A source of sound placed at the open end of a resonance column sends an acoustic wave of pressure amplitude ${\rho _0}$ inside the tube. If the atmospheric pressure is ${\rho _A},$ then the ratio of maximum and minimum pressure at the closed end of the tube will be
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(a)Maximum pressure at closed end will be atmospheric pressure adding with acoustic wave pressure
So ${\rho _{\max }} = {\rho _A} + {\rho _0}$ and ${\rho _{\min }} = {\rho _A} - {\rho _0}$
Thus $\frac{{{\rho _{\max }}}}{{{\rho _{\min }}}} = \frac{{{\rho _A} + {\rho _0}}}{{{\rho _A} - {\rho _0}}}$
So ${\rho _{\max }} = {\rho _A} + {\rho _0}$ and ${\rho _{\min }} = {\rho _A} - {\rho _0}$
Thus $\frac{{{\rho _{\max }}}}{{{\rho _{\min }}}} = \frac{{{\rho _A} + {\rho _0}}}{{{\rho _A} - {\rho _0}}}$
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