Question

Two open organs pipes of fundamental frequencies $\upsilon_{1}$ and $\upsilon_{2}$ are joined in series. The fundamental frequency of the new pipe so obtained will be

• A. $\upsilon_{1} + \upsilon_{2}$
• B. $\frac{\upsilon_{1}\upsilon_{2}}{(\upsilon_{1} + \upsilon_{2})}$
• C. $\frac{\upsilon_{1}\upsilon_{2}}{\upsilon_{1} - \upsilon_{2}}$
• D. $\sqrt{\left( \upsilon_{1}^{2} + \upsilon_{2}^{2} \right)}$

Answer 1

Answer: (B)

$\frac{\upsilon_{1}\upsilon_{2}}{(\upsilon_{1} + \upsilon_{2})}$

Answer 2

$\upsilon_{1} = \frac{v}{2L_{1}}$ …. (i)

$\upsilon_{2} = \frac{v}{2L_{2}}$ ….. (ii)

When these two pipes are joined in series, the fundamental frequency of the new pipe is

$\upsilon = \frac{v}{2(L_{1} + L_{2})} = \frac{v}{2L_{1} + 2L_{2}}$

$= \frac{v}{\frac{v}{\upsilon_{1}} + \frac{v}{\upsilon_{2}}}$

(Using (i) and (ii))

$= \frac{\upsilon_{1}\upsilon_{2}}{\upsilon_{2} + \upsilon_{1}}$