Question

In a resonance tube experiment, an 80 cm air column is in resonance with a tuning fork in the first overtone. Which equation can represent correct pressure variation in the air column? (x=0 is the top point of the tube, neglect end correction, speed of sound = $$320m/\sec$$)

Answer 1

Hint: Using the given data, we have to compute the values of the wavenumber and the angular frequency of the wave that gets generated in the resonance tube. Then, we have to substitute these values in the standard equation to find the expression for the equation that represents the correct pressure variation in the air column.
Formula used:
$$y=A\sin kx\cos \omega t$$

Complete step-by-step solution:
The length of the vibration occurring in the tube is measured in terms of the wavelength of the wave.
$$\begin{array}{rl}& L={\displaystyle \frac{\lambda }{2}}+{\displaystyle \frac{\lambda }{4}}\\ & \therefore L={\displaystyle \frac{3\lambda }{4}}\end{array}$$
Using this, we will represent the equation in terms of the wavelength.
$$\lambda ={\displaystyle \frac{4L}{3}}$$
Substitute the value of the length of the tube in the above equation.
$$\begin{array}{rl}& \lambda ={\displaystyle \frac{4\times 0.8}{3}}\\ & \therefore \lambda ={\displaystyle \frac{3.2}{3}}m\end{array}$$
As we know the relationship between the wavelength, frequency and the speed of sound in air, so, using the same, we will compute the value of the frequency of the wave.
The frequency of the wave equals the speed of sound in air by the wavelength of the light.
$$f={\displaystyle \frac{\nu }{\lambda }}$$
Substitute the values of the speed of sound in air and the wavelength of wave that we have obtained using the length of the tube.
$$\begin{array}{rl}& f={\displaystyle \frac{320}{{}^{3.2}/{}_3}}\\ & \therefore f=300Hz\end{array}$$
The angular frequency in terms of the frequency is given as follows.
$$\omega =2\pi f$$
Substitute the value of the frequency of the wave in the above equation.
$$\begin{array}{rl}& \omega =2\pi \times 300\\ & \therefore \omega =600\pi rad/s\end{array}$$
The wavenumber in terms of the wavelength is given as follows.
$$k={\displaystyle \frac{2\pi }{\lambda }}$$
Substitute the value of the wavelength of the wave in the above equation.
$$\begin{array}{rl}& k={\displaystyle \frac{2\pi }{{}^{3.2}/{}_3}}\\ & \therefore k={\displaystyle \frac{15}{8}}\pi m^{-1}\end{array}$$
The standard wave equation that represents the correct pressure variation in the air column is given as follows.
$$y=A\sin kx\cos \omega t$$
Now substitute the values of the wave number and the angular frequency that we have obtained in the above equation.
$$y=A\sin {\displaystyle \frac{15}{8}}\pi x\cos 600\pi t$$
$$\therefore$$ The equation that represents the correct pressure variation in the air column is, $$y=A\sin {\displaystyle \frac{15}{8}}\pi x\cos 600\pi t$$.

Note: The wavenumber is computed using the wavelength of the wave, whereas, the angular frequency is measured using the frequency of the wave. The length of the vibration that occurred in the tube is measured in terms of the wavelength of the wave.