So, strings are one-dimensional but when we mo-, when we move to rooms, room acoustics we move to a three dimensional system. And what I'd like to do is start here with a rectangular cavity. It could be any inclosed room that you're in. again we can, we can write an expression for the response. we're going to assume that the cavity has dimensions Lx, Ly, and Lz. So we have this rectangular cavity. And we're going to write an expression for the pressure in the sound field as a function of variables in the x, y, and z direction and of course, time. again, using partial differential equations and separation of variables, we can find the solution. which can be represented in the x, y and z dimensions. As well as our harmonic response characteristics here. the assumption that we make is that all the surfaces of the enclosure are rigid. So this is, this defines our boundary conditions. because what happens if we have rigid en-, rigid surfaces in the enclosure of the walls. Then the velocity of the air particles at the wall have to be 0. And for the solution that's expressed above, we end up with a wave number. that's a function of all 3 dimensions. As you can see here, Kx. So we have a wave number in the x dimension, a wave number corresponding to the y dimension and a wave number corresponding to the z dimension. And the the total wave number is the sum of the squares and the root of that. So if we apply boundary conditions the solution to the acoustic wave equation yields a form that's really similar to what we saw for the string. except of course we we now have the response in 3 dimensions. and this is the response, again, this is beyond the scope of this course to derive it. But I'm assuming everybody has dealt with basic sine and cosine waves at this point. And we just have a few subscripts on our variables here, because, you know, again, we have 3 dimensions instead of 1. So, like the string, there are multiple modes, an infinite number of modes, in fact. For the acoustic cavity. But the difference here is that we actually have modes that correspond to three dimensions. So we have a wave number in the x dimension. We have an infinite number indicated by the subscript L. We have a wave number in the y dimension with an infinite number described by the subscript m. And also, a wave of numbering the z dimension, again here with an infinite number of wave numbers described by the end of cn. we have our natural frequency that corresponds to each mode. Of the acoustic cavity, and again we express that as the as the harmonic function here through the complex exponential. Here are the descriptions for the wave numbers as you see and forms are very similar obviously in each direction. the product of the modal n to c L. times 5 or else of x. Again, similarly for the y dimension and again for the z dimension. And, of course, our expression that relates the the natural frequency and the wave number is through the speed of sound. And it's expressed as indicated here. Now, we can solve for the natural frequency, and if we do that in terms of our wave number which we had earlier. k is expressed as the sum of, the root of the sum of the squares. And here's c which corresponds to our speed of sound. Fairly simple expression for the natural frequency that corresponds to each acoustic mode in the cavity. So for example if we let L equal 0 and n equal 0. Let m equal 1. We would get an expression just like the one you see here.
