The whine you are hearing is the energiser charging up a capacitor. It dumps charge into the capaciter in pulses to build up a high voltage charge and the frequency of the pulses generates the acoustic whine. The charge being dumped generates a mechanical force on an internal component which is vibrating. Shouldn't really happen and possibly suggests all isn't right inside the energiser). The capacitor discharges once it reaches a high enough voltage and fires a circuit (the clicking noise) which generates a hi voltage pulse (or spike) which is connected to the net. The Net isn't continuously at a high voltage it is pulsed with a very high voltage at a rate you can often set on some energisers to slow, fast and economy.
I assume you have a tester and if so you should proceed as follows.
1 Test the energiser. Connect the battery as normal, Check the earth wire is securely connected to the earth stake and the earth stake securely driven into the earth which must be wet.
2 Disconnect the hi voltage connector (often a crocodile clip from the net and connect your tester to the hi voltage wire directly from the energiser with no net attached. If the tester has an earth, connect that to the earth rod. Turn on the energier and take a voltage reading. It should be very high (check energiser documentation for no load voltage spec) as there will be no shorting to limit it. If it is the energiser is fine.
Test the net for shorts,
3 Connect ONE net to the energiser and again measure the voltage. If it has dropped significantly it indicates it is shorting somewhere. Check the net at each insulator pole and in particular check the the first live strand (second one from the bottom isn't wrapped round the metal prong of the pole. If it is move it up and tape it up with some insulating tape so it can't droop. Go all the way round the net checking for the First conductor strand touching the ground or being wrapped round a post prong or touching a heavy growth of grass or weeds.
4 Check that where the net is gathered up it is connected together OK.
5 Retest the volatge which if you have cleared the problem should be again high. Your energuiser instructions should say how high but anything of 3500 volts upwards should OK.
Repeat 3 to 5 with the second net connected in series.
Possible problems are.
The battery maybe an issue but I doubt it. The earlier advice about car batteries and leisure batteries seems like nonsence to me. Both are lead acid batteries which when chared will behave the same the difference is their discharge characteristics. Car batteries are built to withstand complete discharge their internal cells distort and they are quickly destroyer when used on nets. Leisure batteries have kevlar or other reinforcing and can stand being completely discharged which is why you should use them for electric fence applications.
It's possible the energiser has damp inside which will prevent the reservoir capacitor charging or cause internal shorting of the high voltage pulse and drying may help. Energisers should be designed for outside use and shouldn't leak and if they do they should be sprayed with a high voltage lacquer to prevent damp effecting their components. So if this is happening its a design or a manufacturing problem and your guarentee should cover you. Covering the energiser from rain can help but ensure it's well ventillated and dries so it doesn't get damp. An old plastic bucket might help ventillation rather than a plastic bag. Neither should be necessary really.
If there's a click from the earth rod it suggests a poor or broken earth connection. Check the green wire from the energiser to the earth rod and that it's securely screwed to the rod and the rod firmly in the ground.
As someone suggests observing any flashing at night (stop smurking there!) will locate a short to earth.
It's possible there's a broken net strand and I would suggest you short all the conducting strands together half way round the net so if there are two breaks it shouldn't isolate a strand. Strip the copper wire from some scrap mains cable and at the mid point of the net wrap it round each conductor several times (3+) and drop it vertically down to the next conducting strand and repeat until all conduting strands have been connected. Snip off any excess. This will help bypass any breaks. You can repair breaks by twisting copper wire around the broken strand too. Easier than trying to use the useles repair kit often provided. Make sure the errected net is in straight runs and nice and taut. Use wooden tree stakes to tie the insulating posts to to anchor the corners and changes of direction, Use nylon or bailer twine to tie the insulator posts to the tree stakes so none of the conductors touch the stake.
Best of luck
HF