A string is made to vibrate at its third harmonic. The diagram shows two points P and Q at a particular instant in time. Which of the following compares the period of vibration of P and Q?(The string is closed at both ends; P is just before the first maximum antinode and Q is at the next minimum antinode).A. Period of P and Q is the sameB. Period of P and Q is differentWhy is it A not B?

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Quinten Cervantes · Accepted Answer

f  (  s  )  =          2        F    1        (    2    s    −    1    ,    s    −          1      2        ;    s    ;    −    1    )    =          2        F    1        (    s    −          1      2        ,    2    s    −    1    ;    s    ;    −    1    )  Even for small values of s, we have a nice logarithmic behaviour  log  ⁡  [  f  (  s  )  ]  ∼  a  −  b    sUsing the data computed for   1  ≤  n  ≤  100, we have, with       R    2    =  0.9999982,                          Estimate                    Standard Error                    Confidence Interval                            a                    1.640373                    0.021344                    {        1.598011        ,        1.682736        }                            b                    1.379308                    0.000367                    {        1.380037        ,        1.378580        }            and, as you noticed, b is quite close to   log  ⁡  (  4  )  =  1.38629Pushing the numerical analysis much further, b is closer and closer to   (  log  ⁡  (  4  )  −  ϵ  ). This is normal since          2        F    1        (    2    s    ,    s    ;    s    ;    −    1    )    =      4          −      s      What is interesting is that, if   log  ⁡  [  f  (  s  )  ] is an increasing function going to infinity while, if   log  ⁡  [  f  (  s  )  ] goes through a maximum value.What is interesting is that      4    s            2        F    1    (  s  −  1  ,  2  s  −  1  ;  s  ;  −  1  )  =  2  +  2      π              Γ      (      s      )              Γ              (        s        −                  1          2                )

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