Near and Far Field Attenuation Inverse Laws

[teknodude]

Homework Statement

near-field distance attenuation follows an inverse cube law (1/d^3), while in the far-field it follows inverse law (1/d). Prove mathematically how we arrive at 60dB/decade from the inverse distance cube relation and 20dB/decade from the inverse distance
relation.

Homework Equations

20 log (D) where D is the distance

The Attempt at a Solution

I'm pretty sure this problem is similar to control theory in converting magnitude to DB scale with 20 log (D). I can actually see the answer just by doing that; however, I'm thinking I'm missing the big picture somewhere by doing that. Mostly I can't seem to justify using the above equation.

 

[berkeman]

Homework Statement

near-field distance attenuation follows an inverse cube law (1/d^3), while in the far-field it follows inverse law (1/d). Prove mathematically how we arrive at 60dB/decade from the inverse distance cube relation and 20dB/decade from the inverse distance
relation.

Homework Equations

20 log (D) where D is the distance

The Attempt at a Solution

I'm pretty sure this problem is similar to control theory in converting magnitude to DB scale with 20 log (D). I can actually see the answer just by doing that; however, I'm thinking I'm missing the big picture somewhere by doing that. Mostly I can't seem to justify using the above equation.

What is log(1/d^3) ?

 

[teknodude]

Maybe I was thinking too much and thought there was more to the proof than just doing 20log(1/d^3) and 20 log (1/d). Thanks.

 

[berkeman]

Glad to help.

 

[paranormal]

As a scientist, it is important to understand the underlying principles and assumptions in any mathematical derivation. In this case, the inverse cube and inverse distance laws are used to describe the relationship between the intensity of a source and its distance from an observer. These laws are based on the assumption that the source is a point source, and the intensity of the source decreases as the distance increases due to spreading of the energy over a larger area.

To understand how we arrive at 60dB/decade from the inverse distance cube relation and 20dB/decade from the inverse distance relation, we can use the equation for sound intensity, which is given by:

I = P/A

Where I is the sound intensity, P is the power of the source, and A is the area through which the sound energy is spreading.

For a point source, the intensity at a distance d is given by:

I = P/4πd^2

Where 4πd^2 is the surface area of a sphere with radius d.

For the near-field distance attenuation, we can use the inverse cube law:

I ∝ 1/d^3

Therefore, at a distance of 2d, the intensity would be:

I' = I/(2d)^3 = I/8d^3

Taking the logarithm of both sides, we get:

log(I') = log(I) - 3log(2d)

Using the logarithmic property log(ab) = log(a) + log(b), we can rewrite this as:

log(I') = log(I) - 3log(d) - 3log(2)

Now, using the equation for sound intensity (I = P/4πd^2), we can write:

log(P') = log(P) - 2log(d) - 2log(2)

Where P' is the power at a distance of 2d.

By taking the difference between the logarithms of the power at two different distances, we can calculate the change in decibels (dB) between the two distances:

Δ dB = 20log(P'/P) = 20log(P) - 20log(P') = 20log(d) - 20log(d') = 20log(2)

This shows that for every doubling of the distance, there is a decrease of 20dB in the sound intensity. In other words, the near-field distance attenuation follows a logarithmic