170 
MESSRS. A. E. H. LOVE AND F. B. PIDDUCK 
“ first middle wave.” When the advancing and receding fronts of the first middle 
wave reach the pistons the original progressive waves are obliterated, reflexions take 
place at the pistons, and new compound waves are generated at the pistons and encroach 
upon the first middle wave. These waves will be described as the “ first reflected 
wave from the left ” (or “ from the right ” as the case may be). The reflected waves 
meet at or near the middle section, from which there then sets out a new compound 
wave called the “ second middle wave.” This wave again has two fronts, one advancing 
and encroaching upon the first reflected wave from the right, and the other receding 
and encroaching upon the first reflected wave from the left. The two fronts eventually 
reach the pistons, and then the second middle wave will have obliterated the first 
reflected waves, and will itself be reflected so as to give rise to new compound waves 
setting out from the pistons. These will be called the “ second reflected wave from 
the left ” (or “ from the right ” as the case may be). The motion goes on in this way 
until a piston reaches an end of the tube if the tube is of finite length. 
In what follows Articles 2-9 are devoted to giving such an account of the theory 
of plane waves of finite amplitude as seems to be necessary for the discussion of the 
problem. Although so much has been written about the subject, it appears to be 
impossible to find what is wanted in a suitable form. Articles 10, 11 contain the formulas 
relating to the two progressive waves. These are already known from the work of 
Gossot and Liouville, but it seemed to be desirable, for the sake of completeness, 
to obtain them anew. Articles 12-17 deal with the first middle wave. Sufficient 
indications of the method of determining this wave have been given by the same writers 
for the case of equal pistons. The really formidable difficulties of the problem begin 
to present themselves when an attenqit is made to discuss the waves reflected from 
the moving pistons. In Articles 18-25 an approximate method of solution is found. 
It seems to be capable of giving results for the first reflected waves correct to any desired 
order of accuracy. In Articles 26-31 the second middle wave is determined. However 
far the approximation to the first reflected waves is carried, the second middle wave 
answering to them can be found by the method here given. Articles 32-40 are devoted 
to the determination of the second reflected waves. The method used for the first 
reflected waves does not give a sufficiently close approximation, and a new method is 
applied. Numerical calculation of a particular example showed that all information 
that can be of practical importance may be obtained from a solution which does not 
go beyond the determination of these waves. The results of this calculation belong 
properly to the second part of the paper. 
Theory of Plane Waves of Finite Amplitude. 
2 . General Equations .—The motion is supposed to take place in an unlimited straight 
tube of uniform cross-section «. Let x be a co-ordinate measured along the tube, 
and specifying the position at time t of a plane of particles, which, when t — 0, is in 
