Question

Two open pipes of different lengths and same diameter in which the air column vibrates with fundamental frequencies 'n₁', and 'n₂' respectively. When both pipes are joined to form a single pipe, its fundamental frequency will be ______.

Options

• `("n"_1 + "n"_2)/("n"_1"n"_2)`

• `("n"_1"n"_2)/(2"n"_2 + "n"_1)`

• `(2"n"_2 + "n"_1)/("n"_1"n"_2)`

• `("n"_1"n"_2)/("n"_1 + "n"_2)`

Answer 1

Two open pipes of different lengths and same diameter in which the air column vibrates with fundamental frequencies 'n₁', and 'n₂' respectively. When both pipes are joined to form a single pipe, its fundamental frequency will be `underlinebb(("n"_1"n"_2)/("n"_1 + "n"_2))`.

Explanation:

For open pipe with frequency n₁ and length l₁,

`"n"_1 = v/(2"l"_1) => "l"_1 = "v"/(2"n"_1)`    ...(i)

Similarly, for open pipe with frequency n2 and length l₂,

`"n"_2 = v/(2"l"_2) => "l"_2 = "v"/(2"n"_2)`    ...(ii)

When both the pipes are joined together then net length of the pipe will be,

L = l₁ + l₂   ...(iii)

Let, after joining the pipes fundamental frequency will be N₀,

So, L = `"v"/(2"N"_0) = l_1 + l_2`

⇒ L = `"v"/(2"n"_1) + "v"/(2"n"_2) = "v"/(2"N"_0)`

Using Eqs. (i) and (Ii), we get

`=> "N"_0 = ("n"_1"n"_2)/("n"_1 + "n"_2)`