How is the voice not distorted when Barry is running at super-speeds?

A few things. Most of the time, even at supersonic speeds, Barry is talking at a normal pace. There are very few times Barry talks really fast (the one time with Iris in his police lab). In the comics and cartoons, he and other speedsters can have conversations at super speed, including the high pitch change that results from it.

A doppler shift here does not apply, for three reasons. Even while running at supersonic speeds, a closely placed microphone will be stationary, relatively speaking, to Barry, so no shift in audio will happen.

A person listening to a car's horn for example, only experience a Doppler Effect, when the source is moving relative to the person listening. If you are standing on the sidewalk with a car beeping passing by at speed, the sound will shift. If you are In the car, moving at the same speed at the same direction, you will not hear a change in pitch.

In this case, it's the microphone that's the listener, moving at the same speed Barry is moving.

And as the microphone is relatively close to Barry, this makes it less likely to experience any issues. The microphone, embedded somewhere in the suit or the comms. It may be a throat microphone, a contact microphone that doesn't even use sound in air. In any case, it is inside the air bubble that the Flash would produce at super-speed.

Barry also doesn't produce shockwaves or sonic booms. This is a very "Speed Force is magic" reason, as he should when traveling at Mach Speeds. But if he were to produce mach speed shockwaves at ground level in a dense urban city, people would be injured, and windows everywhere would be broken, or worse. So there is a super-physics reason anyway.

If Barry did produce a shockwave, the microphone would still be inside the area not affected, so communication is clear. People on a Concord super-sonic plane can talk normally, as the air inside is stationary compared to them.

The last reason is that Cisco, with the show's super-science, would have worked around the issue. Just like he did with the Suit to prevent Barry from catching on fire from running. Some pseudo technobabble later, the issue would be corrected. Or simpler, as mentioned earlier, a throat microphone would be used.

A throat mic is important though, as running outside of an enclosure, like a car/plane, will introduce a lot of wind noise, like when on a motorcycle or car with the windows down. The throat mic is mostly impervious to wind noise.

Most of The Flash's powers come from the Speed Force. Whilst there is a Wiki on it, a lot of the important content has been summed up succinctly by this , which statesL

The Speed Force was the extra-dimensional energy that once powered all of the Flash's superhuman abilities. It is not like gravity or any other fundamental electromagnetic phenomenon. It's origin is likely the same as most superhuman abilities found in the DC Universe, a byproduct of the mysterious omni-energy known as The Source/The Godwave.

Very few people have access to the Speed Force and The Flash is one of them. It enables him to effectively break the laws of physics in multiple ways (including some iterations of The Flash being able to run faster than the speed of light).

Therefore, it stands to reason one of the powers The Speed Force gives him is the ability to communicate in a way that isn't distorted. If memory serves me right, in many of the comics when he is travelling at high speeds, his only communication is with other speedsters who would be able to understand him regardless of the way he was talking (due to their link to the Speed Force).

If he was talking to non-speedsters? I would guess The Speed Force allows this. After all, it would have to.

On a final note, obviously the entire Flash story (like many other superhero stories) breaks the laws of physics. Again, like many other comic stories, the writers have a "god" card they can play to explain all this (in this case The Speed Force). That's as detailed analysis as you're going to get I'm afraid.

On a final final note, if you want an actual, scientific breakdown of what speed he'd need to be travelling at, or exactly how the laws of physics come into affect, you should ask over at the Physics Stack Exchange where physicists as opposed to movie buffs will likely be better help.

I think the key to being okay with the physics behind the Flash's powers is understanding the protective aura that allows him to move at such speeds. Sound waves going from his mouth to his microphone (so long as the microphone is within that aura) wouldn't be distorted.

populationzero has spent a lot of time thinking about these sorts of issues related to the Flash (though, definitely the comic book version), and you'd probably enjoy reading his post (here: http://comicbloc.com/forums/showthread.php?t=78236).

Here's the important part about the Flash's aura:

If a motorcyclist needs protection from windburn you can bet a speedster might need more than a leather jacket to keep the friction from burning him up - and this is to say nothing of the physics of particle collision that would take place at near light speeds. Flash was given a protective aura so that he could save the day without fear of disintegrating while using his powers. One must assume that the aura acts as a sort of wedge in the path of Flash cutting through that resistance but if it also protects him from inevitable accelerated particle collision it must "reseal" the disturbance it created in the quantum foam on the back end of that wedge. [. . .] A Higgs Boson field is the most likely candidate for what his aura is since higgs bosons succumb to no resistance from particles in our material universe. In fact, The Higgs Ocean is my personal favorite as a candidate for the speedforce.

And here are some more of his thoughts on the Flash and sound, which are pretty interesting to think about:

Sound Barrier

If the Flash breaks the sound barrier it stands to reason that he would in fact be traveling faster than sound waves including the ones coming out of his mouth and consequently he wouldn't be able to hear anything - even himself. Everyone knows that superheroes like to talk to themselves and since time and space are relative, it's unlikely that Flash would speak at the same slow pace a normal person would if he were running near the speed of light. He would barely get anything out before he got to where he was going... In the comic book world, this might explain why Flash would be able to get a monologue out while in the midst of a situation that common sense dictates should only take a fraction of a second. Aside from actually speaking at a faster rate, one might argue that as long as the vibration of his voice is inside his protective aura that the sound waves would travel the same relative speed and therefore be audible within his "bubble" thus making it audible to him. It could be possible that when he traveled at extreme velocity and spoke - the frequencies would exit the aura at considerably higher energy than a normal human perceives sound. Sound waves that left his aura might be considerably higher in frequency and take on characteristics of other energies all together...

Problem of Doppler effect

As most any musician can tell you the frequency of a sound wave will determine it's pitch. The Doppler effect is the change in frequency of a wave that results from an object's changing position relative to an observer. If you push a musician off a cliff because he hasn't payed you rent, you're likely to hear the scream lower in pitch as he/she falls away from you, whereas if you fell with him/her - that scream would remain at the same pitch. A more dramatic example of this is the sound of a race car as it passes. Sitting in the car the hum of the engine does not change while the car travels at a static velocity. As Flash moves at super human speed, the length of sound waves will shorten in the direction he travels until he passes the source of the sound at which point the sound waves will lengthen. This would happen at such a dramatic extreme for Flash that it's likely that the sound waves he ran toward would no longer technically even be sound. His sensory perception would have to translate it to something meaningful for him to function at high speeds if he stayed within earshot to hear a particular thing at all. This perception would not simply ramp up to make sense of stuff coming at him faster - but also interpret the waves that are lengthened and pitched down by his superspeed. Flash must effectively hear things at different speeds simultaneously.

It is reasonable to assume that the Flash and his team are communicating by radio in their earpieces. Radio waves travel at the speed of light. Transmission and reception are virtually instantaneous, regardless of the speed at which the Flash is moving.

This is true for all radio transceivers, such as between a dispatch center and police officers, or between a military air control center and a jet aircraft traveling faster than the speed of sound. Bone-conduction microphones in radio-transceiver earpieces do not rely on the sound of the speaker's voice being projected into the air around him or her, although it is still fair to question how Barry can speak at all with the wind rushing into his face. Try talking into a high-speed fan to experience what I mean. :-)

The delay you may have noticed on cellular phones is due to all of the complex circuitry at both ends slowing the audio by a second or two. It is not due to the speed at which the signal travels between cell towers.

This Wikipedia article may be useful:

https://en.wikipedia.org/wiki/Radio_wave