8  Lord  Kelvin  on 
the  unit  of  our  vertical  scale.  Approximate  mensuration, 
with  a  very  rough  estimate  of  the  area  beyond  the  range  of 
the  diagram,  continued  to  infinity  on  the  two  sides,  verifies 
this  conclusion. 
§76.  Fig.  28  is  designed  on  the  same  plan  as  fig.  27  but 
with  eleven  half-wave-lengths  as  the  distance  between  the 
two  forcives  instead  of  one  half- wave-length.     Like  fig.  27. 
it  is  symmetrical  on  the  two  sides  of   the    middle  of   the 
diagram;    but,  instead   of  being  waveless,  as  is  fig.   27,  it 
shows  four  and  a  half  waves,  all  very  approximately  sinusoidal, 
with  two  depressional  halves  of  waves  at  their  two  ends,  and 
elevations  coming   asymptotically  to    zero  beyond  the  two 
ends  of  the  diagram.     The  curve  represented  by  fig.  26  is 
very  accurately  the  right-hand  extreme  of  fig.  28  :  and  the 
same   figure,  turned  right  to  left,  is  the  left-hand  extreme 
of  fig.  28.      If  we  commence  with  the  water  wholly  at  rest, 
and  start  the  forcives  at  the  proper  speed,  with  force  gradually 
(or    somewhat   suddenly)   increasing   up   to    the  prescribed 
amount,   the  motion  produced  will  be  that  represented  by 
fig.  28,  with,  superimposed  upon  it,  a  disturbance   quickly 
disappearing     in    ever    lengthening    waves    of  diminishing 
amplitude,  travelling  away  in  both  directions  from  our  field. 
If  now,  with  the  regular  regime  represented  by  fig.  28,  we 
suddenly  cease  to  apply  the  forcives,   we  have  left  a   free 
procession  of  four  and  a  half  very  approximately  sinusoidal 
waves,  between  a  front  and  a  rear  deviating  from  sinusoidality 
as  shown  in  the  diagram.      From  the  instant  of  being  left 
free,  the  front  of  this  procession   and  its  rear  will  rapidly 
become  modified  ;  while  for  three  periods  the  central  part 
of  the  procession  will  have  travelled  three  wave-lengths,  with 
very  little  deviation  from  sinusoidality.     But,  after  four  or 
five  periods  from  the  instant  of  being  left  "free,  the  whole 
procession  will  hav  e  got  into  confusion.    After  twenty  or  thirty 
or  forty  periods,  the  water  will  be  sensibly  quiescent,  not  only 
through  the  space  where  the  procession  wras,   but  through 
a  considerable  part  of  the  space  ever  which  it  would  have 
travelled  if  its  front  and  rear  had  been  kept  guarded  by  the 
continued  action  of  the  two  travelling  forcives.     At  no  time 
after    the    cessation   of  the    forcives   can  we  reasonably    or 
conveniently  assign  a    "group    velocity"   to   the  group  or 
procession    of    waves   with   which    we    are    concerned.      A 
prevalent  idea  is,  I  believe,  that  such  a  group  of  deep  sea 
waves  could  be  regarded  as  travelling  with  half  the  "  wave- 
velocity  "  of  waves  of  the  length  given  in  the  original  group. 
In  §  30  above,  reasons  are  given  for  accepting  the  theory  of 
"  group  velocity  "  only  for  the  case  of  mutually  supporting 
