What Is the Significance of the Constant c in Wave Equations?

[davidge]

For systems whose motion is discribed by the wave equation

$$ \bigg(\frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \vec{\nabla^2} \bigg)u \big(\vec{x},t \big) = 0$$ $c$ is the speed of light. It corresponds to different quantities depending on what the system under consideretion is. For instance, for a simple vibrating string, $c = \sqrt{T / \rho}$ where $T$ is the tension and $\rho$ is the mass density per unit length.

My question is, What is the meaning of the ratio $T / \rho = c^2$? Maybe, tension propagates at the speed of light throughout the string?

 

[NFuller]

$c$ is the speed of light.

Not necessarily, c is the speed of whatever wave you are dealing with.

 

[davidge]

Not necessarily, c is the speed of whatever wave you are dealing with.

Oh, I forgot about that. Thanks.

But, still, could the ratio $T / \rho = v^2$ be interpreted as the tension propagating across the string at the speed $v$?

 

[NFuller]

But, still, could the ratio T/ρ=v2T/ρ=v2T / \rho = v^2 be interpreted as the tension propagating across the string at the speed vvv?

$v$ is the speed at which a disturbance in the string would travel. If you plucked the string, the distortion in it's shape would travel down the string at this speed.

 

[Preston Leggett]

c is speed of light that has a numerical value of 3*10^8 m/s.

 

[weirdoguy]

c is speed of light that has a numerical value of 3*10^8 m/s.

Did you even read OP and other posts? c does have more general meaning in the context of OP.

 

[rumborak]

If I understand the OP post correctly, he is trying to use the string derivation of the wave equation to glean insight into the "cause" of c, i.e. apply that insight into space-time.
I don't think you can. I think you just have to take the 1/c^2 as a factor that, while eventually causing the wave speed, can have very different and unrelated derivations.

 

[davidge]

Thank you all