This answer try to address your doubts solving the differential problem of the transverse motion $w(x,t)$ of the points of a string with uniform linear mass density $m$, uniform axial pre-stress $N_0$, with length $\ell$, with one extreme point at rest, $w(x=0,t) = 0$, and the other extreme point with prescribed motion, $w(x=\ell,t) = u(t)$. This section is organized as follow: 1) differential problem is set up; 2) normal modes - or standing waves - are introduced as (a basis for) all the possible solutions of the unforced problem, i.e. free motion of the string is a linear combination of normal modes; 3) the differential problem is solved here exploiting a common technique used in mechanical engineering, named static mode recovery, mode acceleration, or acceleration of modes that consists in splitting the general solution as a sum of a static (or fast) and a dynamic (or slow) part; modes are used as a basis for the dynamic part of the solution, and their orthogonality allows to get a closed-form of the solution, particularly simple in some special case of initial and boundary conditions 4) eventually, the solution of the problem is discussed.
Differential problem
The differential problem governing the problem is made of the momentum balance equation, 2 boundary conditions at the extreme points (here, essential or Dirichlet boundary conditions), 2 initial conditions (Cauchy problem),
$$\begin{cases} m \partial_{tt} w - N_0 \partial_{xx} w = f \ , \qquad x \in [0, \ell], \ t \ge 0 \\ w(0,t) = 0 \\ w(\ell,t) = u(t) \\ w(x,0) = w_0(x) \\ \partial_t w(x,0) = v_0(x) \ . \end{cases}$$
Modes - Standing waves
Standing waves are usually defined as the simple harmonic solutions with homogeneous boundary conditions and no volume force, i.e. no external forces acting on the system. With this definition, standing waves for the problem of interest are the solutions of the problem
$$\begin{cases} m \partial_{tt} w - N_0 \partial_{xx} w = 0 \ , \qquad x \in [0, \ell], \ t \ge 0 \\ w(0,t) = 0 \\ w(\ell,t) = 0 \ , \end{cases}$$
having neglected here the initial conditions that will be recovered later. This problem can be solved with the method of separation of variables, $w(x,t) = n(x) g(t)$. It's easy to prove that an infinite number of standing waves $w_i(x) = \sin \left( k_i x \right)$ exists, with $k_i = \frac{i \pi}{\ell}$, $i \in \mathbb{N}$.
Solution of the problem: fast + slow parts
As the standing waves are solutions of the differential problem with homogeneous boundary conditions, they can't exactly solve a problem with a non-homogeneous boundary condition, like the b.c. $w(\ell,t) = u(t)$. If we try to write a solution that's only a linear combination of modes,
$$\widetilde{w}(x,t) = \sum_i \widetilde{a}_i(t) n_i(x,t) = \sum_i \widetilde{a}_i(t) \sin \left( \frac{i \pi x}{\ell} \right) \ ,$$
it's easy to prove that $\widetilde{w}(\ell,t) = 0$, for any value of $\widetilde{a}_i(t)$. If you're a bit familiar with Fourier series, you may expect that some Gibbs' oscillations occur, and that's what happens even if you're not familiar with them:).
A common approach used in mechanical engineering is to exploit the linearity of the problem and principle of superposition to write the solution as a sum of a fast part and a slow part, $w(x,t) = w_f(x,t) + w_s(x,t)$. The fast part is defined as the solution of the problem
$$\begin{cases} -N \partial_{xx} w_f = 0 \\ w_f(0,t) = 0 \\ w_f(\ell,t) = u(t) \ , \end{cases}$$
i.e. $w_f(x,t) = \frac{x}{\ell} u(t)$. The fast part is thus the solution of the differential problem
$$\begin{cases} m \partial_{tt} w_s - N \partial_{xx} w_s = - m \partial_{tt} w_f + N\partial_{xx} w_f = - m \dfrac{x}{\ell} \ddot{u}(t) \\ w_s(0,t) = 0 \\ w_s(\ell,t) = 0 \end{cases}$$
Now the slow part is the solution of a differential problem with homogeneous boundary conditions, so there's hope (and it's a good idea indeed, if we want an analytical solution of the problem) in using the modes as basis for the slow part of the solution only,
$$w_s(x,t) = \sum_{i=1}^{+\infty} a_i(t) \sin(k_i x) \ . $$
Introducing this expression in the differential equation, and exploiting the orthogonality of modes,
$$\int_{x=0}^{\ell} \sin(k_i x) \sin(k_j x) \, dx = \delta_{ij} \dfrac{\ell}{2} \ ,$$
the PDE reduces to a set of infinite number of ODEs for the coefficients $a_i(t)$ of the expansion of the slow part of the solution,
$$\dfrac{\ell}{2} m \ddot{a}_i(t) + \dfrac{\ell}{2} N k_i^2 a_i(t) = \int_{x=0}^{\ell} - m \dfrac{x}{\ell} \sin(k_i x) \, dx \, \ddot{u}(t) = (-1)^i \dfrac{m}{k_i} \ddot{u}(t) \ , $$
or
$$\ddot{a}_i(t) + \dfrac{N}{m} k_i^2 a_i(t) = (-1)^i \dfrac{1}{k_i \ell} \ddot{u}(t) \ , \quad i \in \mathbb{N} \ .$$
This is a set of 2-nd order ODEs for the coefficients of the expansion of the fast part of the solution. These should be supplemented by the initial conditions, projecting the i.c. on the fast part and the modes. Or we may just assume for now, that every system has at least a small damping that make the initial solution decay and only the forced solution exists after a sufficient amount of time (or any other justification to avoid extra-algebra here).
The solution can be thus written as
$$w(x,t) = \dfrac{x}{\ell} u(t) + \sum_i a_i(t) \sin \left( k_i x \right) \ ,$$
being $a_i(t)$ the solutions of the set of ODEs. Now, we're ready to discuss the expression of the solution and the appearance of normal modes (up onto - or summed to - the fast part of the solution), as a function of the frequency content of the forcing $u(t)$.
Comments
Every second order ODE above is formally equivalent to the equation of a undamped harmonic oscillation with:
natural frequency $\omega_{n,i} = \sqrt{\dfrac{N}{m}} k_i = \sqrt{\dfrac{N}{m}} \dfrac{i \pi}{\ell}$
transfer function G_i(s) from $\ddot{u}(t)$ to $a_i(t)$ (in Laplace domain)
$$a_i(s) = G_i(s) u(s) = \dfrac{1}{s^2 + \omega^2_{n,i}} (-1)^i \frac{1}{k_i \ell} \mathscr{L}\left\{\ddot{u}(t)\right\}(s) \ , $$
Being a superposition (linear combination) of contributions experiencing resonance at increasing frequency, $\omega_{n,i} = i \sqrt{\frac{N}{m}} \frac{pi}{\ell}$, it should be not a surprise that if the forcing frequency is close to a resonance of a mode, that mode dominates in the response.
