The lecturer with a monotone voice of 440 Hz and you are both 2m away from the long, smooth and tall wall. You will hear sound that comes directly to you from the lecturer along with sound that comes back from the wall on it’s way to reaching you. You need the path here needs to have the same angle onto the wall as it has after reflection. How far from the lecturer will you pick your seat so you can’t hear the voice at all and find your well-deserved rest?
To find the distance from the lecturer where you will not be able to hear their voice at all, you need to consider the concept of sound interference and reflection.
First, let's calculate the time it takes for the sound to travel from the lecturer to the wall and back to you. The time it takes for sound to travel a given distance can be calculated using the formula:
time = distance / speed
The speed of sound in air is approximately 343 meters per second.
So, the time it takes for the sound to travel from the lecturer to the wall and back to you is:
time = (2 meters + 2 meters) / (343 meters per second) = 0.0116 seconds
To achieve the condition where the path onto the wall has the same angle as the path after reflection, we need to consider the concept of the angle of incidence being equal to the angle of reflection. In this scenario, since the wall is smooth and tall, we can assume that the angle of reflection will be equal to the angle of incidence.
Now, let's consider the wavelengths that will cancel each other out due to interference. When two waves interfere constructively, they reinforce each other, resulting in a louder sound. When they interfere destructively, they cancel each other out, producing silence. In this case, to cancel out the sound from the lecturer's voice, we need destructive interference to occur.
For destructive interference to happen, the path length difference between the direct sound and the reflected sound must be equal to half a wavelength. This condition can be represented by the equation:
path length difference = (N + 1/2) * wavelength
Here, N is an integer representing the number of complete wavelengths.
The wavelength of a sound wave can be calculated using the formula:
wavelength = speed / frequency
In this case, the frequency (pitch) of the lecturer's voice is 440 Hz (cycles per second).
So, the wavelength of the sound wave is:
wavelength = 343 meters per second / 440 Hz = 0.7795 meters
Now, let's calculate the path length difference:
path length difference = (N + 1/2) * 0.7795 meters
To achieve complete destructive interference and cancel out the sound from the lecturer's voice, we need the path length difference to equal half a wavelength, which is:
0.5 * wavelength = 0.5 * 0.7795 meters = 0.3897 meters
By equating the path length difference equation with the half-wavelength equation, we can solve for N:
(N + 1/2) * 0.7795 meters = 0.3897 meters
Solving this equation, we find:
N = 0.5
Since N must be an integer, the nearest integer to 0.5 is 1. Therefore, N = 1.
Now, let's calculate the distance from the lecturer where you should pick your seat by substituting N into the path length difference equation:
path length difference = (1 + 1/2) * 0.7795 meters = 1.169 meters
Keep in mind that this distance represents the extra path length traveled by the sound wave that bounces off the wall. To find the total distance from the lecturer where you can't hear their voice at all, you should subtract this value from the total distance (2 meters):
Total distance = 2 meters - 1.169 meters = 0.831 meters
Therefore, you should pick your seat approximately 0.831 meters away from the lecturer to enjoy your well-deserved rest without hearing their voice.