To determine the distance from the wall at which you would hear no sound at all, we need to consider the phenomenon of sound interference.
When the sound wave emitted by the directional loudspeaker reaches the wall, it will reflect off the wall and create a reflected wave. At a certain distance from the wall, the direct and reflected waves will interfere destructively, canceling each other out and resulting in no sound being heard at that location.
To calculate this distance, we can use the concept of path difference. The path difference between the direct wave and reflected wave should be half of the wavelength for destructive interference to occur.
The formula for path difference is:
Path Difference = 2d
Where: Path Difference is the path length difference between the direct wave and reflected wave. d is the distance from the wall to the listener.
To find the distance from the wall, we need to determine the wavelength of the sound wave. The formula for the wavelength is:
Wavelength (λ) = Speed of Sound (v) / Frequency (f)
The speed of sound in air is approximately 343 meters per second at room temperature.
Using the given frequency of 204 Hz, we can calculate the wavelength:
λ = v / f = 343 m/s / 204 Hz ≈ 1.68 m
For destructive interference, the path difference should be half of the wavelength. Therefore:
2d = λ / 2
Substituting the value of λ:
2d = 1.68 m / 2 = 0.84 m
Dividing both sides by 2:
d = 0.84 m / 2 = 0.42 m
Therefore, you would need to stand approximately 0.42 meters (or 42 centimeters) away from the wall to hear no sound at all.