Question

An open pipe of certain length produces fundamental frequency f₁. A closed pipe of some other length produces fundamental .frequency f₂. When the two are joined to form a longer close tube, its fundamental frequency will be ____________.

Options

• `("f"_1"f"_2)/("f"_1 + "f"_2)`

• `(2"f"_1"f"_2)/("f"_1 + "f"_2)`

• `("f"_1"f"_2)/("f"_1 + 2"f"_2)`

• `("f"_1"f"_2)/(2"f"_1 + "f"_2)`

Answer 1

An open pipe of certain length produces fundamental frequency f₁. A closed pipe of some other length produces fundamental .frequency f₂. When the two are joined to form a longer close tube, its fundamental frequency will be `("f"_1"f"_2)/("f"_1 + 2"f"_2)`.

Explanation:

`"For open pipe," "f"_1 = "v"/(2"L"_1) or "L"_1/(2"f"_1)`

`"For closed pipe,""f"_12= "v"/(4"L"_2) or "L"_2/(4"f"_2)`

After joining the two pipes we get,

L = L1 + L1

Since it is a closed pipe,

`"f" = "v"/(4"L") = "v"/(4("L"_1 + "L"_2)) = "v"/(4(v/(2"f"_1) + "v"/(4"f"_2)))`

`= (8"f"_1 "f"_2)/(4(4"f"_2 + 2"f"_1))`

` = ("f"_1 "f"_2)/(2"f"_2 +"f"_1)`