The fundamental frequency of an open organ pipe corresponds to B above middle C (493.9 Hz on the chromatic musical scale). The third resonance of a different closed organ pipe has the same frequency. What are the lengths of the two pipes? (Assume that the speed of sound in air is 343 m/s.)

1. open pipe

m

2. closed pipe

m

For open end the length of pipe is 0.347 meter.

The length of pipe for closed end is 0.521 meter.

Explanation:

Given

fundamental frequency = f 1 = 493.9 Hz

Velocity of sound = v = 343 m/s

1. For open end pipe the formula for frequency = (n*v) / 2*l

v stands for velocity

n represent number of harmonics or resonance

l stands for length of the pipe

f 1 = (n*v) / (2*l)

we want to find out the length of open end pipe so

f 1 * 2*l = n*v

l = (n*v) / (2*f 1)

l = (1*343) / (2*493.9)

l = 343/987.8

l = 0.347 m

For open end the length of pipe is 0.347 meter.

Part B

For close end pipe the formula for frequency = f = (n*v) / (4*l)

f = (n*v) / (4*l)

we want to find the length of closed pipe

f * 4*l = n*v

l = (n*v) / (4*f)

n=3 for third resonance

frequency will be same as fundamental frequency=f 1=f 3 = 493.9 Hz (given)

v = 343 m/s

l = (n*v) / (4*f)

Now put the values in above equation

l = (3*343) / (4*493.9)

l = 1029/1975.6

l = 0.521 m

The length of pipe for closed end is 0.521 meter