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For example, if you're in a room with a light bulb on the ceiling, the closer your hand is to the wall, the more defined the shadow of your hand will be.

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There are two factors that cause a shadow to have blurred edges which get sharper the shorter the distance between the object causing it and the surface upon which it falls.

One factor is diffraction, which causes light to spread when it passes an edge.

The other factor, which will be the dominant one with typical light sources in the home, is that the light does not emanate from a point but from a wider area- for example, a regular incandescent bulb might be a couple of inches in diameter. That means that there will be an area around the edge of the shadow in which the object has blocked the light from one side of the source but not from the other. That area is known as the pen-umbra, while the darker central part of a shadow, where the object blocks light from across the full width of the source, is known as the umbra. The pen-umbra is a angular distribution of gradually fading intensity, so the nearer the object is to the surface the less opportunity there is for the pen-umbra to spread.

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Here's another perspective (literally) that may be easier to understand than Marco Ocram's answer.

Imagine you're tiny, or everything else is huge, and you put the pupil of your eye just in front of the shadowed surface, looking in the direction of the light source. The total amount of light that you see with that eye (averaged over your field of view) is roughly proportional to how bright that pupil-sized part of the surface was before you got in front of it.

The umbra is the part of the surface where, if your eye were there, you would see the shadow-casting object completely blocking the light source. The penumbra is the part where you would see it partially blocking the light source. The whole umbra is roughly the same darkness, while the darkness of the penumbra varies with distance from the umbra, since the disc of the light bulb is exposed gradually as your eye moves away from the umbra.

The closer the object is to your eye, the faster it moves in your field of view relative to the light source for each unit of distance you move your eye across the surface (parallax). Therefore, the closer the object is, the narrower the penumbra is.

As you move the surface away from both the object and the light source, the object's angular size drops off more quickly than the light source's, since the object is closer. If the object is small enough, then beyond a certain distance it won't completely block the light source when viewed from anywhere on the surface, and so there will be no umbra.

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This is not generally true. If the light comes from a single point then the shadow is sharp. The further away your hand is form the light source the better it may resemble a point source. The effect of the light source extension, better known as half shadow, is larger the further your hand is from the wall.

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Reflection: light reflects off surfaces. If you are far away from the wall, light can still arrive at the wall by reflecting off other walls. Therefore your shadow is both 1) not as dark and 2) relatively indistinct. If you are close to it, then much of this reflected light is blocked, and you get a sharper shadow.

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Take away the affects of a penumbra or rough surface and you still see blurring at the shadows edge. This is caused by diffraction from the edge. A single edge Diffraction pattern is unique where the bright and dark fridges get closer and closer together the farther out they go. This can be mathematically calculated and looks like this.enter image description here

I talk about this in my paper “Single Edge Certainty” at billalsept.com

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  • $\begingroup$ For clarification this only works for monochromatic light (light of one wavelength). The light of a lightbulb has many different wavelengths. $\endgroup$ Jan 26 at 9:47
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This is due to the diffraction (not reflection) of lightwaves at the edges of your hand. Diffraction is the spreading of waves eg. from an edge. The refracted wave spread out as it pass your hand, and the smearing of your hand is therefore bigger when the light travels a long distance after your hand.

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