pure resonance
Take an idealized, frictionless oscillator and drive it at exactly its own natural frequency, with no damping to bleed off energy. Now every single push lands perfectly in time, pumping in more energy than the last, with nothing to remove it. The swings do not settle into a steady size — they grow, and grow, and grow without limit. This runaway, ever-larger oscillation is pure resonance.
Mathematically pure resonance is the special case m x'' + k x = F_0 cos(omega_0 t), with no damping (c = 0) and the driving frequency exactly equal to the natural frequency omega_0 = sqrt(k/m). The usual steady-state formula breaks down because its denominator (k - m omega^2) hits zero. The particular solution instead picks up a factor of t: x_p(t) = (F_0 / (2 m omega_0)) t sin(omega_0 t). That factor of t is the whole story — the amplitude grows linearly with time, the oscillation getting bigger every cycle forever.
Pure resonance is an idealization no real system reaches, because every real system has some damping, and that damping always eventually balances the energy input, capping the amplitude at a large but finite peak (ordinary resonance). Still, pure resonance is the right mental model for why resonance is dangerous: it is the limiting case the real peak approaches as damping shrinks. The telltale fingerprint is the t out front — whenever forcing matches a natural frequency of an undamped system, the solution grows linearly rather than staying bounded.
x'' + 4 x = cos(2t) drives the undamped system (omega_0 = 2) at its own frequency; the solution is x(t) = (1/4) t sin(2t), whose envelope ±(1/4)t fans open without bound.
The factor t in front of sin(2t) is the signature of pure resonance: linear, unbounded growth.
Linear unbounded growth is a property of the idealized undamped model only. In any real system, damping (and eventually nonlinear effects when amplitudes get large) caps the growth — pure resonance is the cautionary limit, not literally what a bridge does.