Theory of Long Waves. 257 



where T is very large. In the case of rectangular section 

 this reduces to 



£T = f 2>/^<fc-UT (IT) 



In a subsequent section a discussion of some of the tidal 

 phenomena of rivers and estuaries will afford a somewhat 

 detailed illustration of the character of the motion which is 

 indicated by the equations we have obtained in this section, 

 but meanwhile a brief sketch of its general nature is desirable. 



From (8) we see that any point on the wave-surface whose 

 ordinate is z may be regarded as advancing without change of 

 elevation with the velocity ( \^ gyjr (z) — k) ; thus in general 

 the waves will change in form as they advance, owing to the 

 difference in the velocities of higher and lower parts, becoming 

 steeper or less steep in front according as ^fr(z) increases or 

 decreases as z increases, but they will be propagated with a 

 constant velocity and without change of form if ^r(z) is 

 constant, a case to be specially discussed in the next section. 

 From (8) we also see that though the form may change, the 

 height of the wave-crests and troughs does not alter. From 

 (9) we see that at any point the component of the velocity 

 parallel to the axis of x depends only on the surface-height at 

 the point ; the vertical component of the velocity will of course 

 be dz/dt at the surface, and, since it vanishes at the bottom, 

 it will for intervening points be proportional to their height 

 above the bottom. 



An important consequence of the continual change in the 

 form of the advancing wave must here be noted. As the 

 velocity of advance of each point of the wave-surface depends 

 only on its ordinate and not at all on the time, it follows that, 

 the case where yfr(z) is a constant alone being excepted, the 

 inclination at some points on the surface, however small 

 initially, will ultimately become so great that the conditions- 

 under which our equations have been obtained no longer hold 

 there. It follows, therefore, that we must be careful only to 

 apply our results throughout those regions and during those 

 times in which the condition of small surface-inclination holds, 

 and we must infer nothing as to the further motion without 

 special investigation. It seems to be generally held that the 

 distortion of the surface-form will go on till discontinuity 

 results ; but such an inference can hardly rest on secure 

 grounds so long as no equations are known which will 

 represent even approximately the motion when the steepness 

 of the surface becomes considerable ; on the contrary, from 

 the known fact of the permanent propagation of certain forms 



