Continuity in the case of sound works very similarly. As sound is longitudinally polarized, the direction of motion is now along the x direction, and we thus consider this component of the momentum. Following the interpretation of quantities in (3), q(x) should now represent the volume density of momentum, and F(x) should represent the rate of flow of momentum per cross sectional area from left to right across point x.
The volume density of momentum is the mass of each chunk times its
velocity divided by its volume, namely the mass per unit volume
times the velocity. Thus,
From the discussion of the string, we know that force is the measure of the rate of momentum flow. Because sound occurs in three dimensions, F(x) should measure flow per unit area. Thus, F(x) should be force per unit area, the pressure
To verify the correct sign, note that the pressure at x pushes in the positive direction on the chunk to its right, and thus F(x)=+P(x).
Substituting these results for q(x) and F(x) into (3), the three-dimensional continuity equation for momentum for sound is
which we know to be true because we again recognize it as the wave equation!!!
