Both drawings show the same square, each of which has a side of length L = 0.86 m. An observer O is stationed at one corner of each square. Two loudspeakers are located at corners of the square, as in either drawing 1 or drawing 2. The speakers produce the same single-frequency tone in either drawing and are in phase. The speed of sound is 343 m/s. Find the single smallest frequency that will produce both constructive interference in drawing 1 and destructive interference in drawing 2.
To find the single smallest frequency that will produce constructive interference in drawing 1 and destructive interference in drawing 2, we need to consider the path differences between the two speakers and the observer in each scenario.
In constructive interference, the path difference between the two speakers and the observer must be an integer multiple of the wavelength. In destructive interference, the path difference must be an odd multiple of half the wavelength.
Let's analyze the two scenarios separately:
1. Drawing 1: Constructive Interference
In drawing 1, the observer is located at one corner of the square. The path difference between the observer and the two speakers will depend on the angle of observation.
Assuming the observer is at the top left corner, the path difference between the top speaker and the observer is zero since they are at the same corner. The path difference between the bottom speaker and the observer is equal to the path traveled along one side of the square.
Since the side length of the square is L = 0.86 m, the path difference for drawing 1 is L.
2. Drawing 2: Destructive Interference
In drawing 2, the observer is still located at one corner of the square. However, the path difference will now depend on the angle of observation.
Assuming the observer is at the top left corner, the path difference between the top speaker and the observer is equal to the distance traveled along the diagonal of the square. This distance can be calculated using the Pythagorean theorem.
The diagonal of a square with side length L is d = √(2L^2). Substituting the given side length L = 0.86 m, we get d ≈ 1.22 m. Therefore, the path difference for drawing 2 is approximately 1.22 m.
Now, we can calculate the wavelength of the sound using the formula: wavelength = speed of sound / frequency.
For constructive interference:
L = n * wavelength (where n is an integer)
For destructive interference:
1.22 m = (2m + 1/2) * wavelength (where m is an integer)
To find the smallest frequency that satisfies both conditions, we need to find the lowest common multiple of the integers n and (2m + 1/2).
The lowest common multiple (LCM) of n and (2m + 1/2) will give us the smallest frequency that satisfies both conditions.
So, we need to find the LCM of n and (2m + 1/2).
Unfortunately, you haven't provided any specific values for n or m in your question, so it's not possible to calculate the exact frequency. However, you can follow the steps above and substitute the values of n and m to find the LCM and calculate the desired frequency.