Let's tackle each part of the question step by step.
### Part (a) - Empty Soda Bottle as a Closed Tube
To determine the resonant frequency of an empty soda bottle, we need to consider the bottle as a closed tube. In acoustics, a closed tube is modeled as a tube that is closed at one end and open at the other. The bottle is closed at the bottom and open at the top where you blow across.
For a closed tube, the lowest resonant frequency, also known as the fundamental frequency, corresponds to a quarter-wavelength standing wave. This means the length of the tube is effectively one-quarter of the wavelength of the sound wave that will resonate.
The formula to find the fundamental frequency (f) of a closed tube is:
\[ f = \frac{v}{4L} \]
where:
- \( v \) is the speed of sound in air (approximately 343 meters per second at room temperature, or 20°C)
- \( L \) is the length of the tube, which in this case is the depth of the soda bottle.
Given that the bottle is 24 cm deep, we need to convert that length into meters for our calculations:
\[ L = 24 ~cm = 0.24 ~m \]
Now we can calculate the frequency:
\[ f = \frac{343 ~m/s}{4 \times 0.24 ~m} \]
\[ f = \frac{343}{0.96} \]
\[ f \approx 357.29 ~Hz \]
So, blowing across the top of an empty 24 cm deep soda bottle, treated as a closed tube, you would expect a resonant frequency of approximately 357.29 Hz.
### Part (b) - Soda Bottle One-Third Full of Soda
If the soda bottle is one-third full, the length of the air column is reduced to two-thirds of the original depth because the soda fills the bottom third. The new effective length (L') of the air column is:
\[ L' = \frac{2}{3} \times 24 ~cm = 16 ~cm \]
\[ L' = 0.16 ~m \]
Using the formula for the fundamental frequency of a closed tube again, we find the new resonant frequency:
\[ f' = \frac{343 ~m/s}{4 \times 0.16 ~m} \]
\[ f' = \frac{343}{0.64} \]
\[ f' \approx 535.94 ~Hz \]
Therefore, if the soda bottle is one-third full, the resonant frequency will increase, and in this case, you would expect it to be approximately 535.94 Hz. This is higher than the frequency of the empty bottle because the shorter air column has a higher natural frequency.