When the source and the wave move at the same velocity

In the last several videos, we assumed that the velocity of the source, of the object that's emitting the wave, we've assumed that that's less than the velocity of the wave. And we saw what happens with the Doppler effect and all of that. But what I want to do in this video is not make this assumption. In particular, let's see what happens, at least, at first for our formulas, and then get a conceptual understanding. Let's see what happens when the velocity of the source is equal to the velocity of the wave. The first thing we might try to do is just apply this new assumption into the formulas that we had in the last video. And those formulas were here. These are the observed period and frequency for an observer that's in the direction of the object. And if we make this assumption, that the velocity of the sound-- and the velocity of the source-- we're not necessarily dealing with soundwaves, although that might be an easy visualization for you. That tends to be how I visualize it. But what happens to these formulas when the velocity of the source is equal to the velocity of the wave? If these two quantities are equal up here, you have something, subtract the same thing from it. This numerator right here becomes 0. So it'll turn this whole thing into being 0. So the period, or the observed period, will be 0, which means you don't have to wait any time at all between successive crests. The entire waveform just gets infinitely bunched together. So it's kind of like one impulse. And if we look at the frequency, we can either look directly at the formula, and you'll see, you have something divided by 0 right here. So you can say this is 1 over 0, or you could just say that the frequency is 1 over the period, and you get this thing that's undefined. But if you want to think about, what does the frequency approach as the velocity of the source approaches the velocity of the wave, if this thing is only a little bit less that thing, this is going to be a very, very, very small, very, very small positive number. So when you divide these quantities by that very, very, very small positive number, you're going to approach infinity. So the frequency is undefined at the speed, at the velocity of the wave, but it's going to approach infinity. It will approach infinity as the source approaches the velocity of the wave. Not necessarily a soundwave. I keep using soundwaves, because that's how I tend to visualize things. And we'll talk in future videos specifically about soundwaves. And we'll touch on it a little bit in this video. So what is this telling us? Does this make any sense? And if you think about it, at least to me, it starts to make sense just what you saw in the last couple of videos. The last couple of videos, when something was moving slower than the speed of sound, you had, OK, I'm here now. And I'm about to release the next crest. If I go one period ago, maybe I was right there. And the crest that I had released at that time period maybe has traveled this far, just like that. If you go a period before that, I would have been over there. And crest that I released then would have traveled that far. We saw this in the last two videos. And if you go the period before that, I would have been there. And the crest that I would have released would have been that far. This was the whole reason why the Doppler effect happens. Because the observer sitting right here-- let me do this in a separate color-- the observer sitting right here is going to experience these crests more frequently than an observer sitting out here. Because the wavelength gets compressed, because every time this guy releases a new crest, or a new cycle, he has moved forward. He's moved forward in the direction of this motion right here. So let's think about what happens when he is exactly moving at the speed of the wave. So let's say that the source is here now. This is right where he is. And he's right about to release a new crest. So where was he at one period ago? So let's say he was here one period ago. So one period ago, if he's right going to release crest or cycle right now, one period ago, he released another cycle. And where has that cycle gone? Well, we're assuming that the wave is traveling at the same velocity as this guy. But it's going radially outward. So whatever he released then, it will have traveled at the same velocity as himself. So it will have gotten this far. He released it one period ago, and that's where he was one period ago. Over the course of the next period, he traveled there and so did the wave. The wave also traveled there. Now, where was this character two periods ago? When I talk about the period, I'm talking about the actual period of the wave. Every period, or how long does it take between similar points in the cycle? And I like to think of them as the crests in the cycle. So two periods ago, he was here. And he released a-- you can imagine, a pulse, or crest. And where will that be now? Well that will have traveled as far as he did. He traveled that far, and so will the pulse that he released. It will have traveled-- actually I have to make it a little bit more symmetric. It will have traveled that far. And if you go three periods ago, I think you get the idea. If you go three periods ago, he was here. And he released a pulse then, or crest, or a cycle of the wave. And where will that be now? Well, it will have traveled as fast as he's gotten. So it will have gotten this far. Of course, it's traveling that velocity in every direction, radially outward. Now think about the situation for the observer. Think, in particular, about the observer who's sitting right here. Let's say he's just out of the way so that this thing doesn't run into him and kill him, and what whatever else. But he's just out of the way, just enough to experience the sound, but not directly collide with this object that's emitting the-- well, I shouldn't say sound-- emitting the wave. I want to be general right here. We're not assuming that this is a soundwave. What's he going to experience? Well, he's not going to-- well, if we assume this is a soundwave, he's not going to hear anything until the thing passes right there. And right when the thing passes, it has all of the sound that it generated coming with it at exactly that moment in time. Instead of hearing things periodically, all of the wave fronts smack this guy all at once. And perceptually, instead of hearing a pitch, because you're hearing something periodic, you're just going to hear a big thump, because all of that sound energy is getting to you at the same time. You're just going to hear a thump. Because it's no longer really a frequency. All of the energy is coming to you at the same time. And when you are talking about sound, specifically, and especially, when you are trans-sonic, which means you're around the speed of sound, or parts of you are above or below the speed of sound, and you move into supersonic speeds, that's what people relate to the sonic boom. And we're going to talk a little bit more about that in the next video and mach numbers, all that, because I just find all of that fascinating. But I think this is intuitive. Because when you look at this, everything is just reaching you at exactly the same time. And this was the case of soundwaves, but it would be true of any type of waves.