## speed… of what (0)

I conducted some measurements about elastic wave speed.  They are expected to be ground truth against which my finite element code is tested.

Here I have got an ultrasonic transducer (bolted-Langevin type). Cylindrical shape (length: 65mm, radius: 10mm).  The first resonant frequency is 38.8KHz.

A cylindrical rod is attached to the transducer with a grab-screw.  Made of stainless, type 304.  The same radius as the transducer. Length more than 100mm. Its material properties:

• Young’s modulus (E): 197GPa,
• Poisson’s ratio ($\nu$): 0.3,
• density ($\rho$): 8000Kg/m^3.

AC voltage is applied to the transducer.  Current is measured for various AC frequencies.  When the current hits local peak, the rod ( transducer + stainless rod) is thought to be resonated.

Cutting the stainless rod to several lengths, the 2nd resonance is sought for each length.  Here are some results:

Though it seems like there are a few glitches around length=105mm, I thought these data are good enough for my initial testing purpose.

Since these frequencies are of 2nd mode, wave length $\lambda = L$ (length of transducer + stainless rod).

What about wave velocity.  For frequency f, velocity $c = \lambda f$. And here is the plot:

This says that wave velocity is not constant but depends on shape, size.  Periodic tables like webelements is implicit about this.  Some of text books  say that speed $c = \sqrt{E/\rho}$ as if it were constant regardless of media shape. More elaborate ones say speed of p-wave $c_p = \sqrt{(\lambda + 2\mu)/\rho}$ and speed of s-wave $c_s = \sqrt{\mu/\rho}$, where $\lambda$ and $\mu$ are so called Lamé’s parameters. But even these ones sometimes are silent about those properties hold only for infinite, homogeneous, and isotropic media.  Confusing.