Chta-Shun Yth 
the maximum or average depth. The solution for edge waves (Stokes 
1839, or Lamb 1932, p. 447) is exact if the region occupied by the 
water is semi-infinite, bounded only by the free surface and a plane 
of constant slope serving as the only solid boundary. If the channel is 
finite in both depth and width, Stokes'solution is nevertheless valid 
for each shore if the wave length (in the longitudinal direction) is very 
short, since the variability of the depth has then an important effect 
only near the shore lines. 
Aside from these three categories, and interrelating them, 
are the exact solutions for water waves in a symmetrically placed 
triangular channel of vertex angle a/2 (Kelland 1839, or Lamb 1932, 
p. 447-449) or of vertex angle 22/3 (MacDonald 1894, or Lamb 
1932, p. 449-450). These exact solutions are useful because they pro- 
vide a check for any approximate theory. 
In this paper internal waves in channels of variable depth are 
studied. The differential system governing the flow of a system of 
superposed layers of homogeneous fluids is formulated by first con- 
sidering a single layer. Then for the layered system it is proved by 
the use of comparison theorems that the frequency 97 of the waves 
increases but the wave velocity c decreases as the wave number k 
increases. Then the differential system governing the flow of a con- 
tinuously stratified fluid is derived, and the increase of o and de- 
crease of c as k increases are again proved in general, 
After giving a few solutions in closed form (under the res- 
triction of the Boussinesq approximation), a general method of solu- 
tion for wave motion in stratified fluids is given, In the form given 
the method is for application to continuously stratified fluids, but it 
can be adopted to deal with homogeneous fluids, and the manner of 
adoption is briefly indicated in the last paragraph of Section 7. 
Finally we study long waves in some detail, and both conti- 
nuously stratified fluids and layered systems are considered. A few 
examples are given, and the connection of the theory to the classical 
shallow-water theory for long waves ina single fluid is shown. 
Il. THE DIFFERENTIAL SYSTEM FOR THE CASE OF CONSTANT 
DENSITY 
If viscous effects are neglected and the motion is supposed 
to have started from rest, and if the density of water is constant, the 
motion is irrotational and a velocity potential $ exists, the gradient 
of which is the velocity vector. We shall use the Cartesian coordinates 
(x, y, z), with z measured longitudinally, y measured vertically, 
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