Question

Prove that a pipe open at both ends of a length of 2L has the same fundamental frequency as another pipe of closed at one end of length L.

Answer 1

1. Let L_o and L_c be the lengths of a pipe open at both ends and a pipe closed at one end, respectively.
2. Let n_o and n_c be their corresponding fundamental frequencies.  
3. n_o = `"v"/(2"L"_"o")` and n_c = `"v"/(4"L"_"c")`
   where v is the velocity of the sound in the air.
4. Since L_c = L and L_o = 2L 
   ∴ n_o = `"v"/(4"L")` and `"n"_"c" = "v"/(4"L")`
   ∴ n_o = n_c 
5. Hence, both the pipes have the same fundamental frequencies.