Question

The physics of wind instruments is based on the concept of standing waves. When the player blows into the mouthpiece, the column of air inside the instrument vibrates, and standing waves are produced. Although the acoustics of wind instruments is complicated, a simple description in terms of open and closed tubes can help in understanding the physical phenomena related to these instruments. For example, a flute can be described as an open-open pipe because a flutist covers the mouthpiece of the flute only partially. Meanwhile, a clarinet can be described as an open-closed pipe because the mouthpiece of the clarinet is almost completely closed by the reed

Part A

Consider a pipe of length \(80.0 \mathrm{~cm}\) open at both ends. What is the lowest frequency \(f\) of the sound wave produced when you blow into the pipe?

Part B

A hole is now drilled through the side of the pipe and air is blown again into the pipe through the same opening. The fundamental frequency of the sound wave generated in the pipe is now

Part C

If you take the original pipe in Part A and drill a hole at a position half the length of the pipe, what is the fundamental frequency \(f^{\prime}\) of the sound that can be produced in the pipe?

Part D

What frequencies, in terms of the fundamental frequency of the original pipe in Part A, can you create when blowing air into the pipe that has a hole halfway down its length?

What length of open-closed pipe would you need to achieve the same fundamental frequency \(f\) as the open-open pipe discussed in Part A?

Answer 1

Part A )

L = 80 cm

= 0.8 m

V = 343 m/s ( sound speed in air )

V₁ = n V / 2 L

= 1 X 343 / 2 X 0.8

V₁ = 214.375 Hz

Part B )

using above equation

V_f = V / 2 L

V_f 1 / L

so the new frequency is high than the previuos or before.

Part C )

now L ' = L / 2 = 0.8 / 2

L ' = 0.4 m

V_n ' = n V / 2 L '

= 1 X 343 / 2 X 0.4

V_n ' = 428.75 Hz

so the new frequency is 428.75 Hz

Part D )

V_n ' = n V / 2 L '

L ' = L /2

V_n ' = 2 n V / 2 L '

V_n ' = 2 n V_n

n ' = 2n

V_n ' = n' V_n

even multiples of fundamental frequency

Part E )

for open- closed pipe

f_n ' = n V / 4 L '

and for open- open pipe  

V_n = n V / 2 L

n V / 2 L ' = n V / 2 L

2 L ' = L

L ' = L / 2

So half the length of open - open Pipe