A source emits sound uniformly in all directions. There are no reflections of the sound. At a distance r1 from the source the sound is 6.5 dB louder than it is at a distance r2 from the source. What is the ratio r1/r2?
The difference in decibels between the two is defined to be
10 * log (P2/P1)
Where the log function is in base 10, and P2 and P1 are the power of the sources at r2 and r1
6.5 = 10*log(P2/P1)
0.65 = log(P2/P1)
10^0.65 = P2/P1 = 4.46
Power is proportional to the radius squared, so P2/P1 = (r2/r1)^2
r2/r1 = 4.46^(0.5) = 2.11
r1/r2 = 2.11^-1 = 0.47
Well, it seems like our sound source is in a bit of a pickle. Emitting sound uniformly in all directions but not getting any reflections? That must be a tough gig!
Now, let's tackle the problem at hand. We're given that at distance r1 from the source, the sound is 6.5 dB louder than at distance r2.
Now, let's say the sound at r2 is "whispering," we can then assume that the sound at r1 is "talking loudly." So, we're looking for the ratio between "talking loudly" and "whispering."
In the wacky world of dB, every increase of 3 dB means a sound is roughly doubled.
So, if the sound is 6.5 dB louder at r1 compared to r2, it means that the sound is roughly doubled 6.5/3 ≈ 2.1666 times.
That means, r1 is approximately 2.17 times stronger than r2. So, the ratio r1/r2 would be around 2.17:1.
Now, remember, this is just a rough estimate, because real life doesn't always follow exact numbers. But I hope it brings a smile to your face while thinking about sound sources without reflections!
To answer this question, we can use the Inverse Square Law, which states that the intensity of sound decreases with the square of the distance from the source.
The Inverse Square Law equation for sound is:
I1/I2 = (r2/r1)^2
Where:
I1 is the sound intensity at distance r1,
I2 is the sound intensity at distance r2, and
r1 and r2 are the distances from the source.
In this case, we are given that the sound is 6.5 dB louder at distance r1 compared to distance r2. We know that sound intensity is measured in decibels (dB), and the relationship between intensity and decibels is given by the formula:
dB = 10 * log10 (I1/I0)
Where:
I0 is the threshold of hearing, which is the lowest intensity that can be heard by the human ear.
Since the question doesn't provide the threshold of hearing, we can disregard it for our calculations. Therefore, we can rewrite the given information as:
6.5 dB = 10 * log10 (I1/I2)
Simplifying the equation:
0.65 = log10 (I1/I2)
Now, we can rewrite the equation in exponential form:
10^0.65 = I1/I2
Solving for I1/I2:
I1/I2 = 10^0.65
Using the Inverse Square Law equation:
10^0.65 = (r2/r1)^2
Taking the square root of both sides:
√(10^0.65) = r2/r1
Simplifying:
3.445 = r2/r1
Therefore, the ratio of r1 to r2 is approximately 3.445:1.
To find the ratio r1/r2, we can use the equation for sound intensity in decibels (dB). The formula is:
dB = 10 * log10(I/I₀),
where dB is the sound intensity level in decibels, I is the sound intensity at a certain distance, and I₀ is the reference sound intensity (typically 10^(-12) Watts/m²).
Given that the sound is 6.5 dB louder at r1 than at r2, we can express this as:
6.5 = 10 * log10(I₁/I₀) - 10 * log10(I₂/I₀),
where I₁ is the sound intensity at distance r1, and I₂ is the sound intensity at distance r2.
Since the sound level difference is given in decibels, we can use the equation to solve for the ratio of intensities:
10 * log10(I₁/I₀) - 10 * log10(I₂/I₀) = 6.5.
First, let's simplify the equation:
10 * log10(I₁/I₀) - 10 * log10(I₂/I₀) = 6.5,
log10(I₁/I₀) - log10(I₂/I₀) = 0.65,
log10((I₁/I₀) / (I₂/I₀)) = 0.65.
Now, we can simplify the logarithmic expression:
(I₁/I₀) / (I₂/I₀) = 10^(0.65).
We can cancel out the I₀ terms:
(I₁/I₂) = 10^(0.65).
Therefore, the ratio r1/r2 is equal to the square root of 10 raised to the power of 0.65:
r1/r2 = √(10^0.65).