Why 3 Hz? - MV Audio Labs

We already know, that our cartridge is velocity sensitive device, and this velocity is proportional to groove amplitude and angular frequency:

$v_{L}=A\, \omega_{L}=A\, 2\Pi f$

Here we are talking about groove velocity of stylus so let’s call it lateral velocity V_L. We consider only the case of constant amplitude static frequency tone. Then groove amplitude is always constant too.

$A=const$

But our groove lives on a record, which is also spinning and has it’s own velocity.

Let’s call this record velocity tangential V_T. It is proportional to groove radius and angular frequency of a record.

$v_{T}=r\, \omega_{T}$

Tangential velocity is always decreasing, because radius of the groove is shrinking when record is being played. Now comes the important part. Because LP record is constant velocity, ratio between tangential and lateral frequencies is always the same!

$\frac{\omega _{T}}{\omega_{L}}=const$

Then it becomes obvious, that ratio between tangential and lateral velocities is only function of groove radius.

$\frac{v _{T}}{v_{L}}=f(r)$

It means that every variation in tangential speed, because of radius or tangential frequency change, will produce proportional increase in lateral velocity. Groove amplitude can’t change, it’s already there pressed as is, so only thing that can change is frequency of our tone. Now that we established this, our problem becomes quite trivial. Our track lies 96mm from record center. It’s tangential frequency is:

$\omega_{T} =2\,\pi \, f=2*3.14*0.55=3.456\, rad/s$

Then it’s tangential velocity:

$v_{T1}=r_1\, \omega_{T}=96*3.456=331.752\, mm/s$

When we have center hole 0.5mm off center, on one side of the record radius becomes 96.5mm, and tangential velocity increases to:

$v_{T1}=r_1\, \omega_{T}=96.5*3.456=333.504\, mm/s$

On the other side of the record, radius becomes 95.5mm, and tangential velocity decreases to:

$v_{T1}=r_1\, \omega_{T}=95.5*3.456=330.048\, mm/s$

That’s a ratio of:

$\frac{r_{2}}{r_{1}}=\frac{96.5}{95.5}=1.0105\: =\: 1.05\, \%$

Which from 300Hz is:

$\frac{300*1.05}{100}=3.15\, Hz \approx3\, Hz$

From another perspective, that’s a wow of 0.5%.

So visual tone sharpness inspection can give us crude stability estimation, but that’s about it. If we want more meaningful and accurate information, we have to go down the rabbit hole.