The Speed of Sound — Question 10

Question 10

A gas spring exposes the adiabatic speed correction

A piston of area $A$ and mass $m$ encloses gas of length $L$ at equilibrium pressure $p_0$. Small rapid compression has $B=\gamma p_0$; slow isothermal compression has $B_T=p_0$. Neglect gas inertia in the piston oscillation. Take $\gamma=1.40$.

1. Derive the effective spring constant and compare rapid and isothermal piston frequencies.

2. For acoustic propagation at the same density, compare adiabatic and isothermal sound speeds. If the isothermal model predicts $290\,\mathrm{m/s}$, find the corrected speed.

Question 10 - Solution

A gas spring exposes the adiabatic speed correction

(a) Method and calculation

An inward displacement $x$ changes volume by $-Ax$ from $AL$, hence $\Delta p=Bx/L$. The restoring force magnitude is $A\Delta p$, so$$k_{\mathrm{gas}}=\frac{BA}{L},\quad\omega=\sqrt{\frac{BA}{mL}},\quad\omega_{\mathrm{rapid}}/\omega_T=\sqrt\gamma=1.1832.$$

(b) Interpretation and further calculation

The same stiffness ratio enters the acoustic law:$$\frac{c}{c_T}=\sqrt{\frac{B}{B_T}}=\sqrt\gamma,\qquad c=290\sqrt{1.40}=343.13\,\mathrm{m/s}.$$Rapid compression has less time to exchange heat and a larger restoring pressure in the supplied model. The lumped piston calculation and distributed wave calculation share a modulus but have different inertia distributions.

Physical check

With $\gamma=1$ the two responses coincide. The neglected gas inertia requires a nearly uniform pressure within the piston chamber.