THE 
LONDON,  EDINBURGH,  and  DUBLIN 
PHILOSOPHICAL     MAGAZINE 
AND 
JOURNAL    OF    SCIENCE. 
[SIXTH    SERIES.] 
FEBRUARY   1906. 
XVII.  On  the  Diffraction  of  Short  Waves  by  a  Rigid  Sphere. 
By  J.W.  Nicholson,  B.  A.,M.Sc,  Scholar  of  Trinity  College, 
Cambridge* ' . 
THE  effect  of  a  rigid  sphere  upon  incident  plane  sound- 
waves has  been  examined  by  Lord  Rayleigh  f,  who  has 
obtained  a  complete  solution  o£  the  case  in  which  the  waves 
have  a  length  great  in  comparison  with  the  dimensions  of 
the  sphere.  In  the  more  general  case,  Lord  Rayleigh  has 
expressed  the  disturbed  motion  in  the  form  of  an  infinite 
series,  whose  terms  consist  of  products  of  surface  zonal 
harmonics  and  functions  derived  from  Bessel  functions. 
The  summation  of  this  series  cannot  be  effected ;  but,  as 
Lord  Rayleigh  has  shown,  when  the  waves  are  very  long,  it 
practically  reduces  to  its  first  two  terms.  In  this  paper,  a 
close  approximation  to  the  diffracted  waves  will  be  obtained 
when  the  waves  are  very  short  in  comparison  with  the  radius 
of  the  spherical  obstacle.  The  method  adopted,  before  the  intro- 
duction of  the  approximations,  is  that  used  by  Lord  Rayleigh. 
The  final  result  appears  in  the  form  of  a  definite  integral 
which  may  be  evaluated  to  the  same  order  of  approximation. 
The  motion  external  to  the  sphere,  whether  due  to  the 
passage  of  sound,  or  to  the  presence  of  small  waves  in  a 
frictionless  fluid,  is  supposed  to  possess  a  velocity  potential. 
*  Communicated  by  the  Author. 
t  '  Theory  of  Sound,'  1896,  §  384  ;  Proc.  Loud.  Math.  Soc.  1872. 
Phil  Mag.  S.  6,  Yol.  11.  No.  62.  Feb.  1906.  0 
