= Solution
The physical <Jeans length> is the wavelength for which pressure and gravity balance:
$$
\boxed{\lambda_J=v_s\sqrt{\frac{\pi}{G\rho_m}}},\qquad
k_J^{\mathrm{phys}}=\frac{\sqrt{4\pi G\rho_m}}{v_s}.
$$
The physical wavelength of the comoving mode is $2\pi a/k$. Modes with wavelength larger than $\lambda_J$ have a gravitationally destabilizing coefficient and can grow; shorter wavelengths are pressure-supported and oscillate, with expansion also damping the motion. A time-dependent background makes the exact growth history depend on the epoch, not just an instantaneous sign. For pressureless matter the <Jeans length> vanishes.
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