4 and 13.8 are all dimensional, with the dimension of inverse time. The additional impulse-response functions in the output file .KR1, defined as the time-derivatives of the IRF’s Lij, have the dimension of inverse time squared.

For Options 1-4 the normalized outputs from F2T are defined as follows:

                              ∫ ∞ [                ]              ∫ ∞
Option  1 :      Lij(t) =   2-     Aij(ω) - Aij(∞ )  cosωt dω = -2    Bij(ω )sinωt dω
                            π ∫0                                π  0
                           1-  ∞ [                                  ]
Options  2,3 :   Ki (t) =   π  0   Re (Xi(ω )) cosωt - Im (Xi (ω ))sinωt  dω
                            1 ∫ ∞ [                                  ]
Option  4 :      Ki (t) =   --     Re (ξi(ω ))cosωt - Im (ξi(ω))sinωt  d ω
                            π  0
In these equations the nondimensional frequency-domain parameters are defined in Sections 3.2-4.

For Options 5 and 6 the pressure and velocities are of diffraction type when INUMOPT5,6=0. In this case the normalized outputs from F2T are defined as follows:

                                  1 ∫ ∞
Options  5p,6p :        K (t) =   --    [Re (p(ω ))cosωt - Im  (p(ω ))sinωt] dω
                                  π ∫0
                                 1-  ∞ [                                ]
 Options 5v*, 6v* :     K (t) =   π  0   Re (V (ω ))cosωt - Im  (V (ω )) sin ωt  dω
Here p is the nondimensional total pressure, defined in Section 3.5, and V is the nondimensional total velocity vector defined in Section 3.7.

When INUMOPT5,6=1 the separate components of the diffraction and radiation potentials and velocities are output by WAMIT and transformed by F2T. In this case the normalized outputs from F2T are defined as follows:

                              1-  ∞
 Options  5p,6p :  KD (t)  =   π  0  [Re (φD (ω))cos ωt - Im (φD (ω))sinωt ] dω
                               2 ∫ ∞                                 2∫ ∞
                    Kj(t)  =   --    Re (φj(ω) - φj(∞ )) cosωt dω =  --   Im  (φj(ω )) sin ωtd ω
                               π ∫0∞ [                               π  0     ]
Options  5v*, 6v* : K (t)  =   1-     Re(∇φ (ω)) cosωt - Im (∇ φ  (ω))sinωt  d ω
                     d         π  0          D                    D
                              2-∫ ∞    (            )             2-∫ ∞    (     )
                    Kj(t)  =   π  0  Re  Vj(ω) - Vj(∞ )  cosωt dω =  π  0  Im  Vj(ω ) sinωt dω

In these equations the nondimensional frequency-domain functions are defined in Sections 3.5 and 3.7. Note in the transforms Kj that the frequency-domain functions φj and V j = φj differ from the corresponding outputs in the WAMIT numeric output files by the factor 1∕KL. Thus, before applying the Fourier integrations of these functions in F2T, the WAMIT outputs are divided by KL = ω2L∕g where L=ULEN and g=GRAV. In all other cases above the frequency-domain functions are the same as the WAMIT outputs in the corresponding numeric output files. PICT PICT PICT PICT PICT PICT

Examples of structures analyzed by WAMIT

(for more details click on one of the structures)
Cylinder NavExp Test09 Test 22 Test 25 TLP2ndOrder WitFig6