154 Long Waves in Canals and Standing Waves in Closed Basins 



Also the travel time, where a distinction should be made between that of the 

 wave crest and that of the high tide, are different from those obtained from 

 Green's hypothesis, as can be seen from Table 19 for the travel times of high 

 tide. The difference as against those computed from the Green-Du Boys 

 equation is considerable, especially for the ones computed from the mean 

 depth h. 



Thorade's papers are important for tidal waves coming from the deep 

 ocean on to the continental shelf. Unfortunately, there are hardly any ob- 

 servations available to test the theoretical results. Tidal waves passing over 

 irregularities of the ocean bottom, as for instance a wavy bottom, shallow 

 places and the like, will show variations in their wave length and amplitude. 

 It would certainly be well worth while to make systematical observations 

 on such variations. 



2. Standing Waves in Closed Basins 



(a) Constant Rectangular Section 



The equations of motion as formulated in (VI. 8) also has a solution in 

 the form of a standing wave, for the reason that the superposition of two 

 progressive waves with corresponding phase difference always causes a stand- 

 ing wave. 



This standing wave can also result from the superposition of an incoming 

 wave and its reflection from a vertical wall. If the reflection occurs at x = 

 at the vertical wall, and if t\ x is the incoming, rj 2 the reflected wave then 



rj! = lacos(at — xx) , tj 2 — %acos(ot + xx) , 



V = ?h + *?2 = acosxxcosot . (VI. 33) 



Equation (VI. 8) required that c = ajx = j (gh), or that the period of 

 the oscillation T = ?./\ gh or 



T = -^j— , (VI. 34) 



n) (gh) 



when n is a positive integer greater than zero and / the length of the rectangular 

 basin (Merian's formula). It is readily seen that n is the number of nodes 

 of the standing wave. 



For (n = 1) we obtain the longest period of the free oscillating water-mass 

 T = 2l/y/(gh); in the centre of the basin (x = \l) there is one nodal point 

 (,; = 0) with maximum values of f, which also follows directly from solv- 

 ing for 



£ = —7— sin xx cos at and u = -i-sinxxsinot . (VI. 35) 



tiK n 



For n = 2, 3, etc., we have two, three, etc., node oscillations. For the nth 

 nodal oscillation, the node is situated at 



