Need help with rewriting log function


I need some assistance with rewriting a log function.

Please see attached Word document with details/background, incl. log functions.

Thank you for your help in advance.

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d-glitchConnect With a Mentor Commented:
>> The value for "r" -- cylindrical spreading -- appears to be incorrect.
     You have =10^((10^(I_S/I_D))).     It should be =(10^(I_S/I_D))

The detection threshold is apt to be much lower, but it will depend on the equipment (hydrophone, amplifier, etc) you are using.

For example, this is a toy (you can tell because it doesn't list the right sort of specs), but it might have a detection threshold of -10dB.

And if your data is in dB, then you subtract rather than dividing.

So for spherical spreading of an 80 dB signal, you could observe it with -10dB hydrophone at
          r = sqrt[ 10^(80-(-10))/10)  =  sqrt( 10^9)  =  31 km  =  19.6 miles

This completely ignores the issue of signal to noise.
The document just has formulas.
And the calculations are wrong.

  20*log10(1000) is 60  (not 59.223)
  10*log10(1000) is 30  (not 29.61)

To calculate detection range, you need to know the source level and the  detection threshold.

The units will be some form of power per area.
Are you talking about sound or RF or light?
I am unclear as to what you want. You said solve for r but you give r as 1000.
In any case r = I/Io and
r = 0.1 antilog (I/Io)
9the max range depends on the sensitivity of the equipment.)
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ExpExchHelpAuthor Commented:

You're correct... the functions in my XLS converts yards to meters (spreading functions requires meters vs. yards).   Anyhow, I looked at the wrong cells (i.e., 1000 yards)... so, again, you're correct w/ respect to "60" and "30".

"To calculate detection range, you need to know the source level and the  detection threshold"... here are my questions:

- Could you please elaborate on the thresholds?   What input parameters do I need to detect MaxRange?
- WRT frequency, this problem is based on sound (transmission) in seawater.

I welcome any ideas (incl. required inputs) for calculating MaxRange.



Thanks for the additional feedback.   Yes, I may have not been very clear.   I may not always have "r" but know the dB level.   So, I wanted to find out about "reverse math"... if I know the dB level, can I calculate "r".    Ideally, "MaxRange".

I hope this makes sense.

How loud is the sound you are trying to observe?
A pin drop, a fire cracker, a car engine, a human voice, ...

If you know what the sound is, you can look up or measure the intensity.

What are you using to listen with?
Your ear, an audio amplifier and headphones, a parabolic microphone, an audio DSP computer board, ....

Each of these will have a detection threshold you can look up or measure.

What is the noise environment like?  
Wind, rain, traffic, WWIII, ...

All the details matter.
>>  Your ear, an audio amplifier and headphones, a parabolic microphone, an audio DSP
      computer board, ....

   Each of these will have a detection threshold and/or gain you can look up or measure.

There is a Typical Sound Level Chart here:
r is not the range
if you you want the range you need to know the initial strength (Io), The max sensitivity of your equip (I) and the attenuation, that is the reduction in signal strength per unit distance.
I think you equations assumed that the signal was reduced because of spreading either spherical of cylindrical.  The attenuation of sea water was not included and is not zero.
ExpExchHelpAuthor Commented:
d-glitch, aburr:

Thanks... initial strength (Io) may vary between 65-100 dB.  

Basically, I'm trying to work on a model (submarine listening to "target signals"... e.g., other ships on surface or other submarines).

WTR to attentuation of seawater, I'm using Mackenzie's equation (based on temp, depth, and salinity).

I welcome suggestions for calculating the MaxRange (how far sound can travel given values from other inputs).   So, right now, I'm just trying to come up w/ a formula.

Are you talking about active or passive sonar?

I did some sonar work back in the 70's, and I may still have the book I used at home.  
I will check.
ExpExchHelpAuthor Commented:
d-glitch -- passive sonar.

And, yes, I already looked at this site.

Question remains... how do I rewrite the equation to solve for lower "r"?

d-glitchConnect With a Mentor Commented:
You need to know your source intensity:        I_S
You need to know your detector threshold:   I_D

If the sound you are trying to observe is loud,  I_S will be large.
If your detection equipment is sensitive, I_D will be small.

The ratio of the two I_S/I_D tells you how much loss you can tolerate.

For spherical spreading:
              20 log( r) =  10  log( I_S/I_D)
                    log( r) =  1/2 log( I_S/I_D)
                            r  =   sqrt[ 10^(I_S/I_D)]

For cylindrical spreading:
              10 log( r) =  10  log( I_S/I_D)
                    log( r) =        log( I_S/I_D)
                            r  =        10^(I_S/I_D)
ExpExchHelpAuthor Commented:
d-glitch-- excellent... thank you for this info.

I've entered some sample data (for I_S and I_D) into a spreadsheet (see attached).

The value for "r" -- cylindrical spreading -- appears to be incorrect.   Should this have been, e.g., 10*(I_S/I_D)... vs. 10^(I_S/I_D)?

Also, can I calculate the max range where sound can be observed?

Again, thank you for your assistance!!

ExpExchHelpAuthor Commented:
Thank you for your help on this question.
ExpExchHelpAuthor Commented:

Quick follow-up question... based on the information listed on


... the rewriting of the equation to solve for "r", I'm using the following example values:

I_S = 80 db
I_D = 65 db

For spherical spreading, the Excel function will be: =SQRT((10^(I_S/I_D))) = 4.125
For cylindrical spreading, the Excel function will be: =(10^(I_S/I_D)) = 17.013

My question:  What's the unit of measurement?   Miles?   Nautical miles?

Maybe I'm not fully tracking where the "-10dB hydrophone" comes into play.

How far away can I hear a sound?  is actually three or four questions.

How loud is the sound?
How does it spread?
How faint a sound can I hear?
How noisy is the environment?

Look at this passive sonar example:

Look at the value of the Source Level in particular:
        SL =  195 dB re . . .
A value in dB is a ratio, not an absolute value.  
Since the reference here is in meters, the value for range will also be in meters.

What is reference for your value of Source Level:  I_S = 80 dB  ?
ExpExchHelpAuthor Commented:

Thank you for your continued support on this question... I appreciate it.

I think you're right about the proposed 4 questions.    For the current state of this research, I probably want to consider only #1 and #2.  

At this time, I'm selecting arbitrary values for testing the calculations.    

Assuming, the two example dB values (80 and 65) are valid, I'm trying to get a better understanding on the radius.    Although I have limited knowledge on this subject (i.e., sound/sonar), it seems odd that the radius would only be a few meters.

Are potentially other inputs missing?    So, if a submarine picks up a signal from a ship (on surface), I'm trying to figure out how far the sound from the "target"  (at X level) would travel before the sub cannot hear it any longer.

Makes sense?
Here are some additional data and some final sample calculations.

I thought your 65 dB number for the threshold of hearing was way off.
But it turns out the hearing in the air and under the water are measured differently, and the threshold in water is something like 62 dB higher than in air.

Wikipedia actually lists a value of 67 dB re 1 µPa.

This does make the value you used for the Source Level  SL=80 dB somewhat suspect.
This chart suggests that 80 dB is the background level due to wind, waves, and distant ocean traffic.

This site has more detailed information on sources:

Page 314 in particular:  
Small craft and boats and medium-sized vessels (e.g., recreational craft, jet skis, speed boats, operational work boats, hovercraft, support and supply ships, many research vessels, fishing vessels) produce source levels of approximately 160 to 180 dB re: 1µPa, depending on speed and other operational characteristics.

So, if you assume a small speed boat [SL=160 dB] with transmission loss due to spherical spreading [TL=20*Log(r)] and a detection threshold DT=67 dB and solve for the maximum range:

     DT  =  SL - TL    ==>    TL  =  SL - DT  =  160 - 67  =  93 dB

     20*Log(r) = 93    ==>  r  =  10^(93/20)  =  10^(4.65)  =  44.7 km  =  27.8 miles

But this is ignoring the ambient noise.

If the signal is not larger than the background noise, you won't be able to detect it.

If you require a Signal-to-Noise Ratio SNR=6 db and assume a noise level NL=80 dB, then sonar equation becomes:

     NL + SNR  =  SL - TL    ==>    TL  =  SL - (NL + SNR)  =  160 - 86  =  74 dB

     20*Log(r) = 74    ==>  r  =  10^(74/20)  =  10^(3.70)  =  5.01 km  =  3.1 miles
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