Question

66) The fundamental wavelength for standing sound waves in an empty pop bottle is 0.88 m. (a) What is the length of the bottle? (b) What is the frequency of the third harmonic of this bottle?

Answer 1

$\begin{gathered} L=0.44\text{ m} \\ f=389.77\text{ Hz} \end{gathered}$

Explanation

Step 1

Let

$\begin{gathered} \text{fundamental wavelength=0.88 ,} \\ \lambda=2L \\ \text{hence, replace} \\ 0.88m=2\text{ L} \\ \text{divide both sides by 2} \\ \frac{0.88m}{2}=\frac{2\text{ L}}{2} \\ 0.44=L \end{gathered}$

hence, the length of the bottle is 0.44 meters

Step 2

What is the frequency of the third harmonic of this bottle?​

For the third harmonic, the length of the string is equivalent to three-halves of a wavelength,so

$\begin{gathered} third\text{ armonci= }\frac{\text{3}}{2}\cdot0.88\text{ m} \\ third\text{ armonci= }1.32\text{ m} \\ \end{gathered}$

so

$\begin{gathered} 1.32=2L \\ \end{gathered}$

replace in the quation and solve for

Let speed of sound = 343 m/s

$\begin{gathered} f=\frac{v}{\lambda} \\ \text{replacing} \\ f=\frac{343}{0.88} \\ f=389.77\text{ Hz} \end{gathered}$

I hope this helps

you