PhD., University of Maryland
Jonathan is a published author and recently completed a book on physics and applied mathematics.
The standing sound waves are known as harmonics that involved sound. Standing sound waves associate with the boundary conditions at the boundaries of the medium. When the traveling wave is reflected back into the medium, energy stands in the way. The two boundary positions are open boundary condition in which the air is open and closed boundary in which the air is blocked from the surrounding. The position of standing waves varies accordingly to the boundary conditions, pressure, and displacement.
So let's talk about standing sound waves, now these are similar to just ordinary standing waves except they involve sound. Now like we know standing waves always are associated with boundary conditions at either side of the medium or at the boundaries of the medium. So at these boundaries, the wave is reflected back into the medium, for that reason the energy doesn't really escape. It's just kind of stands in the wave which is why we call them standing waves. Now for sound we have 2 major types of boundary conditions that we can impose at the boundary. We've got open boundary conditions; at an open boundary condition the air inside the medium is just open to the room, to the just atmosphere at large. And at a closed boundary condition, we've got something blocking the air from leaving the little bit of medium that we're interested in, so we're blocking it from the rest of the room. So it's kind of obvious why we call it open and closed.
Let's see how we can describe standing sound waves using these boundary conditions. Alright now as we know sound waves are associated with 2 different types of waves that exist at the same time. The pressure wave and the displacement wave, so what we're going to do is we're going to draw standing wave patterns associated with what's happening with the pressure wave and also what's happening with the displacement wave. Alright so let's start off with open, open, so what you should picture is a tube like this one where both sides are open to the atmosphere at large. Alright in the pressure since it's open to the atmosphere at large the pressure at the boundaries must equal the pressure of the atmosphere at large. So that means that it can't change, if the pressure out of this boundary would've tried to get bigger than the pressure of the atmosphere, then it just leaves because it's in contact with the atmosphere. So that means as far as the pressure wave is concerned open boundary conditions are nodes, so we've got node, node.
Now what's happening in between? Well we got to have a standing wave so we've got something like this that's going on. This is half of a wavelength, the sound wave is vibrating back and forth like this as far as pressure is concerned with 1 anti-node. 1 pressure anti-node right here alright. So if that length is l, then the wavelength of the sound wave according to this fundamental open, open standing wave will be 2 times l, because this is half of the wavelength alright easy enough. What does that look like as far as displacement is concerned? Well it's open to the boundary and so that means that, the air molecules are free to move back and forth. So that means that these boundaries are going to be anti-nodes in the displacement wave. Now of course that also jives with what we, with our understanding of pressure and displacement waves as being 90 degrees out of phase with one another. So that means wherever this guy is a node this guy got to be an anti-node. So how is this one going to look? Well we've got anti-node, anti-node, anti-node and little node in the middle.
Again this is a whole or a half wavelength, so again the wavelength will be 2 times l alright so that's open, open. The wavelength of the fundamental standing wave frequency in an open, open standing sound wave is twice the length of the pipe. Alright what about open closed what happens here? Well with open closed again in the pressure wave we've a node at the top because we already know what open does it's node at the top in a pressure wave. What about the closed end, well at the closed end that's like I've got my hand right here. So here the pressure doesn't have to equal the pressure in the room because it's separated form the room. So the pressure at a closed boundary is going to be an anti-node. So the fundamental looks like that, now that's only a quarter wavelength, so that means that the wavelength for an open closed standing wave will be 4 times the length twice as big.
Alright and of course over here in displacement, it just looks kind of the opposite because whenever you're an anti-node in pressure you must be a node in displacement and of course that also makes sense because if my hand's here the air molecules can't vibrate into my hand so they got to just sit there. So it's got to be a node in the displacement wave. So again lambda equals 4l now this is fairly straight forward and it's something that you can do just on your own. All you're going to do is blow into a little pipe like this. So if I blow into it open, open I get a certain frequency that comes out that was the fundamental that you were hearing with little bits of over tones from the higher harmonics. So now let's see what happens when I make it open closed it's the same note but much lower frequency. That's because the wavelength is twice as big so that means the frequency must only be half as big. Same note different octave one octave down. And you can hear that just by blowing into a coke can or whatever you want. Anyway those are standing waves sound waves.