So we obtained that the vibration of string. So Y is the amplitude of vibration of string. That can be expressed as something related with the space and something related with time. Normally, we call this is frequency domain analysis. If we plug this one into the governing equation, later on we will find out this term is equal to one over C-squared. The C-squared is the speed of propagation on the string. If I plug in this one into this equation, that will give us the following expression, Y double-dot meaning, the differentiating of this with respect to space is equal to one over C square, and then I have to differentiate this with respect to time twice that will give me J omega squared. So what I have as interesting form, Y double-dot minus omega over c squared, that has to be Y, and Y, and j times j is minus and returning over there that has to be plus, is equal to zero. Okay. From this expression, we can immediately see that depending on omega, the solution of Y will vary. So this is mathematical expression. Physical expression would be this corresponds to I have clamped a string. What is the lowest frequency? The lowest frequency if I said that is omega one, then the string will vibrate with certain form, and intuitively, it can be like this. For another frequency, we can intuitively think that vibration share will look like and so on. So we can get solution of this for each vibration of I or we will get some function that satisfy the boundary condition. Therefore, we can argue that generally, the vibration of string will be, some function corresponding to what we obtained from this argument, and then exponential j omega t, and some weighting factor. In other words, depending on the excitation, the contribution of each phi I would be different, and summing up all this. So that could be the general solution that governs the vibration of a string. Okay. But you can argue that, "Hey, this is the solution of a homogeneous differential equation." That is not a solution of in homogeneous. In other words, that is not the solution that can express the how the string will vibrate when we have a certain excitation. We are in the domain of linear differential equations or linear system. That means, Principle of Superposition always hold. How two apply the principle of superposition concept to this string vibration case. Okay, now back to general excitation case. There is this excitation. Say is zero and this is L, this is not excitation could be say, excitation of this plus excitation of that plus, excitation of this plus, and so on. That is very interesting observation, and thus have very convenient way to be used in practical sense. In other words, if I know the solution of this excitation case, the other cases are just the result of shifted excitation position. We may use the solution that obtained of this case to other cases and the summing all, then we will have the solution of this generally excitation case. That is somewhat challenging, and exciting approach. Therefore, all I would say, that motivated us to explore how the solution of this case, look like.