In a lecture hall you sit 12 m directly in front of one of the two speakers on the front of the hall. The second speaker is 8m from the first in line with the front of the room, so that the speakers and you form a triangle with a 90° angle. What frequencies in the human hearing range will sound soft to you? Just get the first three!
To determine the frequencies in the human hearing range that will sound soft to you, we need to consider the phenomenon of sound intensity and the inverse square law.
The inverse square law states that the intensity of sound decreases as the square of the distance from the source increases. In other words, if you double the distance from a sound source, the intensity decreases by a factor of four.
In this scenario, we can calculate the intensity of sound at your location relative to the first speaker to determine the frequencies that will sound soft.
Let's assume that the first speaker emits a sound with a constant intensity. This means that the intensity of sound at a distance of 12m from the first speaker will be 1/4th of the intensity at a distance of 4m (8m - 4m).
By applying the inverse square law, we can find the ratio of intensities as follows:
(12m ^ 2) / (4m ^ 2) = 144 / 16 = 9
So, the sound intensity at your location is 1/9th (or 1/3 squared) of the intensity at a distance of 4m from the first speaker.
Now, let's consider the frequencies in the human hearing range that would sound soft to you. Softness or loudness perception varies among individuals, but generally, softer sounds fall within the lower end of the frequency spectrum.
The human hearing range is typically considered to be from 20 Hz (low-frequency) to 20,000 Hz (high-frequency). However, for this explanation, we'll consider the soft range to be on the lower end of the spectrum.
Here are the first three frequencies that may sound soft to you:
1. 20 Hz: This is the lowest frequency within the human hearing range. It is perceived as a deep, rumbling bass sound.
2. 30 Hz: Slightly higher than 20 Hz, this frequency still falls within the bass range and may be perceived as a subtle rumble.
3. 40 Hz: Another low-frequency sound that can be perceived as a gentle rumble or vibration.
Please note that individual perception may vary, and the concept of what sounds soft to one person may differ from another. The range provided here is a general guideline based on the assumption that softer sounds tend to be in the lower frequency range.
To determine the frequencies in the human hearing range that will sound soft to you, we can use the concept of the inverse square law.
The inverse square law states that the intensity of sound decreases inversely with the square of the distance from the source. Mathematically, it can be expressed as:
I2 = I1 * (r1 / r2)^2
Where:
I1 = intensity of the sound at distance r1
I2 = intensity of the sound at distance r2
Given that you are sitting 12 m directly in front of the first speaker and the second speaker is 8 m from the first, we can calculate the intensity ratio between the two speakers:
I2 / I1 = (r1 / r2)^2 = (12 m / 20 m)^2 = (3 / 5)^2 = 9 / 25
This means that the second speaker will have an intensity that is 9/25 (or 0.36) times that of the first speaker at your position.
Now, let's consider the frequency range for human hearing, which is generally considered to be between 20 Hz and 20,000 Hz.
To determine the frequencies that will sound soft to you, we need to consider the threshold of audibility. The threshold of audibility is the minimum sound intensity that can be heard by a typical human ear at a given frequency. It varies with frequency.
For simplicity, let's assume that the soft threshold occurs when the sound intensity is half the threshold of audibility.
Based on this assumption, we can calculate the soft threshold intensity for each frequency using the given intensity ratio:
Soft threshold intensity = (1/2) * threshold of audibility
Now, let's find the three frequencies in the human hearing range that will sound soft to you:
1. For a frequency of 20 Hz:
Threshold of audibility for 20 Hz = 0.00002 Pa (approximate value)
Soft threshold intensity for 20 Hz = (1/2) * 0.00002 Pa = 0.00001 Pa
Since the soft threshold intensity at your position is 0.36 times the intensity of the first speaker, the minimum intensity at your position for a 20 Hz frequency to sound soft is:
Minimum intensity = 0.36 * 0.00001 Pa = 0.0000036 Pa
2. For a frequency of 1000 Hz (1 kHz):
Threshold of audibility for 1000 Hz = 0.02 Pa (approximate value)
Soft threshold intensity for 1000 Hz = (1/2) * 0.02 Pa = 0.01 Pa
Minimum intensity = 0.36 * 0.01 Pa = 0.0036 Pa
3. For a frequency of 20,000 Hz (20 kHz):
Threshold of audibility for 20,000 Hz = 20 Pa (approximate value)
Soft threshold intensity for 20,000 Hz = (1/2) * 20 Pa = 10 Pa
Minimum intensity = 0.36 * 10 Pa = 3.6 Pa
Therefore, the first three frequencies in the human hearing range (20 Hz - 20,000 Hz) that will sound soft to you, given your position in the lecture hall, are approximately:
1. 20 Hz (minimum intensity = 0.0000036 Pa)
2. 1000 Hz (minimum intensity = 0.0036 Pa)
3. 20,000 Hz (minimum intensity = 3.6 Pa)