Question

On a hot day with a temp of 35 celsius, you blow across the open end of an empty test tube and produce the fundamental standing wave for the air column inside the test tube. The test tube acts like a 22 cm pipe open at one end and closed at the other.

What is the fundamental frequency?

What is the frequency of the next possible Harmonic?

A person standing nearby records a sound level of 50 dB. What is the sound intensity measured by that person?

Answer 1

Velocity of sound in air at 35 degree celsius is 352 m/s .

The basis for drawing the standing wave patterns for air columns is that vibrational antinodes will be present at any open end and vibrational nodes will be present at any closed end. If this principle is applied to closed-end air columns, then the pattern for the fundamental frequency (the lowest frequency and longest wavelength pattern) will have a node at the closed end and an antinode at the open end. For this reason, the standing wave pattern for the fundamental frequency (or first harmonic) for a closed-end air column looks like the diagram below.

The distance between node and antinode is    .

So, from the figure   = length of the pipe = 22 cm

wavelength = 22*4 = 88cm = 0.88m

velocity = frequency * wavelength

352 = f * 0.88

f = 400 Hz.

fundamental frequency = 400 Hz

2) Unlike the other instrument types, there is no second harmonic for a closed-end air column. The next frequency above the fundamental frequency is the third harmonic (three times the frequency of the fundamental). In fact, a closed-end instrument does not possess any even-numbered harmonics. Only odd-numbered harmonics are produced, where the frequency of each harmonic is some odd-numbered multiple of the frequency of the first harmonic.

frequency of the next possible Harmonic = frequency of third harmonic = 3 * 400 = 1200 Hz

3)

We have

- intensity level in dB (deciBels)

I_{o -} sound intensity level at the threshold = 10^-12 W/m²

So ,

sound intensity measured by that person = 10 ^-7 W/m²