A couple and their two children were walking along Cathedral Walk, Chelmsford, at 2.30pm on Monday July 11 when a man sitting on a bench near the cathedral started throwing coins and pebbles at them.
When challenged, he made a racially abusive comment.
The suspect was described as white, aged about 50, with and dark blond hair. He wore a red T-shirt and glasses. He walked in the direction of Waterloo Lane.
Investigating officer Pc Alun Causaker said: “This was a disgraceful and unprovoked attack on a couple and their children.
“This happened during a busy time of day and lots of people were in the area.
“If you saw anything, please help me identify this man.”
Witnesses or anyone with information are asked to contact Pc Alun Causaker at Chelmsford police station on 101.