I successfully identified my problem. I will post the information here in case anyone else has this same issue and runs across this thread.
The company I work for has a PC technician who comes by our office to work on our computers. I told him about my situation with this noisy external drive. He told me that it was probably not the hard drive itself. He said that noise usually comes from the fan.
I called the manufacturer tech support line. I did not tell them about my conversation with my PC technician. I asked the manufacturer rep what he thought about the noise coming from my drive. He said that the majority of time when they encounter this problem it is the fan......NOT the drive itself.
They both said that there is a fast, easy way to test this. So here it is.
1 - turn off the drive
2 - use something to block the fan from turning(pencil, toothpick, etc.)
3 - turn on the drive. The fan will not spin because you are blocking it
4 - if the noisy sound is gone, then you know the problem has to be the fan and not anything else
When I blocked the fan then the noise stopped completely.
Now that I have pinpointed the problem, I have to decide my next course of action. I have two choices.
OPTION 1 - send the drive back to the manufacturer and file a warranty claim. Let them replace the fan. This would require me to copy the content off of the drive onto some other storage device. After they receive that faulty drive they will send me a replacement.... notice that I said "a replacement" drive -- not necessarily a brand new drive. They will probably me one of their demo units or refurb units. Honestly, that is fine with me. Whatever they send me will still be covered under the original warranty that was left over from the first thing that I bought.
OPTION 2 - I could open the bad hard drive and try to replace the fan myself. There are some pros and cons with that. Pros--I would not have to worry about offloading the content from my drive onto some other device, and I would not have to hassle with a return to the manufacturer, and I would not have to wait, etc. The main "con" is that opening the drive would void the manufacturers warranty. The other con is that I would have to find a replacement fan that would work and I would have to pay for it out of my own pocket.
Not sure what I should do here, but to make a long story short...... when your hard drive is making noise then check the fan first.