Why is speed of sound constant?

This just occurred to me a few minutes ago, sorry if it's a dumb question:'

Why is the speed of sound constant in an ideal gas?

Suppose you have a box with an ideal mono atomic gas and a diaphragm that can oscillate. The diaphragm busts one move at ~3000 m/s. Since collisions with ideal gas particles are modeled elastically, the RMS speed of the particles that were in contact with the diaphragm while moving is ~3000m/s. Consequently, the neighboring particles will be bumped to ~3000m/s also, and so on until the end of the box is reached. Since no energy is lost and the RMS speed in the other two dimensions is zero, won't the pressure wave be traveling at ~3000m/s?