ScotRail services between Edinburgh and Milngavie via Airdrie are affected as a result and may be delayed or revised.
The transport firm said disruption is expected until at least 10:00.
Ticket acceptance is in place on First Scotland East bus services between Edinburgh and Bathgate and also Edinburgh Trams services services between Edinburgh and Edinburgh Park.
ScotRail said on their website: “The points failure at Edinburgh Waverley has re-occurred and our staff are returning to site to investigate. Services into and out of Edinburgh will be subject to disruption.”
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