Dishwasher drain air gap

I've just replaced our 20 yr-old dishwasher with a new one.( I did NOT go for the one with wi-fi hookup that can be controlled with a smart phone.)
The old unit worked just fine but the new ones are very much quieter. However, we are getting a smell from the drain into the new unit and I see that the manual recommends to install an air gap. The drain for the old unit went directly into an inlet just above the trap under the sink and worked just fine. I hooked up the new unit to the same inlet and I get a smell. After reading about air gaps I see that they want the 1/2 inch dishwasher drain to enter the air gap and leave in a 3/4 inch hose that then enters the kitchen sink drain. I suspect that this new dishwasher has a more powerful pump, and that this together with the fact that my old 1/2 inch drain narrows down to a 3/8 inch tube before it enters the kitchen sink drain, contributes to the problem.
The manual suggests, alternatively, a high loop (at least 32 inches above the floor) could be used instead of the air gap, so that is what I chose. My high loop is 33 1/2 inches above the floor but it is not totally effective. There is some relief from the drain smell but I want it all gone.
My question: is there a better alternative as I really do not have counter space for an ugly 'air gap' unit?

Purxhase an air gap and epoxy it to the underside of the sink cabinet. (5 minute epoxy will work well. This should do the trick without drilling a counter top hole. This allows the down movement of the final few
inches of the washer drain tube to flow from gravity into the connection. Running a pint of water down the drain after using the dish washer also
rinses the P trap of food waste. The picture i inserted would be changed so the gap was under the sink top. The only issue would be a plugged drain at the P trap would spill water out the air gap if the water in the plugged sink got to the height of the air gap. If it did there would be water everywhere any way as the sink flooded. Jim

Where or why does the drain line narrow down to 3/8"?

That sounds too small.

The air gap, all I would think that is for is to prevent siphoning if the machine drain is connected directly to the drain line. Since the drain is connected above the trap it shouldn't need an air gap.

If you can you connect the drain line directly to the sink, above the trap, it should not stink.

If there is a disposal, maybe that is what is smelling.

Are you saying the sink drain smells or the dishwasher smells?

Most newer dishwashers have a filter that must be cleaned after each use. They are very quiet because there's no grinder in the pump, but if you don't pre-rinse your dishes there will be a lot of food stuck in the filter. Look for a cylindrical basket in the bottom of the washer; they usually twist out.

Air gaps on a drain can be problematic. When the code changed to require them we had to help customers through a lot of problems with them. Your particular problem is your 3/8 tube. It increases the velocity of the water flow so that it is causing the kitchen sink trap to be nearly emptied of water allowing sewer gases to come up into your kitchen. You probably need a new "Wye" fitting to accommodate the 1/2 inch drain. I wouldn't install an air gap. You will eventually have a flood.

x2

I just had to replace my air gap for my dishwasher. It looks just like the one pictured by janicholson. 2 different size hoses, and I had to go to 3 different hardware stores to find one. The old one leaked from above and I found water under the sink. No leaks now...