For a given system of finite length, like our cable , we have all these modes. And we know something very interesting - they are mechanically orthogonal. Can we use this? Certainly. You know that when you have orthogonal vectors like e_x and e_y here, you can use them to represent any vector say V = V_x e_x + V_y e_y where V_x and V_y are the coordinates of V in the e_x - e_y space. Let use our orthogonal modes to represent any motion! Imagine a motion of the system, y(x,t). Any motion. To use the modal shapes phi_N(x) to represent it I would write y(x,t) = sum on all modes of q_N(t) phi_N(x). Here the q_N are the modal coordinates of the motion y on the modal shapes phi_N. I know the modal shapes phi_N. My unknown are now all the q_N. How can I find them? The displacement y(x,t) satisfies the equation of motion, which may include an external force on the r.h.s, in the most general case m yddot - Ty'' + ay = f. By using the modal representation of y this reads m sum (q_N)ddot phi_N - T sum q_N (phi_N)'' + a sum q_N phi_N = f. Here is the equation again. Let us use our inner product, that was central .in the orthogonality property. Let us compute an inner product between all this and a modal shape phi_P. I can separate this as a sum of inner products sum (q_N)ddot (m phi_N, phi_P) + sum q_N (-T (phi_N)''+a phi_N, phi_P) = (f,phi_P). Now this is interesting! All these products here are zero in the sum except for N=P, because of the orthogonality. The only remaining term is (q_P)ddot (m phi_P, phi_P) + q_P (-T (phi_P)'' +a phi_P, phi_P) = (f,phi_P). What do we have here? We have an equation on the modal coordinate q_P(t). A very simple equation, which I can write m_P (q_P)ddot + k_P q_P = f_P. The coefficient m_P, which is called the modal mass is specific to mode P, its m_P = (m phi_P, phi_P). The modal stiffness of mode P is k_P right here. And the force f(x,t) has been transformed into a modal force f_P(t) = (f,phi_P). I can do this on each and every mode, and as a result, I will have a series of equations for all the modal coordinates q_N(t). Let us summarize. In the physical space we had an unknown y(x,t), which depended on two variables x and t. It satisfied a partial differential equation. We found some particular free oscillating solutions, the modes, defined by their modal shapes and frequencies. Using modal decomposition, we changed of variables, and in the modal space, we have a series of new unknown variables, the q_N, which depend on time only. Each of them satisfies a very simple differential equation. And remarkably, the equations on q_N(t) are independent from one another. They are totally decoupled. If I can solve them I will be able to go back to the physical space by simply reconstructing, or recombining the coordinates with the modal shapes. To have a complete picture I should say something about initial conditions. In the physical space, we have known initial conditions on displacement and velocities y(x,0) = y_0(x) and ydot(x,0)=ydot_0(x). These conditions are needed to solve the equation of motion. They can also be transported into the modal space and become on initial conditions on each and every coordinate, q_N(0) = 1/m_N (m y_0, phi_N) and (q_N)dot(0) = 1/m_N (m ydot(0), phi_N). so that I can solve each and every modal equation. As you can see, we can move between the two spaces by recombining modal coordinates we get physical displacements. By projecting physical forces we get modal forces. All the dynamics that was formulated in the physical space can be reformulated in the modal space. This is a major step, because we have somehow split the problem. Schematically, I had a concrete wall, in the physical space and now I have a wall of stones, in the modal space, well separated stones. Much easier to handle, certainly. This modal space is not very intuitive to imagine. We had free modal vibration, these harmonic motions according to one mode; What I called the pearls. But, in the modal space we can address any motion that may exist on the system, harmonic or not. It is always a combination of motions following each and every modal shape. The q_N(t) have no reason to be harmonic, as you will see in many examples. It just depend on how you force the system. The pearls, the harmonic free motion along a mode, are actually particular solutions of this, when all q_N are zero except one, and there is no forcing. All we have seen here was for tensioned cables with foundations, our model system. But we have waves, and therefore modes on many other systems - beams in bending, acoustics, liquid free surfaces, .. You will see them soon. And these modes will also be mechanically orthogonal. If so, I can use them to represent any motion xi(x,t) = sum q_N(t) phi_N(x). And the modal coordinates will satisfy our simple ODE. We have here a common feature to all dynamical systems in the modal space - the equations to solve always have the same form. Of course, the modes are different, but the form of the equation for the modal coordinate are always the same. Let us summarize. By introducing boundary conditions we constrained waves. We constrained them so much that free harmonic motion only existed in the forms of modes, a discrete series of them. These modes have a remarkable property which is their mechanical orthogonality. As such we can use them to represent any motion, using a set of modal coordinates. And these modal coordinates satisfy simple equations. Can we solve these simple equations? Certainly, they look so simple. And there is a lot of physics there. We shall do that soon.
