Two speakers are wired to emit identical
sounds in unison. The wavelength in air of
the sounds is 6 m.
Do the sounds interfere constructively or
destructively at a point 10 m from both speakers?
1. Destructively
2. Neither
3. Constructively
At the same distance from each speaker, they interfere constructively. This assumes that they are wired "in phase".
Constructively
To determine whether the sounds interfere constructively or destructively at a point 10 m from both speakers, we need to consider the phase relationship between the two sound waves.
If the two speakers are emitting identical sounds in unison, the waves will have the same phase and constructive interference will occur if the path length difference (PLD) between the two speakers is an integer multiple of the wavelength. Destructive interference will occur if the PLD is an odd multiple of half the wavelength.
In this case, the wavelength of the sounds is given as 6 m. The point in question is 10 m from both speakers. To calculate the PLD, we can consider the distance from each speaker to the point.
The path length difference (PLD) between the two speakers is given by:
PLD = |d2 - d1|
Where d1 and d2 are the distances from the first and second speakers to the point, respectively.
In this case, since the point is equidistant from both speakers (10 m), the PLD would be:
PLD = |10 m - 10 m| = |0| = 0 m
Since the PLD is zero, which is an integer multiple of the wavelength (6 m), the sounds interfere constructively at the point 10 m from both speakers.
Therefore, the correct answer is:
3. Constructively
To determine whether the sounds interfere constructively or destructively at a point 10 m from both speakers, we need to consider the phase difference between the two sound waves.
The formula to calculate the phase difference is:
Phase Difference = 2π (Distance / Wavelength)
Given that the wavelength of the sound is 6 m, we can plug in the values to find the phase difference:
Phase Difference = 2π (10 m / 6 m)
Phase Difference ≈ 10.47 radians
Now, let's analyze the options:
1. Destructively: Destructive interference occurs when the phase difference between two waves is an odd multiple of π (180 degrees). In this case, the phase difference is approximately 10.47 radians, which is not an odd multiple of π. Therefore, we can conclude that the sounds do not interfere destructively.
2. Neither: This option would indicate that the sounds do not interfere at all, which is not correct. There will be interference between the sound waves due to their superposition.
3. Constructively: Constructive interference occurs when the phase difference between two waves is an even multiple of π (360 degrees). In this case, the phase difference is approximately 10.47 radians, which is not an even multiple of π. Therefore, we can conclude that the sounds do not interfere constructively.
So, the correct answer is 2. Neither. The sounds do interfere, but neither constructively nor destructively at a point 10 m from both speakers.