Wavelength in a Closed Column
Here is the prompt: We place a speaker near the top of a drinking glass. The speaker emits sound waves with a frequency of 3.75 kHz. The glass is 14.1 cm deep. As I pour water into the glass, I find that at certain levels the sound is enhanced due to the excitation of standing sound waves in the air inside the glass. Find the minimum depth of water at which this occurs (distance from surface of water to bottom of glass). The standing sound wave has a node at the surface of the water and an antinode at the top of the glass. Assume that the antinode is exactly at the top of the glass. The speed of sound in air is 343 m/s
My Work: So the distance between a node and an antinode is wavelength/4, so I calculate wavelength via lambda=v/f, converting 3.75 kHz to 3750 Hz. This gives wavelength equal to 0.091466667m. I then take 14.1 cm, convert it to 0.141 m, and subtract the 0.091466667/4 from it to get the depth of the water, which is 0.118133333 m, or 11.8 m. Which is wrong.
To find the minimum depth of water at which the sound is enhanced, we need to determine the wavelength of the standing sound wave.
First, we can calculate the wavelength using the formula:
wavelength = speed of sound / frequency
Given that the speed of sound in air is 343 m/s and the frequency is 3.75 kHz (which is equal to 3750 Hz), we can substitute these values into the formula to find the wavelength.
wavelength = 343 m/s / 3750 Hz
Calculating this expression gives us a wavelength of approximately 0.091466667 meters.
Now, let's consider the nodes and antinodes in the standing sound wave. In this case, we know that there is a node at the surface of the water and an antinode at the top of the glass.
The distance between a node and an antinode is equal to one-fourth of the wavelength. Therefore, the distance from the surface of the water to the bottom of the glass can be calculated as follows:
depth of water = (antinode to top of glass) - (1/4 * wavelength)
Since the antinode is at the top of the glass, we can use the depth of the glass, which is 14.1 cm (or 0.141 m), as the antinode to top-of-glass distance.
depth of water = 0.141 m - (1/4 * 0.091466667 m)
Evaluating this expression, we find that the minimum depth of water at which the sound is enhanced is approximately 0.118133333 meters, or 11.8 cm.
Based on your calculations, it seems that there was a slight error in the subtraction step. Double-check your arithmetic calculations to ensure accuracy.
To find the minimum depth of water at which the sound is enhanced, we need to consider the formation of standing waves inside the glass. The standing sound wave has a node at the surface of the water and an antinode at the top of the glass.
First, let's calculate the wavelength of the sound wave. We can use the formula λ = v/f, where v is the speed of sound and f is the frequency.
Given:
Frequency (f) = 3.75 kHz = 3750 Hz
Speed of sound (v) = 343 m/s
λ = 343 m/s / 3750 Hz = 0.091466667 m
Now, we know that the distance between a node and an antinode is equal to λ/4. In this case, the distance between the node (water surface) and antinode (top of the glass) is equal to λ/4.
Distance between node and antinode = λ/4 = 0.091466667 m/4 = 0.022866667 m
To find the minimum depth of water (distance from surface of water to bottom of glass), we need to subtract the distance between the node and antinode from the total depth of the glass.
Minimum depth of water = Total depth of the glass - Distance between node and antinode
= 14.1 cm - 0.022866667 m = 0.118133333 m
Therefore, the minimum depth of water at which the sound is enhanced is approximately 0.118 m or 11.8 cm.