Introduction to Acoustics

a bubbly medium (and for the simple case of singlesized
bubbles), an acoustic beam will be altered by the
absorption and scattering out of the beam. For an incident
plane wave of intensity I0, the power removed by
each bubble is I0σe so that the rate of change of intensity
as the beam travels through a bubble medium of N
5.3 SONAR and the SONAR Equation
A major application of underwater acoustics is SONAR
system technology. The performance of SONAR is often
approximately described by the SONAR equation. The
methodology of the SONAR equation is analogous to an
accounting procedure involving acoustic signal, interference
and system characteristics. Figure 5.18 provides
a schematic of passive and active SONARs.
5.3.1 Detection Threshold and Receiver
Operating Characteristics Curves
The detection threshold (DT) [5.16] is a decibel number
that essentially incorporates the SONAR system’s
(which includes the operator) ability to decide that a detection
is made or not made. The detection process
includes the following probabilities:
• the probability of detection (PD): the probability that
a signal is detected if it is present;
• 1-PD: the probability the signal will not be detected
if it is present;
• the probability of false alarm (PFA): the probability
that a signal is detected when it is not present;
• 1-PFA: the probability that the signal will not be
detected when it is not present.
In practical terms, since the signal and noise are fluctuating,
the detection is made (over a time interval)
when the fluctuating sum of the signal and noise exceeds
a threshold that is determined from empirically
derived probability density functions (PDFs) of noise
and signal plus noise. For example, the case that the
noise alone rises above the threshold contributes to the
PFA. Therefore, the process for determining a detection
threshold level will depend on the PD and PFA. Typically
numbers might be a PD of 0.5andPFA of 0.0001.
The probabilities will themselves be a function of the
relation between the signal and noise statistics, as represented
by their mean and variance. The detection index
d succinctly characterizes this relation in that it indicates
Underwater Acoustics 5.3 SONAR and the SONAR Equation 165
bubbles per unit volume is
dI
dx =−I0σeN → I = I0 exp (−σeNx) . (5.23)
Therefore, a bubbly medium changes the sound speed,
absorbs sound, and is dispersive.
how easy it is to observe a signal in noise,
d = (Msn − Mn) 2
σ 2 , (5.24)
n
where Msn is the mean of the signal plus noise, Mn is
the noise mean and σ 2 n is the noise variance. Figure 5.19
shows schematically the implications of the detection
index where the relative proximity of the two probability
density functions (PDFs) determine the detection
Passive
Active
Fig. 5.18 Passive and active SONAR for submarine detection. Passive:
the submarine on the right tries to detect sounds (blue) from
the other submarine using a towed array (antenna). These sounds
are distorted by the shallow-water environment and are embedded
in ocean surface noise (green) and surface shipping noise (red).
Active: the ship on the right sends out a pulse (red) and an echo
(blue), distorted by the shallow-water environment, is returned to
the ship SONAR which tries to distinguish it from backscattered
reverberation (yellow) and ocean noise (green) [5.1]
Part A 5.3