Question

A PVC pipe has a length of 45.132 centimeters.a. What are the frequencies of the first three harmonics when the pipe is open at both ends? Include units in your answers.b. What are the frequencies of the first three harmonics when the pipe is closed at one end and open at the other? Include units in your answers.

Answer 1

ANSWERS

a. f₁ = 380 Hz; f₂ = 760 Hz; f₃ = 1140 Hz

b. f₁ = 190 Hz; f₃ = 570 Hz; f₅ = 950 Hz

Step-by-step explanation

a. For a pipe of length L open at both ends, the frequencies of the first three harmonics are:

`\begin{cases}f_1=(v)/(2L) \\ \\ f_2=2f_1=(v)/(L) \\ \\ f_3=3f_1=(3v)/(2L)\end{cases}`

Assuming that the speed of the wave is the speed of sound: 343 m/s and knowing that the length of the pipe is L = 45.132 cm = 0.45132 m we can find the frequencies of the first three harmonics:

`\begin{cases}f_1=(343m/s)/(2\cdot0.45132m)\approx380Hz \\ \\ f_2=2f_1=2\cdot380Hz\approx760Hz \\ \\ f_3=3f_1=3\cdot380Hz\approx1140Hz\end{cases}`

b. For a pipe of length L closed at one end and open at the other, the frequencies of the first three harmonics are:

`\begin{cases}f_1=(v)/(4L) \\ \\ f_2=DNE \\ \\ f_3=3f_1=(3v)/(4L)\end{cases}`

In a closed pipe, there can only be odd harmonics (1, 3, 5...). Therefore, the second harmonic does not exist and the "third harmonic" would be the 5th,

`\begin{cases}f_1=(v)/(4L) \\ \\ f_3=3f_1=(3v)/(4L) \\ \\ f_5=5f_1=(5v)/(4L)\end{cases}`

Again, the length of the pipe is 45.132 cm = 0.45132 m, so the first three harmonics are:

`\begin{cases}f_1=(343m/s)/(4\cdot0.45132m)\approx190Hz \\ \\ f_3=3f_1=3\cdot190Hz=570Hz \\ \\ f_5=5f_1=5\cdot190Hz=950Hz\end{cases}`