The term directivity refers to the directionality of the source or receiving hydrophone. In the case on a monostatic system the directivity index of the source and hydrophone can be considered to be equivalent. However, in bistatic systems they must be considered separately.

Directivity index of the source (DI_T).
Let us deal with the directivity of the source first. In an active sonar system we will give the directivity of the transmitting source the term DI_T. The source level calculations we have studied previously have assumed that the sound energy from the acoustic projector spreads omnidirectionally outwards away from the source. However, in many applications it is advantageous to focus the source energy into a narrower beam in order to increase the source intensity in the direction of interest. Thus, directivity in this context is a quantitative measure of the focusing of acoustic energy by the source.

The directivity index is defined on the decibel scale as follows:

DI_T = 10 log(Intensity of the acoustic beam / Intensity of an omnidirectional source)
Equation 1

A schematic representation of the problem is given below:

The diagram shows energy from the source (S) being focused into a narrow beam of angular width a. As with the source level calculations the reference location is defined at a range (R) of 1m from the source. The radius of the circular beam here is r. For an omnidirectional source the Power of the system is distributed over an area equal to the surface area of a sphere with radius R (i.e. 4pR²). For the more directive situation illustrated above the source power is distributed over a much smaller area leading to a grater intensity (remember I = P/A). In the directive case the source power is spread over the area of a circle (i.e. pr²) of radius r as shown above. Thus, using our definition of DI_T (equation 1) we have:

DI_T = 10 log({P/pr²}/{P/4pR²})

You will notice that P and p cancel out in the above equation and that by definition R=1. Therefore we can simplify this expression to:

DI_T = 10 log(4 / r²)    (equation 2)

Using some trigonometry it can also be seen from the diagram that:

tan(a/2)  = r / R = r  (for R=1)

Therefore:

r² = tan²(a/2)     (equation 3)

Substituting equation 3 into equation 2 we have an expression for DI_T in terms of the beamwidth only:

DI_T = 10 log(4 / tan²(a/2))    (equation 4)

Some applications like side-scan sonar for example which are designed to sense long narrow strips of the seabed require that the beam pattern is more elliptical than round. An equivalent derivation for an elliptical beam gives:

DI_T = 10 log(4 / {tan(a₁/2) tan(a₂/2)} )    (equation 5)

Here a₁ and a₂ are the beamwidths of the major and minor axes of the ellipse. Another expression for the directivity of a circular piston transducer is:

DI_T = 20 log(pD / l )    (equation 6)

Where D is the piston diameter and l is the wavelength of the sound emitted.

(NB. You will remember that for passive systems the target itself is the source. Thus, the DI in this case is the directivity of the target-source. We give this the term DI_s so that it is not confused with directivity of an active system.)

Directivity of the receiving hydrophone.
There is a two-fold advantage to having a directional hydrophone:
1) The system is able to identify the bearing of the target.
2) In an isotropic noise background, much of the noise is filtered out. That is the system is uneffected by noise from directions other than the 'look direction' of the hydrophone. 

In bistatic sonar systems the source and hydrophone components are separate often having different directivities. Monostatic systems will have have similar directional properties.