Of the various unstable motions observed in solid propellant rocket chambers, the most troublesome currently are those involving oscillatory motions parallel to the axis. Such instabilities are found to arise particularly in larger rockets using propellants which contain aluminum. The problem is formulated here in one-dimensional form and solved for the case of small amplitude standing waves. Both pressure and velocity coupling may be accommodated, although the proper description of the response function for velocity coupling is not yet known. In addition to several special cases, the stability boundary is discussed for a straight chamber having variable cross section. The influences of the mean flow field, the nozzle, particulate matter, and motions of the solid propellant grain are taken into account.