Davis and Zarnick 



Thus, if we observe a transient wave in the water which initially has a very 

 high frequency (associated with slow velocity) and if this frequency linearly de- 

 creases toward zero with constant amplitude and tapers off as in Fig, 4 with 

 time reversed, then at some point in space and time a very large wave would be 

 created for a brief instant, assuming that linear wave theory holds. This phe- 

 nomenon can be viewed as the simultaneous meeting of a large number of wave 

 components whose individual speeds and starting times were properly adjusted 

 so that the faster traveling waves were behind but catching up with the slower 

 ones. 



For the purposes of wave generation in a model-testing facility, it is mani- 

 festly impossible to provide a sinusoidal wave at infinite frequency. However, 

 it is certainly possible to generate a wave train which has a frequency varying 

 linearly from the highest desired value toward zero. Such waves have been the 

 backbone of the Model Basin transient studies and will be described more fully 

 along with experimental results in the following sections. Briefly, the linear 

 theory appears to hold quite well, and in the early exploratory studies very large 

 peaked waves — approximating the impulse — were formed, although they were 

 limited by cresting and other nonlinear mechanisms. 



An important property associated with a transient water wave is that the 

 magnitude of the Fourier transform remain constant regardless of where it is 

 observed and when the origin in time is fixed; i.e., the water transfer function 

 is solely a phase operator. For a pair of moving probes separated by a fixed 

 distance x, the same frequency response relation is applicable. However, 

 transforms of wave height measured at nonzero speed are computed using the 

 frequency-of-encounter time scale, where each wave length component corre- 

 sponding to a stationary frequency ^ is measured at the frequency 



= OJ + O)" 



(20) 



where v is the speed of the wave probes against the direction of wave travel. K 

 a Fourier transform of a transient is computed for one wave height measure- 

 ment, the companion wave measurement in the direction of wave travel will have 

 the same magnitude at each frequency but the phase will be shifted by e" J'^''^' "/^ 

 where w is the stationary frequency of the wave component concerned. For 

 waves traveling in the same direction as the wave probes, ambiguities exist, 

 and special techniques, which are beyond the scope of this report, must be em- 

 ployed. 



The preceding treatment of transient water waves does not follow a conven- 

 tional path in that spatial effects are suppressed and initial conditions or excit- 

 ing forces on the water are not considered. If an instrument measures unidirec- 

 tional wave height at some point for all time, then the time history at any other 

 point is readily estimated through transform techniques. Even though a wave- 

 maker may be generating the transient wave in the testing basin, the height- 

 measuring probe and the ship model considered the wave to be one that has been 

 traveling forever on an infinite surface. 



514 



