Homework H5 Solution - University Of...

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Homework H5 Solution 1. Turn in: A flute tone is played for a duration of 2 seconds at a frequency of =440 Hz (middle A on the keyboard). Note that =2 . a. As a percentage of the frequency , what is the uncertainty in the pitch being played? Now someone hits a drum for 1 millisecond, making a sound centered at =440 Hz (a ‘tuned drum like they use in orchestras). b. As a percentage of the frequency , now what is the uncertainty in the pitch being played? c. Based on your finding above, which of these two instruments would be more useful for playing a melody? d. If you sped up your playing and you could play a note on the flute every 1 millisecond, is it still possible in principle to tell the difference between an A and A# (440 Hz and 466.1 Hz)? Can one play melody on a flute that fast as a matter of basic principle, no matter how nimble? Solution: a. As it is played for 2 seconds, Δ = 2. You learnt that the Fourier conjugate variables and t are related as Δ Δ = 1 4 Therefore, Δ = ! !! = 0.03978 . Δ 100 % = 9.04 10 !! % b. Now, Δ = 10 !! . So, Δ = ! !!!" !! = 79.57 . !! ! 100 % = 18.08 % c. Thus, for playing a melody, the flute would be more useful if the time duration is high like in part a, since the frequency, or the note played would be more precise. d. If you speed up playing the flute so that Δ = 10 !! , the uncertainty in the frequency would again be 79.57 Hz. So in principle it would be impossible to distinguish the notes A and A#, which have a difference of 26.1Hz (< 79.57 Hz !)

Transcript of Homework H5 Solution - University Of...

Page 1: Homework H5 Solution - University Of Illinois"for"playing"a"melody,"the"flute"would"be"more"useful"if"the"time"duration"is"high" like"in"part"a,"since"the"frequency,"or"the"note"played"would"be"more"precise."

Homework H5 Solution 1.  Turn  in:  A  flute  tone  is  played  for  a  duration  of  2  seconds  at  a  frequency  of  𝜈=440  Hz  (middle  A  on  the  keyboard).    Note  that  𝜔=2  𝜋𝜈.  a.  As  a  percentage  of  the  frequency  𝜈,  what  is  the  uncertainty  ∆𝜈  in  the  pitch  being  played?      Now  someone  hits  a  drum  for  1  millisecond,  making  a  sound  centered  at  𝜈  =440  Hz  (a  ‘tuned  drum  like  they  use  in  orchestras).  b.  As  a  percentage  of  the  frequency  𝜈,  now  what  is  the  uncertainty  ∆𝜈  in  the  pitch  being  played?    c.  Based  on  your  finding  above,  which  of  these  two  instruments  would  be  more  useful  for  playing  a  melody?    d.  If  you  sped  up  your  playing  and  you  could  play  a  note  on  the  flute  every  1  millisecond,  is  it  still  possible  in  principle  to  tell  the  difference  between  an  A  and  A#  (440  Hz  and  466.1  Hz)?  Can  one  play  melody  on  a  flute  that  fast  as  a  matter  of  basic  principle,  no  matter  how  nimble?    Solution:    a. As  it  is  played  for  2  seconds,  Δ𝑡 = 2𝑠.  You  learnt  that  the  Fourier  conjugate  variables  𝜈  

and  t  are  related  as        

Δ𝜈  Δ𝑡 =14𝜋  

Therefore,  Δ𝜈 =   !!!= 0.03978  𝐻𝑧.  

∴Δ𝜈𝜈∗ 100  % = 9.04 ∗  10!!  %  

   

b. Now,  Δ𝑡 = 10!!𝑠.  So,  Δ𝜈 =   !!!∗!"!!

= 79.57  𝐻𝑧.                                                                              ∴ !!

!∗ 100  % = 18.08  %  

 c. Thus,  for  playing  a  melody,  the  flute  would  be  more  useful  if  the  time  duration  is  high  

like  in  part  a,  since  the  frequency,  or  the  note  played  would  be  more  precise.    d. If  you  speed  up  playing  the  flute  so  that  Δ𝑡 =  10!!  𝑠,  the  uncertainty  in  the  

frequency  would  again  be  79.57  Hz.  So  in  principle  it  would  be  impossible  to  distinguish  the  notes  A  and  A#,  which  have  a  difference  of  26.1Hz  (<  79.57  Hz  !)  

   

Page 2: Homework H5 Solution - University Of Illinois"for"playing"a"melody,"the"flute"would"be"more"useful"if"the"time"duration"is"high" like"in"part"a,"since"the"frequency,"or"the"note"played"would"be"more"precise."

 

2.  The  inverse  Fourier  transform  is  given  by  𝜓(t)  =  (1/2𝜋)  ∫  d𝜔  𝜓  (𝜔)  exp[-­‐i𝜔t].  It’s  just  like   the  Fourier   transform,  with  a  minus   sign   in   front  of   the   “i”.  The  Fourier   transform  gets  you  from  𝜓  (t)    to  𝜓  (𝜔),  and  the  inverse  transform  gets  you  from  𝜓  (𝜔)  back  to  𝜓  (t)  .    Comparing  the  two,  

𝜓  (𝜔)  =  1  .              ∫  dt  𝜓  (t)  exp[+i𝜔t]  

and  

𝜓   (t)   =   (1/2𝜋)   ∫   d𝜔  𝜓   (𝜔)   exp[-­‐i𝜔t].     It’s   shown   in   the   “T1   Reading”   handout   on   the  course  schedule.    

Use  this  knowledge  to  prove,  using  integration  by  parts  of  the  inverse  Fourier  transform  like  what  we  did  in  lecture  with  the  Fourier  transform,  that  ∂/∂𝜔  →  +it.  

 

Solution:  

Begin  similarly  as  how  you  learnt  in  class  about  proving  !!"  ⟶  −𝑖𝜔.  

                       ℱ𝑇!! !" !

!"=   !

!!𝑑𝜔 !" !

!"  𝑒!!"#!

!!                [  Integrating  by  parts]                                                                                                                                  =   !

!!𝜓 𝜔 𝑒!!"#|!!! +   𝑑𝜔  𝜓 𝜔  𝑖𝑡  𝑒!!"#!

!!                                                                    = !"

!!𝑑𝜔  𝜓 𝜔!

!! 𝑒!!"#                  [lim!⟶±! 𝜓 𝜔 = 0  ]                                                                    = 𝑖𝑡  ℱ𝑇!![𝜓 𝜔 ]                    Thus,      

!!"

→  +𝑖𝑡