Standing wave generation in an hollow tube - a real world application

I'm working on a project to analyse the sounds that are created when bones are subjected to surgical instrumentation.

Our model is highly simplified: we consider the bone to be a hollow tube, closed at one end, and the sound arises from a standing wave. The lengths of the bone and sound velocity in the medium are known quantities (see end of this question for a reference to previous work by our group that explains the model in detail).

I wish to expand the model in the following ways:

1. Might the diameter of the tube, impact on the sound? I am aware that the concept of 'end correction' may apply, but my Google-fu doesn't seem to be strong on this one, and I can't find a definitive reference that properly explains the concept.

1a. Would wall thickness matter (as long as the diameter remains constant?)

2. Bones aren't strictly tubes. They are wider at one end, narrow at the middle, and then widen out again. Would this have a bearing on the validity of the model?

2b. Should I be measuring the narrowest part of the tube? The widest part part? Taking an average?

Ref:

Whitwell G, Brockett CL, Young S, Stone M, Stewart TD. Spectral analysis of the sound produced during femoral broaching and implant insertion in uncemented total hip arthroplasty. Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine 2013;227(2):175–180.

DOI: 10.1177/0954411912462813

• The paper considers resonances in marrow. I do not know anything about this procedure but I would also expect resonances in bone, with fundamental modes at high frequencies (because of higher velocity and $\lambda = 2L$). And also transverse modes. – Pieter May 11 '18 at 20:03
• λ=4L, since it's a closed-end tube. – jsm May 11 '18 at 20:04
• Please don't be put off by the context of the question. I'm keen on keeping things easy to understand by a non-technical audience, and I'd be pleased to have answers that 'assume a spherical cow'. – jsm May 11 '18 at 20:09
• A resonance of sound in the bone itself would have $\lambda = 2L$ (same boundary conditions on both sides). But there could also be bending modes at lower frequencies. The marrow channel is only part of the problem. I do not quite understand why it is not closed at both ends. – Pieter May 11 '18 at 20:11
• We saw off the top end in order to perform the surgery. – jsm May 11 '18 at 20:12

First a few preliminaries, then my best shot at answering your questions.

In general, the allowed frequencies for a standing wave in a half-open cylinder take the following form:

$$f_{nm} = c\sqrt{\left[\frac{\left(n+\frac{1}{2}\right)}{2L}\right]^2+\left[\frac{\beta^{(1)}_{m}}{2\pi R}\right]^2} \ \ , \ \ n,m=0,1,2,\ldots$$

where $L$ is the length of the tube, $R$ is the radius of the tube, $c$ is the speed of the sound wave, and $\beta^{(1)}_m$ is the $m$-th zero of the first order Bessel function of the first kind, $J_1$. For convenience, define $\beta^{(1)}_0 = 0$.

The much simpler 1D analysis is equivalent to setting $m=0$, at which point the frequencies become

$$f_{n0} = \frac{c\left(n+\frac{1}{2}\right)}{2L}$$

On the other hand, if $R$ is nonzero, there are higher order allowed frequencies which correspond to larger values of $m$. Taking these frequencies into account is important if $$n\sim \sqrt{1+\left[1.2\frac{L}{R}\right]^2}$$

If $L\gg R$ then this corresponds to a very large value of $n$, so you could reasonably ignore the additional frequencies (for a femur, this would correspond roughly to $n= 50$). On the other hand, for bones which are thicker relative to their length (fingers/toes, maybe?) this could be as small as $n\approx 5$.

Conduction of Sound

The above analysis is performed under the assumption that the bone itself does not conduct sound, which is obviously not true. If the sound medium (marrow, I guess?) is substantially less dense than the surrounding bone, then the energy transfer from marrow to bone is pretty minimal, but otherwise the bone will vibrate along with the marrow and you'll need corrections for that.

Might the diameter of the tube, impact on the sound?

Only if it is sufficiently large relative to the bone length to make the additional (radial) wave modes important. Very roughly speaking, the 1D allowed frequencies are correct as long as $n \lt \frac{L}{R}$.

Would wall thickness matter (as long as the diameter remains constant?)

Wall thickness matters if you are considering the vibration of the bone along with the marrow. If you make the approximation that the bone does not vibrate, then the wall thickness isn't important.

Bones aren't strictly tubes [...]

If the bowing is small, then this would be a relatively minor effect. If $R_{max}$,$R_{min}$, and $R_{av}$ are the maximum,minimum, and average radii, then "small bowing" means $\frac{R_{max}-R_{min}}{R_{av}}\ll 1$. If you want to include tiny corrections for this, you may be able to do it with some fairly messy perturbation theory (I'm not sure how, though), but more than likely you'd have to solve the equations numerically using a computer model.

Should I be measuring the narrowest part of the tube?

Hopefully all three numbers would be fairly close to one another. If you're ignoring the bowing (or treating it as an extremely small effect) use the average; otherwise, as I said, you'd probably need to use a computer.