He is white, in his early 20s, about 5ft 9ins and slim. He has short, dark, spiky hair and was wearing a grey cardigan with buttons, loose jeans and was carrying a white shoulder bag that may have had a Gola logo on it.
Detective Inspector David Storey said: "This has been a terrible experience for the girl involved. Although she escaped any injury, she has been understandably shaken as a result.
"She has, however, been able to provide us with a good description of the suspect and so I would appeal for anyone who may recognise him to get in touch.
"Our inquiries are at an early stage but we believe the suspect may have also got off the bus at Ratho behind the girl. So we are keen to hear from anyone who may have been on the number 12 from Edinburgh and who has seen this man to contact us."