A72 F. W. Boggs and N. Tokita 
simple and, as a matter of fact, is simpler than the classical problem. The reason for this I 
will point out in a minute, I will first mention that other investigations in this area have 
been published and are on the way to being published. There is a forthcoming paper in the 
Journal of Fluid Mechanics by Dr. Bannerman of Cambridge and I think that his paper is by 
far the most comprehensive and has very interesting physical discussion on it. I said that 
the mathematical problem is really quite simple, however, the difficulties are in the physical 
interpretation of the results which are, I must say, extremely difficult. First of all, the rea- 
son why the mathematical problem is so simple is that you can state this problem as a direct 
problem instead of as an indirect problem. In this particular case you can, instead of asking, 
what is the stability for a particular layer, ask what kind of wall admittance should one 
choose to maintain neutrally stable oscillations for waves of a particular wavelength and 
wave number. This is quite straightforward and as a matter of fact if you straighten out all 
these initial difficulties the results come out to be extremely simple. The results show that 
in order to take care of the instability completely you have to supply energy into the bound- 
ary layer and consequently a passive wall layer will never do the trick. However, it turns 
out that a pure compliance of the surface, a pure spring compliance, that is, a very flexible 
wall, will push the minimum Reynolds number up quite a bit. You can push it up, as Dr. 
Boggs pointed out, as far as possible, but there is one particular thing here I must say I 
don’t quite understand completely yet. It looks as if you say to the boundary layer, I take 
5 percent of the energy out of you, and then the boundary layer says, all right, I give you 
10 percent. The more energy you take out of it the more energy the boundary layer supplies, 
and this is a very peculiar thing, which I think has to be understood more completely before 
you can really design these layers with good knowledge. 
F. S. Burt (Admiralty Research Laboratory) 
At ARL we also discussed this problem of the use of damping in the skin to stabilize 
the laminar boundary layer and came to a similar conclusion, that in the normal instability 
case one might have to put energy into the layer rather than take it out. The only other 
thing I would like to ask Mr. Boggs is if he can give us some figures for the critical Reynolds 
number, both maximum and minimum that he was referring to in his figure, preferably in terms 
of length to the transition point, which is the more normal Reynolds number used for transition. 
F. W. Boggs 
To the question of the energy, I would like to point out one thing. We have found, as a 
practical matter, that when the mechanical loss in the rubber that we use in the coating be- 
comes large, the coating becomes substantially ineffective, maybe even detrimental. As to 
how energy might be transferred I wonder whether it wouldn’t be through the propagation; our 
feeling is, and my impression is also, from an examination of our data that it is only when 
you have propagation that you will have stabilization. If you have propagation of the 
Schlichting wave which differs from the propagation of the wave in the coating (let’s sup- 
pose, just to make things simple, that there is a factor of 2), then if they are in phase at 
one spot, half a wavelength ahead they will be out of phase. So whereas in one case you 
would be transferring the energy into the boundary, a wavelength or half a wavelength away 
the energy transfer would be taking place in the other direction. I would think that this is 
the type of mechanism that might explain it. As to Mr. Lang’s comments, we have discussed 
this and we have seen reports on porpoises that travel at very high speeds, but I don’t know, 
every time you talk to somebody he gives you something higher. The Scientific American, 
where we all get our information now in the United States, had an article on this subject and 
