The force has recently had reports of two incidents involving 12 to 16-year-old youngsters leaping into the water at Arbroath Harbour.
And they say a child could be seriously hurt or killed in the busy water port if the trend continues.
Jamie Sutton, 17, was killed after plunging 30 feet into water near his home at Whitburn, Tyne and Wear earlier this month.
And officers in Arbroath say that as schools finish up and the summer brings in better weather, more teenagers are engaging in the fad – which involves jumping from a high vantage point into open water.
Last year a youth narrowly avoided serious injury after tombstoning into Arbroath harbour and just missing a boat propeller.
A spokeswoman for Tayside Police said: “The harbour is an extremely busy place with a number of boats coming and going or being taken out of or placed back into the water.
“The young people involved are placing themselves in great danger of being injured as a result of jumping into the water and swimming in the harbour area.
“Similar incidents occurred at the harbour last year and we clearly seek to highlight the dangers involved in this kind of activity.
“We don’t want to see anyone seriously injured or even die from what is obviously regarded as a bit of fun.”
The spokeswoman said that this is the second year local inspectors have had worries over the activity – which has claimed six lives in the UK in the last two years.
She said: “Last year one youth narrowly escaped serious injury when he missed the propeller of a fishing boat.
“The next one might not be so lucky.”
The area being used by the youths is monitored by CCTV and police are appealing for parents to check if their children are involved.
The spokeswoman added: “We would urge any parents or guardians who suspects that their child is taking part in this activity to challenge them about it and highlight the dangers involved.
“Likewise, we would ask that any members of the public who may know the identity of any children involved to get in touch.”