A submarine is attempting to detect an aircraft carrier transiting the Straits of Malacca. The aircraft carrier has a source level of 90 dB. The submarine's passive sonar has a directivity index of 20 dB and a detection threshold of 15 dB. Biological noise in the Straits is 54 dB. The submarine's self noise is 50 dB. Given that TL = 10 log r, where r is the range in yards (we will show you where this comes from soon), at what range can the carrier just be detected?

7.94 yards

Explanation:

Given:

Source level = 90 dB

Directivity index = 20 dB

Detection threshold = 15 dB

Biological noise in the Straits = 54 dB

submarine's self noise = 50 dB

TL = 10 log r

where r is the range in yards

Now,

Passive sonar equation is given as:

Detection threshold = Source level - TL - (Noise level - Directivity index)

Also,

Noise level = Biological noise in the Straits + submarine's self noise

or

Noise level = 54 dB + 50 dB = 104 dB

Thus, we have

15 dB = 90 dB - TL - (104 dB - 20 dB)

or

TL = 9 dB

also, we have

TL = 10 * log r

thus,

9 = 10 * log r

or

log r = 0.9

taking anti log both sides, we get

r = 7.94 yards