As shown in the previous section, the material above you doesn't matter. At all times, gravity is determined by the material below you - an inner sphere that is concentric with the Earth.
Assume the Earth has radius m, which is approximately 6.4 million meters. Let the tube run parallel to the z axis, at x = c. Let d = sqrt(m2-c2). Thus the path runs from -d to d. At the height z, the radius r is sqrt(c2+z2). Turn this into volume, and multiply by a constant g that incorporates 4π/3, gravity, and the density of the Earth. Divide this by r2 to get the radial acceleration of a small mass at distance r, and get gr. (Here g is a rather funny constant, having units of reciprocal seconds squared.)
Since gravity at the Earth's surface is 10 meters per second squared, set gm = 10, and g is approximately 1.56e-6.
Only the z component matters, hence the force at height z is -gr times z/r, or -gz. This gives the following differential equation in z(t).
z′′(t) = -gz(t)
Start time at the center of the tube for convenience, and the solution is v×sin(sqrt(g)t), where v×sqrt(g) is the velocity as you fall past the midpoint of the tube.
The top of the sin wave has height d. In other words, sin is multiplied by v, and the chrest has height d. Therefore v = d.
z(t) = d×sin(sqrt(g)t)
So - how long does it take to pass through the Earth? The answer is half a sin wave, or π/sqrt(g). This is about 2,510 seconds, or 42 minutes. Jump into the tube in New York, fall for 21 minutes, zip past the halfway mark traveling 5.5 km/s, rise for another 21 minutes, and emerge in Beijing.
If you started at Denver, the "Mile High City", you would fly out of Beijing with energy to spare. (Don't hit your head on the ceiling.) On your return trip, you'll need a boost to climb the extra mile.
Notice that the travel time does not depend on the chord through the Earth. A tunnel trip from Detroit to Chicago takes 42 minutes, just like Atlanta to Sydney.
As a curiosity, it takes the space shuttle approximately 42 minutes to travel halfway around the Earth. But it's not really a coincidence. The shuttle is never in a polar orbit, but pretend like it is, flying from the south pole to the north. At the same time you are falling in your capsule from the south pole through the center of the Earth to the north. Follow the earlier convention, so that time = 0 at the center of the Earth. How fast does the shuttle go to remain in a perfect circular orbit? Circular acceleration is v2/m, so set this equal to gm and gm2 = v2. In other words, the tangential velocity v = m×sqrt(g). After time t the arc distance traveled is m×sqrt(g)t, and the projected latitude is m×sin(sqrt(g)t). This is the formula for the falling capsule. If you stare out your window, looking through the Earth, you will see the space shuttle at all times. It tracks your motion perfectly.