Question

If you blow air across the mouth of an empty soda bottle, you hear a tone. Why is it that if you put some water in the bottle, the pitch of the tone increases?

Short answer

Answer: The pitch of the tone increases when water is added to the bottle because the length of the air column decreases, which results in a higher frequency (pitch) of the standing wave formed inside the bottle.

Step-by-step solution

Reminder of the standing waves in the air column

When you blow air across the mouth of an empty bottle, the air inside the bottle vibrates and forms a standing wave. This standing wave is created due to the resonance between the incident and reflected waves inside the air column. The pitch (frequency) of the tone that you hear is determined by the natural frequency of this air column.

Understand the relationship between the length of the air column and the frequency

The frequency (f) of the standing wave in the air column is given by the following equation: f = \frac{v}{4L} where v is the speed of sound in the air, and L is the length of the air column. From this equation, it can be observed that the frequency is inversely proportional to the length of the air column. So, if the length of the air column decreases, the frequency (pitch) will increase.

Visualize how adding water affects the air column

When you put some water in the bottle, the water level rises inside the bottle, which decreases the air column's length inside the bottle. As a result, the length of the air column available for the standing wave to be formed is shortened.

Explain the increase in frequency

By adding water to the bottle, we decreased the air column's length, as discussed in Step 3. From the equation mentioned in Step 2, we know that the frequency is inversely proportional to the length of the air column. So, as we decrease the length of the air column, the frequency (pitch) of the tone will increase. In conclusion, when water is added to the bottle, the pitch of the tone increases because the length of the air column decreases, which results in a higher frequency (pitch) of the standing wave formed inside the bottle.