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.