His name is Mud but he doesn't mind.

As Mud Morganfield he has his father's real surname and a contraction of the name the world came to know his father by, Muddy Waters, and he's proud to be following his father's example as a blues singer.

It wasn't always this way. Growing up in Chicago as Larry Williams – he took his mother's surname because his father was away working on the road so much – Mud wouldn't have impressed his peers by telling them what his father did for a living. They weren't into the blues and wouldn't have recognised Muddy Waters as the legend that he was. Even Mud himself wasn't a blues fan; as a teenager and into his 20s he was more interested in the Temptations, Hall & Oates ("I absolutely loved that I Can't Go for That song of theirs") or his all-time favourite, the Walrus of Love himself, Barry White.

"My dad wasn't around much because he was always working, trying to make sure we had a roof over our head, clothes to wear and food on the table," he says. "But when he was at home he'd have people like Otis Spann [Muddy Waters's long-time pianist] come round and play. I just thought Otis was Pops's pal, you know, the guy in his band. I tell people now about him coming round to our house back then and from their responses I realise what I was taking for granted. I'd no real idea at the time what the music they were playing was about."

Before moving to Chicago and becoming a central figure in the development of the city's blues style, as well as one of the most vital cogs in the Chess Records wheel, Muddy Waters had lived and worked on a Mississippi plantation, growing cotton and dealing illicit whiskey on the side. It was a hard life for the man known at the time as McKinley Morganfield. But for his son, life in the city wasn't so much easier.

"I've been down to Clarksdale and the Stovall plantation where Pops worked the fields and I've seen the shack he lived in – it's in a museum now – and I tell you, it's heart-breaking," says Mud. "But the west side of Chicago was no picnic, either. I didn't come up in no mansion house. Pops couldn't afford one of those. We lived two blocks away from a place known as the Bucket of Blood, and that name will tell you everything you need to know about the kind of things that went on there. I'd fall asleep at night with gunshots going off and the sound of the Chicago fire and police departments going about their business."

Sometimes when his father was playing locally his mother would take Mud and, later, his siblings – he's oldest of the family – to the clubs he would play in. Mud knew there was something special going on. He could see the effect his father was having on people and he may well have absorbed some of the power his father could hold over an audience. His earliest memories are of hearing music and although he had no thoughts of following his father into the blues as a profession, he remembers being determined to play music.

"I was going to be a drummer," he says. "From before I went to school I can remember playing along to the radio or the records that were always on in the house. Usually it was on my mother's best table and I remember getting a few punches for that. Eventually Pops bought me a little drum kit and I used to whack away on that. But it was the bass guitar that got me into playing music seriously, probably as a result of the soul music that I was listening to. That became my instrument."

It still is. It's the bass guitar he uses to work on new songs, although he doesn't play it on stage due, he says, to those pesky music critics, who are always looking for a "but".

"You know these guys, they'll say: 'Yeah, he sings good but he's a lousy bass player'. Or: 'He sings OK but he can't play the harp,'" he says with a laugh. "The way I see it, there's nothing you can do that's new in music. It's all recycled. If you play the harp, chances are that Little Walter did whatever you're trying to do before you even started. So I keep the bass for my private time. That way, I still get to enjoy playing it just for myself."

When Muddy Waters died in 1983, Mud and his younger brother Big Bill Morganfield entertained the notion of becoming professional musicians and paying homage to their father. Big Bill is out on the road working as we speak but for Mud it wasn't until 2007 that he felt ready to pick up the baton and perhaps leave himself open to accusations of cashing in on his father's legacy or worse, invite unfavourable comparisons.

Physically, the family resemblance is pretty clear and as his latest album, Son of the Seventh Son, illustrates, he can sound uncannily like his father – to the extent that people have asked him if there's some sort of spiritual connection going on during his concerts.

"I do feel close to my father when I sing," he says. "But it's a natural feeling. I'm 57. I've got me a bunch of blues. I've been blessed in that I look like Pops and I sound like him, and I can live with that. There's no way you can copy what my dad had. You've either got it or you don't and I'm just trying to be part of something that he spent his whole life working on. I do my own stuff; I look for a strong message in my lyrics, that's the most import thing for me. And I sing some of Pops's songs. So I've got the best of both worlds; I can be me and at the same time, I can honour my father, and that's a special feeling."

Mud Morganfield plays Darvel Town Hall on Friday as part of the town's annual music festival. For further information visit www.darvelmusicfestival.org.