632 MR E. M. WEDDERBURN ON THE TEMPERATURE SEICHE. 



§ 27. This equation is of exactly the same form as the equation arrived at by 

 Chrystal where 2(y) is substituted for <r(v), and all the results arrived at by him 

 Avith regard to ordinary seiches may be extended to temperature seiches. We can 

 draw a temperature normal curve for the lake, knowing the depth at which the abrupt 

 change of temperature is supposed to take place. It is the curve whose ordinate and 



abscissa are b(x) -i—^—+ TfT~\C anc ^ Jd x H x )^ it being remembered that b(x) refers to 



the breadth of the surface of separation. It will, in general, not greatly affect the result 



if we take as orclinates b(x) < + y, for in nature p and p' are always nearly 



unity. It is only the difference between them which is important. 



The period of the temperature seiche of small amplitude (following Chrystal's 

 reasoning) therefore depends on the differential equation 



c7 2 P n 2 



— + _ P = ... . (14) 



where w = 2P sin n(£ — t), P is a function of v alone and t is constant. The values of 

 n admissible depend on the circumstances of each case. 



§ 28. It will be seen that whenever A(x) becomes zero 2(<y) also becomes zero, 

 and that therefore the theory takes no account of the water in the upper liquid at the 

 ends of the lake — beyond the point where the surface of separation ends. The tempera- 

 ture normal curve will be exactly the same whether the basin is truncated at the ends 

 of the surface of separation or whether it continues to slope gradually. It is probable 

 that a " mouth " or " end " correction should be applied in cases where the lake is not 

 truncated, and the result of some experiments by Mr Williams and the author bearing 

 on this point are given in Part IV. of this memoir, but it appears that such a correction 

 is very small. 



§ 29. Before passing on to the computation of the period of the Madiisee, it will 

 be well to review some of the assumptions which have been made. (1) It was 

 assumed that the amplitude of the seiche was small. It has been seen that the 

 amplitude amounts to one-third of the depth, and this cannot be counted small. 

 (2) It was assumed that there was no transverse motion. This assumption is more 

 accurate in the case of the temperature seiche than of the ordinary seiche. Gradually 

 shelving shores must, in the case of the ordinary seiche, cause transverse motion, but in 

 the temperature seiche the vertical motion at the free surface is nil, and therefore the 

 assumption is probably fairly accurate. Any gradual shelving at the surface of 

 separation would be a different thing, but this must be a very uncommon occurrence in 

 nature. (3) It was assumed that there was an abrupt discontinuity in temperature. 

 Frequently, as has been seen, the discontinuity is very abrupt — at least at one end of 

 the lake. But oscillations do occur when the discontinuity is anything but abrupt, and 

 when there is a considerable temperature gradient in the surface layers. The motion 

 investigated above depends on the difference of level at the surface of separation 



