Why does air vibration produce a spherical wavefront and not a plane wavefront?

Question

I want to know why the back-and-forth vibration of a material produces a spherical wavefront, for the produced sound wave, and not a plane wavefront.

Answer 1

Assuming that the question is related to sound waves in air, then when an object sets air into vibration, it sends out sound waves in all directions away from the point of vibration.

This creates spherical wavefronts because the sound spreads out evenly in all directions, forming expanding spheres.

Plane wavefronts, on the other hand, arise from the vibration of large, flat sources (like a large speaker) or from viewing spherical wavefronts from a great distance, where they appear flat due to the vast scale.

Answer 2

Introduction

The assumption that the vibration of objects (the shape of the resulting wavefront is not dependent only on the material but other factors too) produces spherically expanding waves is wrong. Spherical waves constitute a, rather convenient, mathematical model of the resulting wavefield of a monopole source. Complex sources (in shape and characteristics, such as non-isotropic materials) do not radiate spherical waves and have complex radiation patterns. Below are some radiation patterns of various objects as found online.

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Simulated loudspeaker.

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Studio monitor and Flat loudspeaker prototype radiated pressure per frequency and angle.

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$\mathbf{3}$D radiation pattern of an iPhone.

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Why?

So, as you can see, most (if not all) real sources do not radiate omnidirectionally (spherical waves). At frequencies where the wavelength is way longer than the largest dimension of the source, spherical wavefronts are a rather good approximation of the actual radiation pattern, since all vibrating particles are positioned are pretty much the same phase of the wave, resulting in vibration with all particles moving with the same amplitude and phase. However, for higher frequencies the modal behaviour of sources makes particles of the source vibrate with different phases and amplitudes at different positions on the source surface. Below is an image with vibration patterns of a square plate where you can also see the frequency for which the pattern occurs. More complex surfaces with complex boundary conditions may exhibit more complex patterns.

Now, when a source vibrates in patterns like those shown above, neighbouring areas with vibrations that are of opposite phase will "cancel” each other out, resulting in what is known to be evanescent waves (see Wikipedia’s page) where there is energy stored in the system and the air molecules oscillate back and forth between the two areas, close to the source. You can see an animation below of a vibrating plate. Notice the movement of air particles/molecules close to the source.

In evanescent waves, no energy propagates far from the source. Of course, this is an approximation and almost always radiation is achieved, but in the evanescent waves case the efficiency is extremely low.

Furthermore, the radiation patterns resulting from sources with complex geometries, such as those shown in the first figures, are non-isotropic and frequency-dependent. As you can see, the radiation of sources is anything but omnidirectional.

Answer 3

"Why" it expands spherically is because that's what it does. Science has equations to describe it, and they can be used to show spherical wavefronts, but fundamentally it does because it does.

You can get some intuition with a sandbox. Use your hand as a speaker, and shove some sand forward. Sure, at first glance the sand moves forward. But look closer at the edges. You'll see that when you move your hand forward, the sand doesn't move forward per-se. It moves in the direction of least resistance. For some of the particles, that is indeed forward, but if you look at the edges, its easier to move out of the way. Indeed, if you repeat this with a pile of sand rather than a sandbox, and push it far enough with a flat surface, eventually all of the sand will have moved out of the way, leaving a snail's trail of sand that moved sideways rather than forward.

Now that analogy has its limits. Sand doesn't move all that similarly to sound. And our hand is really big compared to the space we're looking at. Speakers are typically small compared to the area (if you put a speaker in a tube, the sound waves will indeed be planar). We need something that's closer to the situation you're thinking of. But it is nice to start with because sand particle are visible. It's harder to think of sound waves because the particles are tiny and their motion is even tinier.

It turns out ripples on a pond and sound in a room behave similarly. They both get captured using "wave equations." Go find a pond or a pool, and bring a small piece of flat wood, like a discarded piece of 2x4. Put it in the water and gently push it forward to create a ripple. Watch how that ripple moves. For a short while, the ripple will be planar in front of the wood, but starts to curve near the edges, where the path of least resistance encourages particles to move sideways instead. As they peel away, particles closer to the center start to get a path of least resistence that's a bit sideways as well, so they start to deviate. At some reasonable distance from where you made the ripples, that curve has gotten so similar to a circular wave, that we just call it circular. (pond waves are circular -- two dimensional. sound will be spherical in 3d)

Incidentally, we call this the "far field effects" of your source. The far field effects are what you get when you're far enough away that the size of the original source doesn't really matter much. In the far field, waves are all spherical (well, I'm sure there's some analyses where we don't treat them that way, but hopefully you get my meaning). We also study "near field" effects, where the size does matter. Near field would capture the reality that the wave was almost-planar when it was close to the block of wood.

When you hear "Near Field Communications," NFC, in the cell-phone and bank card world, that's actually what they're talking about. NFC is designed to transmit information using signals that vanish towards 0 in the far field. They're only detectable in the near field. That's why the card reader can't detect your card until it's right up near it.