Long Waves in a Rectangular Trough. 



159 



'the catch was suddenly released, the end dropped, the surface 



of the water was represented for an instant by f = cf x — — ) 



and then began to oscillate about its equilibrium position. A 

 scale was held from above over the surface of the water 

 parallel to the sides of the trough at its middle. Any motes 

 that happened to be in the water were brought to the surface 

 there ; the turning-points of one of them were read on the 

 scale as long as the motion lasted, and the rate of damping 

 'thus obtained. 



The following table gives some results : — 



h. 



Observed 

 period. 



T . 



Difference. 



Difference 

 kT 2 



calculated. 



k calcu- 

 lated. 



K 



observed. . 



cms. 

 10 



sees. 

 10-1 



sees. 

 9-75 



sees. 

 •35 



•45 



•028 



•I 

 •048 



20 



7-12 



6-89 



•23 



•23 



•017 



•032 



30 



5-80 



5-62 



•18 



•14 



•012 



•028 



50 



4-49 



4-3.5 



•14 



•09 



•0084 



•023 



7-0 



3-G8 



3-G8 



•00 



•02 



•0066 



•022 



100 



312 



3 08 



•04 



•0! 



•0050 



•011 



The first column gives the depth of the water and the 

 second the observed period. The third gives T , the 

 theoretical value obtained for the period when viscosity was 

 neglected ; the fourth gives the difference of the second and 

 third, and the fifth gives the effect of the viscosity of the 

 water on the period calculated according to (18). The 

 seventh column gives the observed values of k, and the sixth 

 gives values of k calculated according to (16). If we use 

 (14) slightly better values are obtained. 



The actual value of k is thus much greater than its theoretical 

 value. The ditference is due neither to neglecting the other 

 modes, which die away more rapidly, nor to the vibrations 

 not being small enough, nor to the " residual motion M of the 

 water. The same values were obtained for k when the water 

 had a slight irregular motion in addition to the periodic one. 

 The approximations made in deriving the formula are 

 perfectly justifiable ; for, to take a concrete example, for 

 7i~5 ems. in equation (10) cr' 2 — 2'l and van 2 = 0*00033, and 

 in equation (11) the modulus of Mi is about GO. If we take 



