This is a Silver Dollar. You know that it's solid. This handkerchief appears to be solid as well, but it is made up of individual fibers. There are thousands of minute holes that will actually allow the coin to pass right through.
That was pretty quick so let's try that again. Oops... it didn't pass through, it vanished. Well, easy come, easy go.
HOW-TO TUTORIAL
Begin by holding the coin between the right thumb and first and second fingers.
Drape the hanky over the coin. As you adjust the hanky, push a portion of the material down so it is trapped between the coin and right thumb.
Pull back the front edge of the hanky to expose the coin. Now flick the hanky forward as you drop the hand. All of the edges of the hanky drop forward. At this point, the coin is behind the handkerchief, not underneath, as the audience believes.
Regrip the coin under the hanky with the left finger and thumb. Twist the hanky like this and push the coin up to show the penetration.
For the vanish, start with the coin in the same position as the first time. Cover the coin.
As you apparently take it with the left hand, allow the coin to drop into the right hand and grip it in fingerpalm position. Click on Building Blocks in Coin Vanish and Production for more details.
Remove the right hand and grip one of the corners of the hanky. Let go of the hanky with your left and flick it with your right. Show the hanky then put it in your pocket with the hidden coin.
HOW-TO TIPS
You might also consider the story possibilities if you use a large foreign coin or exotic medallion for this effect.
Make sure that you have enough material trapped between the coin and thumb, so the coin is completely covered when you flip the hanky forward.
Use the left thumb to push the coin slowly up and to your right. For more details on the fingerpalm, click on Building Blocks in Coin Vanish & Production.
To really sell the vanish, you must direct your audience's attention by focusing on where the coin is supposed to be.
Of course, you can retain the coin in your right hand after you put away the hanky, then reproduce it from another location.
PROPS
A large coin, preferably a half dollar or silver dollar
