Derivation of speed of sound in (diatomic) chain using Newton-Laplace formula

I am familiar with the derivation of the speed of sound done in the style of, for example, this question:

Diatomic chain and speed of sound

I would like to derive that same result

$$c^2=\frac{2ka^2}{m+M}$$ Obtained in that problem using $$c= \lim_{q \to 0} \frac{\partial \omega}{\partial q}$$ but instead I want to derive the speed of sound using the Newton-Laplace formula $$\frac{\partial P}{\partial \rho}=c^2$$ What I tried so far:

For the density I put delta functions at each mass $$\rho = \sum_n m_n \delta(x-u_n)$$ where the $$n$$ th particle has mass $$m_n$$ and equilibrium position $$u_n$$. Note that this has correct units since the units of a delta function are the inverse of the units of the argument. For pressure I'm using something like $$P= k( x_n-x_{n-1} )$$ which has the right units so that $$P/\rho$$ has units of speed squared but I need pressure to include some delta functions at each point without messing up the units. Perhaps a kronecker delta function would work?

I'm unsure of both my choice of $$P$$ and of $$\rho$$ and I don't see how to continue with the problem in any way that's likely to produce the $$a^2$$ factor I need in my answer.

This question is essentially equivalent to:

Speed of Sound in 1D using pressure and density

since diatomic vs monatomic is not important. However I don't find the answer posted there to be very useful.

Perhaps If I Fourier expand $$x_n$$ as $$x_n = \int_{- \pi}^\pi \frac{1}{2\pi} dq x_q e^{iqn-i\omega_q t}$$ and use that $$\omega_q= \frac{k}{\mu}\pm \sqrt{ \frac{k^2}{\mu^2}-\frac{4k^2}{mM} sin^2(qa) }$$ To reiterate: how do I derive the speed of sound in a 1d chain using the formula $$\frac{\partial P}{\partial \rho}=c^2$$ Have I chosen my density and pressure correctly? Any assistance would be appreciated!

• It might be worth remembering that the Newton-Laplace formula you provide is a macroscopic formula, and is only strictly valid for continua. I would recommend you define a macroscopic density and pressure (averaging over large numbers of elements), and then apply the formula to those quantities. Feb 11, 2021 at 18:11