My backyard boarders on a rather busy street and brings in more noise that I would prefer. There is a six foot wall but building it higher may not be practical.
What are the best methods for dealing with a problem like this???
A natural fence which can usually be allowed to grow taller than fence restrictions and fill in will deflect and buffer sound waves.
Choose something that stays green year-round.
Where I'm at a stand of aborvitae or other evergreen such as upright juniper works well.
You didn't say where you are but from your same I guess southern california. Don't know what would work in your area that wouldn't require too much water and not be a fire concern. Maybe bamboo but you'd have to build a containment system to keep it from spreading all over the yard and the neighbors!
While in the yard some pleasing white noise might also help mask the traffic noises such as a babbling brook/pond/fountain. You can get pumps that are solar driven but with evaporation/replacing water needs and drought and water restrictions plus the risk of attracting wild life to a source of water might not be an option for you.
Don't expect too much relief from the noise by planting trees , etc . I'm afraid it's not going to be very effective . You would need a DENSE , mature , forest to get any real benefit . Sorry .