So if you have a string that has length L, and the fixed boundary condition. I mean, plausible vibration of the string will be this one, and then that one, Then to avoid any confusion, I can write, so on and so on. Let's say this is first mode, this is the second mode, and this is the third the mode. So on and so on. Then I can argue that, intuitively, the vibration at any space along x-axis and any time, can be superposition of these mode. One plus 2, plus 3, plus and so on. For example plus n. There is no other way that the vibration of a string, vibration of a guitar string can be the mode other than this. But depending on the excitation, the contribution of each mode will be different. So I will say that is contribution of first mode is a_1, second mode is a_2, third mode is a_3, nth mode is a_n plus. So in general expression, I can say, the vibration of the string due to the certain excitation can be expressed mathematically like a_i if I denote the boos like a phi i. That is function of x and i can be one to may be infinity. That has to be axial light with a frequency, a certain frequency. There could be some scale factor. That scale factor also depends on either excitation or the boundary condition of a string. That is general expression. But recall that, we use interesting measure to characterize the vibration of single degree of freedom system, and the multi degree of freedom system. It is interesting to note that, each vibration can be considered as vibration of single degree of freedom system. So continuous systems vibration can be decomposed into many many many single degree of vibration system that has different natural frequency. That observation suddenly remind us, we have look at the vibration of single degree of freedom system using transfer function. But c is the force with respect to amplitude of displacement. Can you see the vibration or can you see the characteristics of vibration of a string in terms of transfer function? Of course yes. It is also interesting to note that this transfer function is not only the function of time or omega, but also function of space. That I will explain in next lecture.
