296 Frederick Guthrie on Stationary Liquid Waves. 



planes are at right -gig. 2. 



angles to the plane 



of the lath (fig. 2). A ^^^ 



But the nodes are 



now at nearly \l 



from the ends, and ^^^zzx^^x-^'" 



the amplitude in 



the middle is approximately twice that at the ends. 



Imagine a thin circular board to consist entirely of such 

 sectorial laths. They would, if strung on a circular wire passing- 

 through their centres of gravity, form a rigid system ; for motion 

 around the wire would cause gaping between the apices and 

 crushing between the bases of the sectors. Nevertheless such 

 motion is possible if the disk is elastic, as is shown by the for- 

 mation of a nodal line around a circular plate at a distance of 

 one sixth of the diameter from the circumference when the disk 

 is bowed through a hole in the middle. The motion of such a 

 disk is analogous in many respects to that of water in a circular 

 trough. We may compare both cases with systems of isosceles 

 triangles oscillating about lines passing through their centres of 

 gravity. With the disk we have momentum and inertia con- 

 trolled by elasticity ; in the wave they are controlled by gravity. 

 If the surfaces of the waves were conical, the following simple 

 comparison might be drawn for an elementary displacement. 

 Let fig. 3, D E, be the Fio . 3 



axis of the cylindrical 

 trough. Let D G, 

 D B be the limiting- 

 radii of the displaced 

 cone -sector D G B. 

 When D sinks to E, 

 GB will be at FH. 

 If the sector turns on the line C A parallel to its base and one 

 third BD from it, then DE = 2FG. When D sinks to E, all 

 the water in the tetrahedron (A C J) E) leaves it, and as much 

 water has to enter the wedge FGBAHC. These volumes 

 must be equal. It is easy to show that they are so. This is in 

 accordance with the general law that when two heavy planes in 

 one plane turn together round any axis passing through their 

 common centre of gravity, they sweep out solids of equal 

 volume. 



As the water is constrained neither to separate in the middle 

 nor from the walls, the sectors change from conical to circular 

 sectors and back again. And with conical waves the actual 

 volume of water concerned in the motion is less than that traced 

 out by the rotating sector; but this defect is the same in volume 

 with the wedge as with the tetrahedron. 



