462 F. W. Boggs and N. Tokita 
It is apparent that e, and e, both pass through the origin when B is equal to zero. It 
follows that they will both pass through the origin when C vanishes. Furthermore, when @ 
is small e, will be negligible and e, will be a function of 8 only. These two characteris- 
tics will have some important practical consequences which we will consider in the next 
section of this paper. It can also be seen by substituting that similar conditions will be ob- 
tained when the propagation constant of waves over the surface is proportional to the veloc- 
ity of flow. 
5. CURVE OF NEUTRAL STABILITY FOR A FLEXIBLE WALL 
If Eq. (3.16) is rewritten in terms of e, and e,, we obtain 
e 
1 
F(t) - G(a,C) eg Mideg YGCa CT ge ee 
Sung OR; eae ay Sele Beep wae e al y-c4)) 
a,H(o) a Corr) ae 
2 a 
The left-hand side depends on the Reynolds number only through its dependence on ¢, 
whereas the right-hand side depends on the Reynolds number through the frequency 8 and 
wave number 0, which depend on both the velocity and the Reynolds number. The curve of 
neutral stability can be obtained by identifying the real and imaginary parts of both sides of 
Eq. (5.1) and eliminating either @ or C. For each value of U,, there will be a different curve 
of neutral stability and a corresponding critical Reynolds number which, if it exists, will 
depend on U,,. The procedure outlined above represents a formidable amount of work which, | 
even with the aid of calculators, would be forbidding. Can we find a procedure for obtaining 
bounds for the Reynolds number which will simplify the general discussion? 
For a given velocity the right hand-side of Eq. (5.1) will be a function of two variables 
only, & and 8. Furthermore, as we saw in Section 4 all the curves will pass through the 
origin and be tangential to each other when © and B are small. 
If we hold constant we can plot the right-hand side of Eq. (5.1) as a function of C 
treating R as a parameter. For a given value of © the right-hand side of Eq. (5.1) will bea 
single curve passing through the origin while the left-hand side will give a family of curves, 
one for each Reynolds number. The intersection of the curve corresponding to the right-hand 
side of (5.1) with each one of the family of curves for the left-hand side will correspond to a 
value of @ and of R on the curve of neutral stability. If the family of curves representing the 
left has an envelope through the elimination of R, the value of the frequency corresponding 
to the intersection of this envelops with the right plotted for % and U,, constant will be an 
extreme value for C. The corresponding value of the frequency for each value of & will form 
a boundary separating areas of stability and instability. The value of the frequency along 
this boundary will be a function of the velocity and of @, and its value will be given by an 
equation of the form 
B60 koi 22 00, (5.2) 
