Question

You blow across the open mouth of an empty test tube and produce the fundamental standing wave of the air column inside the test tube. The speed of sound in air is 344 m/s and the test tube acts as a stopped pipe.(a) If the length of the air column in the test tube is 14.0 cm, what is the frequency of this standing wave? (b) What is the frequency of the fundamental standing wave in the air column if the test tube is half filled with water?

Answer 1

Answer:

(a) Frequency of the standing wave is 614.28 Hz.

(b) Frequency of the standing waves is 1228.56 Hz.

Explanation:

Frequency of standing wave in the case of one end open and other end closed of pipe (also known as stopped pipe) is given by the relation :

$f_{n} =n\frac{v}{4L}$     .....(1)

Here $f_{n}$ is the frequency of nth harmonic, v is speed of wave, L is length of the pipe and n is the harmonic which only takes values 1,3,5.. and so on.

(a) Given :

Speed of sound in air, v = 344 m/s

Length of tube, L = 14 cm = 0.14 m

n = 1

Substitute these values in equation (1).

$f_{1} =1\times\frac{344}{4\times0.14}$

f₁ = 614.28 Hz

(b) In this case, half of the tube is filled with water. So,

Length of tube having air, L = (0.14)/2 = 0.07 m

Substitute the suitable values in equation (1).

$f_{1} =1\times\frac{344}{4\times0.07}$

f₁ = 1228.56 Hz

Answer 2

A) The frequency of the standing wave is;

f_n = 614.29 Hz

B) The frequency of the fundamental standing wave in the air column if the test tube is half filled with water is; f_n = 1228.57 Hz

Since we are told that the mouth of the test tube is opened, it means we can use the formula for frequency of a standing wave when one end of a pipe is open and another closed. Thus;

f_n = nv/4L

Where;

v is speed of sound in air = 344 m/s

n is number of harmonic and can only be odd integers like 1, 3, 5 e.t.c

A) We are given;

v = 344 m/s

Length of air column; L = 14 cm = 0.14 m

There's only 1 harmonic and so; n = 1

Thus;

f_n = (1 × 344)/(4 × 0.14)

f_n = 614.29 Hz

B) The test tube is half filled with water. Thus;

L = 0.14/2

L = 0.07 m

Thus;

f_n = (1 × 344)/(4 × 0.07)

f_n = 1228.57 Hz

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