NODAL DISTANCES IN ACOUSTICS 
G. W. STEWART 
Our elementary texts discuss the nodal planes produced by two 
plane waves of the same amplitude and frequency travelling in 
opposite directions and leave us with the impression that the dis- 
tance between nodal surfaces is usually one-half of the wave 
length. Yet this is far from the truth. Nodal surfaces having 
such a distance apart exist only for plane waves, the most com- 
mon case being the standing waves in pipes. The purpose of this 
note is to call attention to a case wherein the distance from a nodal 
to a loop surface can be made very short, almost as short as you 
please, and thus to contribute to a greater clearness of under- 
standing. 
Consider the case of a conical horn used as a receiver and 
closed at the vertex. The horn is approximately one-half a wave 
length of the resonating frequency, yet it has a node at the vertex 
and a loop at the open end. Now cut a tip off the vertex. There 
are now two loops, one at each end, and the node very close to 
the one at the incomplete vertex. Here then, is a nodal surface 
inside a conical horn which can be brought very close to a loop 
surface, indeed within a certainly very small fraction of a wave 
length. Moreover there is in the one instrument a nodal surface 
at two widely different distances from two loop surfaces, show- 
ing the possibility of having a distance from node to loop either 
less or greater than one-fourth of a wave length. A distance be- 
tween nodal planes not one-half wave length can be secured by 
using an overtone. 
To put the general statement a little more bluntly, a nodal 
surface is produced by opposition of phase with equality of ampli- 
tude, and traversing a distance is only one of the ways of pro- 
ducing the appropriate changes in phase. No limitation can be 
placed on the forms of apparatus which will give nodal surfaces 
separated at distances other than half a wave length. 
Moreover, it may be added that one cannot expect that the 
difference in phase between two points in any kind of enclosure, 
room or otherwise, is determined by the wave length only. 
State University oe Iowa. 
