Question

Analyze the standing sound waves in an open-closed tube to show that the possible wavelengths and frequencies are given by Equation 17.18.

Short answer

The conditions of having one node and one antinode at the end the formulas that one could get are,

Step-by-step solution

The concept of node and antinode 

Node is a point which's amplitude is 0. The point in which amplitude is maximum is known as antinodes.

Finding out the frequencies based on the information 

In an open-closed tube, the presence of node and antinode is a must. so, the possible arrangements can reach to $\frac{1}{4}\lambda ,\frac{3}{4}\lambda ,\frac{5}{4}\lambda$

Thus the outcome is $L=\frac{m\lambda }{4}$

Here m= 1, 3, 5 and the arrangements will be ${\lambda }_m=\frac{4L}{m}$

In order to make the connection with frequency, the wave has speed v and wavelength $\lambda$and the frequency could be found as $v=\lambda f\Rightarrow f=\frac{v}{\lambda }$

The substitution of the wavelength isrole="math" localid="1649148923810" $fm=m\frac{v}{4L}$