Answer
378.3k+ views
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.
\[\begin{align}
& L=\dfrac{\lambda }{2}+\dfrac{\lambda }{4} \\
& \therefore L=\dfrac{3\lambda }{4} \\
\end{align}\]
Using this, we will represent the equation in terms of the wavelength.
\[\lambda =\dfrac{4L}{3}\]
Substitute the value of the length of the tube in the above equation.
\[\begin{align}
& \lambda =\dfrac{4\times 0.8}{3} \\
& \therefore \lambda =\dfrac{3.2}{3}m \\
\end{align}\]
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.
\[f=\dfrac{\nu }{\lambda }\]
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.
\[\begin{align}
& f=\dfrac{320}{{}^{3.2}/{}_{3}} \\
& \therefore f=300\,Hz \\
\end{align}\]
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.
\[\begin{align}
& \omega =2\pi \times 300 \\
& \therefore \omega =600\pi \,{rad}/{s}\; \\
\end{align}\]
The wavenumber in terms of the wavelength is given as follows.
\[k=\dfrac{2\pi }{\lambda }\]
Substitute the value of the wavelength of the wave in the above equation.
\[\begin{align}
& k=\dfrac{2\pi }{{}^{3.2}/{}_{3}} \\
& \therefore k=\dfrac{15}{8}\pi \,{{m}^{-1}} \\
\end{align}\]
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.
\[y=A\sin \dfrac{15}{8}\pi x\cos 600\pi t\]
\[\therefore \] The equation that represents the correct pressure variation in the air column is, \[y=A\sin \dfrac{15}{8}\pi x\cos 600\pi t\].
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.
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.
\[\begin{align}
& L=\dfrac{\lambda }{2}+\dfrac{\lambda }{4} \\
& \therefore L=\dfrac{3\lambda }{4} \\
\end{align}\]
Using this, we will represent the equation in terms of the wavelength.
\[\lambda =\dfrac{4L}{3}\]
Substitute the value of the length of the tube in the above equation.
\[\begin{align}
& \lambda =\dfrac{4\times 0.8}{3} \\
& \therefore \lambda =\dfrac{3.2}{3}m \\
\end{align}\]
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.
\[f=\dfrac{\nu }{\lambda }\]
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.
\[\begin{align}
& f=\dfrac{320}{{}^{3.2}/{}_{3}} \\
& \therefore f=300\,Hz \\
\end{align}\]
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.
\[\begin{align}
& \omega =2\pi \times 300 \\
& \therefore \omega =600\pi \,{rad}/{s}\; \\
\end{align}\]
The wavenumber in terms of the wavelength is given as follows.
\[k=\dfrac{2\pi }{\lambda }\]
Substitute the value of the wavelength of the wave in the above equation.
\[\begin{align}
& k=\dfrac{2\pi }{{}^{3.2}/{}_{3}} \\
& \therefore k=\dfrac{15}{8}\pi \,{{m}^{-1}} \\
\end{align}\]
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.
\[y=A\sin \dfrac{15}{8}\pi x\cos 600\pi t\]
\[\therefore \] The equation that represents the correct pressure variation in the air column is, \[y=A\sin \dfrac{15}{8}\pi x\cos 600\pi t\].
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.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)