The 13-year-old victims were assaulted at around 5pm on Friday, July 3 at the Cramond waterfall, a generally “safe” area which is popular with families and dog-walkers.
The man, who has been described as being of Asian appearance, possibly from the Indian sub-continent, approached and then indecently assaulted the pair before walking off towards Cramond beach.
The girls reported the assault to police who have appealed for witnesses after an initial attempt to track him down proved unsuccessful.
Police said they had been following a “positive line of investigation” but have now asked for the public’s assistance to help identify the attacker.
It is understood that police thought they would be able to identify him from information they had received which included a detailed description from his victims.
The alleged attacker has been described as being around 25 years old, of large build, with short dark hair. He had a full beard and was wearing a white polo shirt, black dress trousers and black shoes.
Detective Sergeant David Brady, from Edinburgh’s Public Protection Unit, said: “This was a distressing experience for the young girls and we have been pursuing various lines of investigation over the past week to identify the suspect.
“However, we have so far been unable to establish who he is and I am keen to hear from anyone with information that can assist with our inquiry.
“In particular, we would like to speak with anyone who was walking between Cramond waterfall and Cramond beach last Friday evening and may have seen this male.”
He added: “Similarly, anyone with any further information in relation to these assaults is also asked to come forward.”
Those with information can contact Police Scotland on 101 or the charity Crimestoppers on 0800 555 111.
Lindsay Paterson, the Conservative councillor for the ward, said the community would be particularly shocked by the attack because this was generally thought of as a “safe area” and that incidents of this kind were “extremely rare”.
She said: “Something like this is very rare and that’s partly what makes it so worrying.
“Certainly, from my memory, I don’t recall a similar incident there.
“Cramond waterfall and its surrounds are very popular with walkers and dog-walkers and I think people will be very surprised and shocked that this has happened, particularly because the attack involves two young girls.
“It is very worrying that this man is still at large and I would urge anyone who might have seen anything to contact the police.”
Ms Paterson added: “Fortunately, the victims have provided quite a detailed description of their attacker and we just have to hope that the police catch the person responsible as soon as possible.”