Final answer:
The smallest distance between the speakers for perfect destructive interference is equal to the wavelength of the sound waves.
Step-by-step explanation:
In order for the interference of sound waves to be perfectly destructive, the waves need to be completely out of phase. This means that the path difference between the two speakers must be equal to a multiple of the wavelength. The formula for the path difference is given by:
Δx = m * λ
Where Δx is the path difference, m is an integer, and λ is the wavelength of the sound waves. In this case, the interference is destructive, so we are looking for the smallest distance between the speakers that satisfies the condition. Since the speakers are in phase, the path difference is equal to one wavelength, so:
d = λ
Using the formula for the speed of sound, which is given by:
v = λ * f
Where v is the speed of sound, λ is the wavelength, and f is the frequency of the sound waves, we can rearrange the formula to solve for λ:
λ = v / f
Substituting the values given, with v = 343.00 m/s and f = 720 Hz, we can calculate the wavelength:
λ = 343.00 m/s / 720 Hz = 0.476 m
Therefore, the smallest distance d between the speakers for perfect destructive interference is equal to the wavelength of the sound waves, which in this case is 0.476 m.