- #1

thomas19981

## Homework Statement

A naval towed-array sonar comprises a line of ##80## transducers, equally spaced over a total length of ##120 m##, that is towed behind a ship so that it lies in a straight line just below the surface of the water. An adjustable phase delay can be introduced electronically for each transducer, allowing the sonar beam to be steered without physically moving the array. The speed of sound in salt water may be taken to be around ##1520 ms^{-1}##.

If the transducers are used in phase at a constant frequency ##f##, estimate the angular width of the (zeroth order) sonar beam.

A phase delay ##\delta \phi## is now introduced between successive transducers. Determine how the angle ##\theta## through which the beam is steered depends upon ##\delta \phi##

Find the maximum frequency that may be used if only one diffraction order is ever to be present as the beam is scanned from ##\theta = -90º## to ##\theta = 90º##.

## Homework Equations

##n\lambda=dsin(\theta)##

##v=f\lambda##

##dsin(\theta)-dsin(\delta \phi)=n\lambda##

## The Attempt at a Solution

So for the first part I used ##n\lambda=dsin(\theta)## and set ##n=1##. ##\theta=arcsin(\frac{n\lambda}{d})## so the angular width would be ##2\theta=2arcsin(\frac{n\lambda}{d})##. Subbing in the values given and using ##v=f\lambda## gives ##2arcsin(\frac{1520*80}{120f})##. Is this the right approach.

For the second part I considered it as plane waves incident on a double slit which was at an angle ##\delta \phi##. I know that this is not a double slit but I guessed the fact that the formula for maxima for a double slit and a diffraction grating are the same it would be ok. Anyways this eventually came to be one of the "relevant equations": ##dsin(\theta)-dsin(\delta \phi)=n\lambda##. So I just rearranged this for ##\theta## which came to ##\theta=arcsin(\frac{n\lambda+dsin(\delta \phi)}{d})##.

The third part is where I get stuck. Any help would be very much appreciated.