Let's review what we learned in the previous lecture, the vibration of a string. Okay, we consider the very infinitesimal element of a string that combines the tension, and we suppose that there is some excitation force that is disturbing the string and that has to be balanced by the motion of a string. If we neglect the motion in x direction, the motion would be the tension per unit length, multiplied by dx and then d square y, dt square, that is the acceleration that is mass of this element of string. From this pictorial expression of Newton's second law, we obtain the following very interesting governing equation. That can be written as tension multiply by, interestingly, the curvature of string, and then mass of unit length, and then the acceleration. That has to be equal to the excitation force acting on the string, that we we can write. This is partial differential equation. That often looks like a simple but not easy to solve it for general cases. As you know, there are very straight forward way to handle partial differential equation. But let's remind you that this course is not mathematics course. This course is to understand vibration of general body. We start with the simplest case, which is the string vibration. So we will invite the physical intuition to tackle this problem, including mathematical background. The solution of this partial differential equation can be tackled as following: for example, a simple boundary condition, when we have fixed boundary condition, and when we have some excitation force over here, this string may vibrate in a certain form. The simplest case of this excitation force should be some force, point force. Okay, if there is a point excitation for the string will vibrate. But we can easily say that the vibration form, in this case, would look like this or may look like that. So on, so on. So we can see that solutions are determined mainly by the boundary condition, as well as where the excitation force is. Okay, if excitation force is located over there, then this kind of shape of vibration is not likely possible. Right? Also, the other thing we can observe from this understanding, the solution due to any excitation force may be composed by this shape of vibration, and that shape of vibration, and some other vibration. That vibration shape has to satisfy boundary condition. So in that sense, we can say the solution of this partial differential equation has to follow the solution of this partial differential equation, assuming that there is no excitation. Physically meaning that, I excited system and let it go. Then the vibration under that circumstance will follow this equation. So this is homogeneous differential equation. In summary, what I am arguing that is that, in terms of mathematical point of view, in homogeneous partial differential equations, solution can be composed by the solution of homogeneous partial differential equation. How the solution of this partial differential equation look like? You can see that the displacement of string that is function of x and time. If you assume that this solution is separable in space and time, then I can write the solution x is composed by two component. One is what related with this placement and one is what is related with the time. Everybody can remember that's often recall this kind of approach is the separation of a variable. If we further assume that the time dependent can be expressed by exponential j Omega t. In other words, that can be cosine Omega t or sine Omega t. Physically meaning that, if I excite the whole system by giving cosine Omega t all over here, that means I am giving him a time t equals zero. The amplitude has to be one. Therefore, I am giving him sort of initial displacement. If I use a sine Omega t over here, that means that I'm giving him zero initial displacement but giving him initial velocity. So depending on the time excitation, it could be the combination of cosine Omega t and sine Omega t. In that sense, using exponential j Omega t expression for time excitation is very likely possible. So I can write again using this expression y, x, Omega t. For convenience, let me change this notation, not x to y. That will look more convenient. Then I can say this can be expressed by y, x, and exponential j Omega t. That is interesting. That is interesting. Let's look at some physical meaning of this expression. What it means? This means, when I excite the system with a certain Omega, say Omega one. At a certain point, this system will be vibrating with y, x. Y, x. For example, when I excite a system, as I demonstrated in the previous lecture using guitar, I plot one point, ting!. Then it'll vibrate toward Omega one, as well as that excitation, that Omega one excite the system of the string with a certain shape of a vibration. Later on, we will call this as mode shape. Okay.