The air column in a pipe closed at one end is made to vibrate in its second overtone by a tuning fork of frequency `440 Hz`. The speed of sound in air is `330ms^(-1)`. End corrections may be neglected. Let `P_(0)` denote the mean pressure at any point in the pipe, and `DeltaP` the maximum amplitude of pressure variation.
(a) What the length `L` of the air column.
(b) What is the amplitude of pressure variation at the middle of the column?
( c ) What are the maximum and minimum pressures at the open end of the pipe?
(d) What are the maximum and minimum pressures at the closed end of the pipe?

Explanation

(i) 93.75 cm (ii) `(triangleP_0)/(sqrt2)` (iii) `P_0` always (Iv) `(P_0+triangleP_0),(P_0-triangleP_0)`
(i) for II overtone `L=5/4lambda=5/4v/f=5/4xx330/440=15/16m=93.75cm`
(ii) At the middle of the column (i.e at C),distance of C from the closest node N, `NC=(lambda)/8=9.375 cm`
At C pressure amptitude `triangleP=triangleP_0 sin kx=trianglep_0sin((2pi)/(lambda)(lambda)/8)=(triangle P_0)1/sqrt2` (iii) No variation in pressure at the open end takes place as node lies there.At open end,presure `=P_0` (always)
(iv) At closed end,max presure `=P_0+triangleP_0` Minimum presure `=P_0-triangle P_0`