Outside my office window, with a background roar of rushing water, the weir at Teddington Lock churns the Thames into a creamy grey foam. Just beyond, over on the opposite bank and half hidden by foliage, a small obelisk marks the point where the Environment Agency hands over responsibility for the river to the Port of London Authority. It's the beginning of London's River, and the starting point for a sonic exploration of our capital's great artery by American sound artist Bill Fontana, which he’s developed in the subterranean lightwells of Somerset House.

The late-18th century, neo-classical former tax office on the Strand once lay alongside the river, its great arch the embarking point for Navy barges heading down river to inspect the dockyards. The construction in 1864 of Joseph Bazalgette's sewer and the Victoria Embankment above it left Somerset House marooned; it now stands separated from the Thames by racing traffic. Fontana has returned the river to the building: the great arch once again echoes to the sound of lapping water.

Most of the work – supported by Somerset House Trust and one of the building's resident organisations, Sound and Music – is installed in the alley-like lightwells, where the low-ranking clerks once sat looking across a metre of gloom at the coal holes opposite. Speakers now project the sound of water, ships' horns, bells and engines around these forgotten corners, while the coal holes house related film projections. It all conjures up the spirit of a passageway crammed between the warehouses of Rotherhithe or Wapping during the heyday of London's ports.

This is an intimate portrait of the river: the Thames, up close and personal, we – and it – caught off guard. One film shows the dirty brown water glimpsed through the gap in the road midpoint over Tower Bridge. Another, the churned foam at Teddington Lock with which we began. And when we do see something familiar – traffic and crowds crossing Tower Bridge – we don't hear the cars or conversations, but the internal sound of the structure as it responds to the vibrations, captured using an accelerometer. This technique was also used to record the structural responses of HMS Belfast's hull to the passing river.

Other sounds are taken from the mighty wheel at Kew Bridge Steam Museum, from water lapping on the riverbank, and from many bells – from HMS Belfast, from the National Maritime Museum's collection, and, on the quarter hour, from the clock of Somerset House, its ring drawn by microphone down into the lightwells. The internal sound of the river itself has been captured by hydrophones suspended beneath its surface. Then at journey's end, beyond where Southend Pier projects precariously into the estuary, out where the Thames gives itself into the North Sea, we hear the lonely, haunting call of buoys, captured in the fog.

These disembodied, eloquent, eerie sounds are not our aural impressions of the Thames – they are how the river might hear things, if it were alive. But as it is, we might think of the Thames as a river of ghosts. Of the victims of its worst disaster, when the Princess Alice sunk in 1878 drowning 600, or of the 58 residents of Canvey Island claimed by its waters in the 1953 flood, or of the 17th-century gentleman usher whose tomb we encounter in a passageway buried beneath Somerset House's courtyard. But also the ghosts of abandoned warehouses, relics of obsolete industries, that line the river’s banks, or of the once mighty, now emasculated, tributaries – the Fleet, the Westbourne, the Tyburn – now reduced to trickling pathetically out of their outlets in the embankment walls. Perhaps this is what the buoys’ loyal, mournful toll is for.

I found it all strangely moving. Perhaps it's a sign that, after a decade in the city, both living and working moments from the brooding waters of our great river, I've finally become a Londoner.

River Sounding by Bill Fontana is at Somerset House until May 31. Admission is free.