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Scientists have turned their attention to investigating that most annoying of human habits - the sound made when you crack your knuckles.

The characteristic pop can be explained by three mathematical equations, say researchers in the US and France.

Their model confirms the idea that the cracking sound is due to tiny bubbles collapsing in the fluid of the joint as the pressure changes.

Surprisingly, perhaps, the phenomenon has been debated for around a century.

Science student Vineeth Chandran Suja was cracking his knuckles in class in France when he decided to investigate.

He developed a series of equations with his lecturer, Dr Abdul Barakat of École polytechnique, to explain the typical sound that accompanies the release of the joint between the fingers and the hand bones.

"The first equation describes the pressure variations inside our joint when we crack our knuckles," he told BBC News.

"The second equation is a well-known equation which describes the size variations of bubbles in response to pressure variations.

"And the third equation that we wrote down was coupling the size variation of the bubbles to ones that produce sounds."

The equations make up a complete mathematical model that describes the sound of knuckle cracking, said Chandran Suja, who is now a postgraduate student at Stanford University in California.

"When we crack our knuckles we're actually pulling apart our joints," he explained. "And when we do that the pressure goes down. Bubbles appear in the fluid, which is lubricating the joint - the synovial fluid.

"During the process of knuckle cracking there are pressure variations in the joint which causes the size of the bubbles to fluctuate extremely fast, and this leads to sound, which we associate with knuckle cracking."

Conflicting theories

The model neatly bridges two conflicting theories. The idea that a collapse of bubbles causes the cracking sound was first put forward in 1971.

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This was challenged 40 years later when new experiments showed that bubbles persist in the fluid long after knuckles have been cracked.

The new mathematical model appears to resolve this by showing that only a partial collapse of the bubbles is needed to produce the sound. Thus, tiny bubbles can hang around in the joint fluid after the knuckle has been cracked.

The study, published in the journal Scientific Reports, shows that the pressure generated by the collapse of the bubbles produces acoustic waves that can be predicted mathematically, as well as measured experimentally in three volunteers.

It also confirms why some people are unable to crack their knuckles. If you have a large space between the bones in the knuckles, the pressure in the fluid does not drop low enough to trigger the sound.

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