Question

A pipe open at both ends has a fundamental frequency of 220 Hz when the temperature is 0°C. (a) What is the length of the pipe? (b) What is the fundamental frequency at a temperature of 30°C?

Answer 1

Explanation:

It is given that,

Fundamental frequency, f = 220 Hz

(a) We know that at 0 degrees, the speed of sound in air is 331 m/s.

For open pipe, $\lambda=2l$

l is the length of pipe

Also,

$v=f\lambda$

$l=\dfrac{v}{2f}=\dfrac{331}{2\times 220}=0.75\ m$

(b) Let f' is the fundamental frequency of the pipe at 30 degrees and v' is its speed.

$v'=331\sqrt{1+\dfrac{T}{273}}$

$v'=331\sqrt{1+\dfrac{30}{273}}$

v' = 348.71 m/s

So, $f'=\dfrac{v'}{\lambda}$

$f'=\dfrac{348.71}{2\times 0.75}$

f' = 232.4 Hz

Hence, this is the required solution.

Answer 2

Answer:284.4 Hz

Explanation:

At $o^{\circ}C$ speed of sound is 331 m/s

and we know

$Velocity\left ( v\right )=frequency\left ( f\right )\times wavelength\left ( \lambda\right )$

$331=220\left ( 2L\right )$------$\left ( \lambda =2L\right )$

L=0.613 m

$\left ( b\right )$

$v=v_0\sqrt{1+\frac{T}{273}}$

where velocity of sound at$t=0^{\circ}C$

$v=331\sqrt{1+\frac{30}{273}}$

v=348.7 m/s

for  frequency

$v=f\lambda$

$348.7=f\times 2\times 0.613$

f=284.4 Hz