138 A. W. Duff — Secondary Undulations 



Another possible source of periodic fluctuations under certain 



circumstances. 



Can any other explanation be found for the definite period 

 that characterizes the regular undulations at each place ? I can 

 think of but one other. 



Most people who have, from the shore, watched the progress 

 of a ship have noticed the regular series of waves thrown off 

 from the bow. If the ship be at anchor in a rapid current, a 

 similar series of waves are produced, reaching any point on the 

 shore at equal intervals. In general it may be stated that " a 

 line of pressure athwart a stream flowing with velocity c pro- 

 duces a disturbance consisting of a train of waves of length 

 27r<? 2 /<7 lying on the down-stream side."* The effect of inequali- 

 ties in the bed would be similar. Would an island, shoal, bank 

 or headland, past which a current is setting, cause a similar 

 series of waves of a length and period commensurate with the 

 size of the obstruction ? For example, St. Paul's Island divides 

 two opposite currents between the Atlantic and the Gulf of 

 St. Lawrence ; at Yarmouth the direction of the tidal current is 

 reversed with the tide and the Lurcher shoal is a prominent 

 obstruction. 



I do not attempt to consider this suggestion in detail. It 

 seems to me a much less likely explanation than the theory of 

 stationary undulations or seiches ; but no adequate attempt, in 

 a brief space, to test its applicability would be possible, even 

 if data as regards currents, etc., were to be had.f 



/Summary of preceding. 

 A. Facts. 



1. Secondary undulations occur at most of the places where 

 observations have been made, the undulations at any place 

 being sometimes irregular but at other times of a regular 

 periodic nature. 



2. When the undulations are regular, the period of undula- 

 tion has a distinct and characteristic value for each place, 

 varying, from place to place, from less than a minute to over 

 an hour. 



3. At some places at least two systems of regular periodic 

 undulations of different periods are found ; they usually occur 



* Lamb's Hydrodynamics, § 228, where illustrations of such waves from experi- 

 mental papers by W. and R. K. Froude and theoretical papers by Lord Kelvin and 

 Lord Rayleigh are given. 



•f Moreover, in making this somewhat vague suggestion, I am aware that the 

 formula quoted above for the length of such waves is independent of the size of 

 the obstruction : but this does not preclude the possibility that hydrodynamical 

 theory, applied to the case of very large obstructions, might be able to account 

 for all the phenomena. 



