Very few computer speakers are going to be under 3 inches tall. By definition, they are designed to be larger to accommodate the uses of said PC.
For example
http://www.amazon.com/dp/B00GHY5F3K/ref=psdc_172471_t2_B007772TTO
Even a small pair like this that is designed for laptops ins't small enough to meet your demand. The only thing smaller is going to be soundbars and speaker designed for use with tablets.
A soundbar might be your only option as they are made to go under the monitor but the problem with that is they are very long, far exceeding you 4 inch width limit. Another problem is that PC soundbars aren't plentiful (as full speakers are much better on desks) and the few that are around are designed to clip onto the monitor.
http://www.amazon.com/Logitech-Z205-Portable-Computer-Speaker/dp/B0043TB7OC/ref=sr_1_4?s=electronics&ie=UTF8&qid=1431671322&sr=1-4&keywords=logitech+soundbar
This is one of the cheaper models and it's still out of your price range.
Not that any of these smaller speakers are a good choice. They all sacrifice sound quality and price for size. If I were you, I would suggest that you loosen your requirements. You aren't going to find speakers in that price range and size that sound remotely good. If you can't, you might want to consider headphones.