Q31E - University Physics with Modern Physics

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 act 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?

Step-by-Step Solution

Answer

1. The frequency of the fundamental standing wave in the air is $614\text{\hspace{0.17em}Hz}$ .
2. If the test tube is half filled with water, the frequency of the fundamental standing wave is $1229\text{\hspace{0.17em}Hz}$ .

STEP 1 Concept of the length of the air column and the frequency of standing wave

The length of the air column is given as $\mathbf{L}\mathbf{=}\frac{\lambda }{4}$  and the frequency of standing wave is given as $\mathbf{f}\mathbf{=}\mathbf{v\lambda }$   where  $\mathbf{\lambda }$ is the wavelength of the wave.

STEP 2 Calculate the frequency when the length is 14.0cm

The length of the closed pipe is  $14.0\text{\hspace{0.17em}cm}$ .The frequency is given by  $\mathbf{f}\mathbf{=}\mathbf{v\lambda }$.

$\begin{array}{c}\frac{l}{4}=\lambda \\ l=4\lambda \\ =4\times 0.14\text{\hspace{0.17em}m}\\ =0.56\text{\hspace{0.17em}m}\end{array}$ 

  

$\begin{array}{c}f=vl\\ f=344\text{\hspace{0.17em}m}/\text{s}\times 0.56\text{\hspace{0.17em}m}\\ =614\text{\hspace{0.17em}Hz}\end{array}$

Therefore, the frequency of the fundamental standing wave in the air is $614\text{\hspace{0.17em}Hz}$ .

Step 3 Calculate the frequency when the length is half filled

The length of the closed pipe is $7.0\text{\hspace{0.17em}cm}$ .

$\begin{array}{c}\mathrm{\lambda }=4\times 0.07\mathrm{m}\\ =0.28\mathrm{m}\\ \mathrm{f}=344\mathrm{m}/\mathrm{s}\times 0.28\text{\hspace{0.17em}m}\\ =1229\mathrm{Hz}\end{array}$

Therefore, the frequency of the fundamental standing wave in the air column if the test tube is half filled with water is  $1229\mathrm{Hz}$.