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XXII.  On  the  Production  of  Vibrations  by  Forces  of  Relatively 
Long  Duration,  with  Application  to  the  Theory  of  Col- 
lisions.    By  Lord  Bayleigh,  O.M.,  P.R.S.* 
THE  problem  of  the  collision  of  elastic  solid  bodies  bus 
been  treated  theoretically  in  two  distinct  cases.  The 
first  is  that  of  the  longitudinal  impact  of  elongated  bars, 
which  for  simplicity  may  be  supposed  to  be  of  the  same 
material  and  thickness.  Saint-Venantf  showed  that,  except 
when  the  lengths  are  equal,  a  considerable  fraction  of  the 
original  energy  takes  the  form  of  vibrations  in  the  longer 
bar,  so  that  the  translational  velocities  after  impact  are  less 
than  those  calculated  by  Newton  for  bodies  which  lie  called 
perfectly  elastic.  It  will  be  understood  that  in  Saint- 
Vena  nt's  theory  the  material  is  regarded  as  perfectly  elastic, 
and  that  the  total  mechanical  energy  is  conserved.  The 
duration  of  the  impact  is  equal  to  the  period  of  the  slowest 
vibration  of  the  longer  bar. 
The  experiments  of  YoigtJ,  undertaken  to  test  this  theory, 
have  led  to  the  conclusion  that  it  is  inapplicable  when  the 
bars  differ  markedly  in  length.  The  observations  agree  much 
more  nearly  with  the  Newtonian  law^in  which  all  the  energy 
remains  translational.  Further,  Hamburger  §  found  that  the 
duration  of  impact  was  much  greater  than  according  to 
theory,  though  it  diminished  somewhat  as  the  relative  velocity 
increased.  I  do  not  think  that  these  discrepancies  need  cause 
surprise  when  we  bear  in  mind  that  the  theory  presupposes  a 
condition  of  affairs  impossible  to  realise  in  practice.  Thus  it 
is  assumed  that  the  pressure  during  collision  is  uniform  over 
the  whole  of  the  contiguous  faces.  But,  however  accuratelv 
the  faces  may  be  prepared,  the  pressure,  at  any  rate  in  its 
earlier  and  later  stages,  must  certainly  be  local  and  be  con- 
nected with  the  approach  by  a  law  altogether  different  from 
that  assumed  in  the  calculation.  Since  the  region  of  first 
contact  would  yield  with  relative  ease,  we  may  expect  a  pro- 
longation of  the  impact,  and  in  consequence,  as  we  shall  see 
more  in  detail  presently,  a  diminished  development  of  vibra- 
tions. Possibly  with  higher  velocities  and  longer  bars  a 
nearer  approach  might  be  attained  to  the  theoretical  con- 
ditions. 
*  Communicated  by  the  Author. 
t  Liouville's  Journal,  xii.  (1867).     See  also  Love's  f  Treatise  on  the 
Theory  of  Elasticity,*  vol.  ii.  p.  1 : > T  (1893). 
X  Wied.  Ann.  xix.J1883). 
§  AVied.  Ami.  xxviii.  (1886). 
