The Stagecoach vehicle was carrying passengers when it was hit in Perth last week.
The window was smashed at about 4pm on Tuesday February 5 when the bus was making its way along Western Avenue, near Perth Royal Infirmary, towards Rose Crescent.
Tayside Police said the bus passengers may have seen those responsible, who have been described as two children aged between five and 12.
It is not yet clear which window was hit or how many people were on the bus at the time of the incident.
Officers are appealing for witnesses and have asked anyone with information about what happened to get in touch.
In a separate incident, Tayside officers are also working to trace two teenagers who smashed a lorry window, injuring the driver.
The 38-year-old behind the wheel of a gritting lorry had a "lucky escape" after the pair shattered his cab window, police said.
The man was gritting in Dundee's Lochee Road, adjacent to Dudhope Park, at about 7.30pm last night when two youths standing on a nearby footpath threw something at his cab.
The window glass shattered and cut the driver's cheek.
The two, who fled after the incident, are described as being aged around 14. One was wearing a white jacket.
A force spokeswoman said: "The implications of throwing any type of missile at a moving vehicle can be very serious indeed for anyone within that vehicle, or anyone nearby for that matter.
"Those who get involved in this reckless behaviour may view it as a bit of harmless fun, but they should consider the potential consequences of their actions as they have the potential to cause serious injury.
"Anyone with any information to assist police with their inquiries regarding these incidents should contact Tayside Police on 0300 111 2222."