140
THE EFFECT OF JUNCTIONS ON THE
The wave in AB is to be regarded as propagated onwards round corner at A rather than as reflected. As was to be anticipated, the reflec wave F' is smaller, the smaller is AB. It will be understood that validity of these results depends upon the assumption that the region ro-A through which the waves are irregular has dimensions which are neglig in comparison with X.
An even simpler example is sketched in fig. 3, where for the present
Fig. 3.
various lines represent planes or cylindrical surfaces perpendicular to paper. One bounding plane G is unbroken. The other boundary cons mainly of two planes with a transition at AB, which, however, may b any form so long as it is effected within a distance much less than X. ^ a notation similar to that used before, fOA may denote the incident posi wave and F the reflected wave, while that propagated onwards in CB is "We obtain in like manner
f cs —
GB + GAJ w'
AB
/'c
OB + GA
When AB vanishes we have, of course, f'CB -f'CA, F' = 0. A little 1 we shall consider the problem of fig. 3 when the various surfaces ar revolution round the axis of z.
'Leaving the two-dimensional examples, we find that the same gen method is applicable, always under the condition that the region occu by irregular waves has dimensions which are small in comparison wit! Within this region a simplified form of the general equations avails, thus the difficulty is turned.
An increase in X means a decrease in p. When this goes far eno it justifies the omission of dfdt in equations (1), (2), (3), (4). Thus P, ( become the derivatives of a simple potential, function <£, which itself satited by fGA and that therein reflected by F, while the waves propagated along CB, AB be denoted by/^,/^, we have