Training your dog to wave is an advanced trick that is learned more quickly if the foundation of a solid "shake" exists. If your dog already knows how to shake you are ready to begin, but if not, begin by training your dog how to Shake.
Teaching your dog how to wave can be achieved by following these 5 steps. Remember the two most important factors in successfully training your dog are to make sure you have their attention, and to always use positive reinforcement upon getting the desired results.
Make sure your dog is focused. Get your dog's attention via a piece of food, snapping your fingers, a clicker, or just calling its name. Once you have your dog's focus, you can begin training.
The wave involves the same movement as a shake, so begin by extending your hand out as if you were going to shake your dog's paw. Give the command "shake", and your dog should reach out to shake your hand. As the dog extends its paw, don't accept it, but rather stop short and allow your dog to continue trying to shake. This will cause your dog's paw to search around looking for your hand, but its also the foundation of the "wave" motion.
Get your dog to understand the command "wave". As your dog reaches his paw out as if to shake your hand (reference step 2), give the command "wave". Let your dog know this is the exact motion you are requesting through the use of positive reinforcement ("good", petting, treat, and/or whatever makes him/her happiest)
Use praise and rewards to reinforce the command "wave". Each time your dog starts to "wave" reward him or her with its favorite treat to help repeat the behavior, or if the dog likes games and playing, play more with the dog after each time it performs the "wave". Remember it's important to associate positive reinforcement with the command, "wave".
Repetition is the key to success. Doing this activity several times during the day will help form a pattern and help make the dog know when to "wave" upon request. It's an advanced trick, so make sure you practice, and are patient with your Dog not immediately understanding the command.