Why a clamped string rings instead of running
The last two guides let waves run free. On the infinite line, d'Alembert's formula split any disturbance into a left-mover and a right-mover, u(x,t) = F(x - c t) + G(x + c t), each gliding off forever at the wave speed c. But tie the ends down — clamp a guitar string at both bridges — and the waves have nowhere to go. A right-mover races to the far end, reflects, comes back as a left-mover, reflects again, and so on. Trapped between two walls, the travelling story stops being the useful one. The waves interfere with their own echoes, and what you see instead is a shape that stands still and merely breathes up and down.
That breathing shape is a standing wave. Its profile along the string never travels left or right; instead the whole pattern swells and sinks in place, like a jump rope held by two children. The points that stay pinned at zero are called nodes — the clamped ends are always nodes — and the points that swing with the largest amplitude are antinodes. The crucial new feature, compared to the running waves of the infinite line, is that the walls select which shapes are allowed. Not every wiggle fits between two fixed ends; only a discrete list survives, and finding that list is the whole game of this guide.
Separating variables to find the natural shapes
To find those allowed shapes systematically we reuse the move that solved the heat equation a rung ago: separation of variables. We hunt for solutions of the product form u(x,t) = X(x) T(t) — a fixed spatial shape X(x) whose amplitude is dialed by a single time factor T(t). That is exactly what a standing wave looks like, so the ansatz is not a wild guess here; it is a description of what we already see. Substitute it into the wave equation u_tt = c^2 u_xx and the two derivatives peel apart cleanly.
Substituting gives X T'' = c^2 X'' T. Divide both sides by c^2 X T and the same magic as before appears: T''/(c^2 T) on the left depends only on t, while X''/X on the right depends only on x. A function of t alone equals a function of x alone, which can only happen if both equal the same constant. Write that separation constant as -lambda. The single PDE splits into two ordinary differential equations: X'' + lambda X = 0 in space, with the clamped ends forcing X(0) = 0 and X(L) = 0, and T'' + c^2 lambda T = 0 in time.
u(x,t) = X(x) T(t) into u_tt = c^2 u_xx
X T'' = c^2 X'' T
divide by c^2 X T:
T'' X''
------- = ------ = -lambda (same constant for both)
c^2 T X
=> X'' + lambda X = 0 (space, with X(0)=X(L)=0)
T'' + c^2 lambda T = 0 (time, an oscillator)The space equation X'' + lambda X = 0 with X(0) = X(L) = 0 is a spatial eigenvalue problem, the very same one the heat equation handed us. Its verdict is identical: only lambda_n = (n pi / L)^2 for n = 1, 2, 3, ... admit a nonzero shape, and those shapes are the sine modes X_n(x) = sin(n pi x / L). Each is a clean standing pattern pinned to zero at both ends — half a wave, a full wave, three half-waves, and so on. These are the only spatial shapes a clamped string is allowed to hold steady.
Time supplies the pitch, not decay
Here is where the wave equation parts ways sharply from the heat equation, and it is worth savoring the contrast. For heat, the time equation was first order, T' + k lambda T = 0, whose solution is a dying exponential e^(-k lambda t): heat leaks away and shapes fade. For the wave equation the time equation is second order, T'' + c^2 lambda_n T = 0 — the equation of a simple oscillator. Its solutions do not decay at all; they oscillate forever, T_n(t) = a_n cos(omega_n t) + b_n sin(omega_n t), with angular frequency omega_n = c sqrt(lambda_n) = c n pi / L.
Multiplying the matching space and time pieces gives one elementary vibration, a single normal mode: u_n(x,t) = sin(n pi x / L) · [a_n cos(omega_n t) + b_n sin(omega_n t)]. A normal mode is a motion in which every point of the string moves in step, at one shared frequency omega_n, never changing its shape — only its overall size pulses. The shape is the eigenfunction; the frequency is set by the eigenvalue. Strike the string just right and you can excite essentially one mode alone: the whole string then hums at a single pure pitch, swelling and sinking as one.
Harmonics and the timbre of a real note
Look at the frequencies again: omega_n = (c pi / L) · n, an exact whole-number multiple of the fundamental. The second mode is twice the pitch, the third is three times, and so on, with no gaps and no irrational offsets. This perfectly evenly spaced ladder is the harmonic series, and it is special to the simple wave equation on a uniform string. The fundamental is the first harmonic; the higher modes are its overtones. Their dead-on integer relationship is precisely why a plucked string sounds musical rather than clangy — the ear hears all those frequencies as fitting together into one pitch.
A real pluck never excites a single mode in isolation. The hammer or fingertip hands the string some arbitrary initial shape, and superposition takes over: any sum of normal modes is again a solution, so the true motion is u(x,t) = sum over n of sin(n pi x / L) · [a_n cos(omega_n t) + b_n sin(omega_n t)]. To meet the initial displacement and velocity you expand the starting shape and starting speed as Fourier sine series — exactly the matching step from the heat-equation guide, with two families of coefficients now because the time equation is second order and needs both a position and a velocity to start.
The recipe of coefficients is the timbre of the note. Which modes are loud and which are faint is decided by where and how you pluck — pluck near the bridge and you wake up many high harmonics, giving a bright, twangy tone; pluck near the middle and the gentle fundamental dominates, giving a round, mellow tone. Same string, same pitch, different soul. This is why a violin and a flute playing the identical note still sound nothing alike: they are the same harmonic ladder climbed with utterly different coefficient mixes.
Resonance: pushing a mode at its own frequency
So far the string rings on its own. Now drive it: add a forcing term, say a periodic push f(x) cos(omega t), turning u_tt = c^2 u_xx into the inhomogeneous u_tt = c^2 u_xx + f(x) cos(omega t). Expand both the response and the forcing in the same sine modes, and each mode's amplitude obeys a forced-oscillator ODE: T_n'' + omega_n^2 T_n = (forcing on mode n) · cos(omega t). The question of resonance is now a question about a single ordinary oscillator, one you met long before this rung.
A forced oscillator usually responds with a bounded wobble at the driving frequency omega. But if you tune the drive so that omega lands exactly on one of the natural frequencies omega_n, the response no longer stays bounded — in the idealized frictionless string the matching mode's amplitude grows without limit, linearly in time, sweeping ever larger. That runaway is resonance: energy is pumped in perfectly in phase, push after push adding rather than fighting, so the amplitude climbs and climbs. It is the same effect as a child on a swing pushed in rhythm, or — famously and destructively — a bridge driven at its natural frequency.
Be honest about the idealization here. Unbounded linear growth is what the lossless, exactly linear model predicts; it is a feature of the equations, not of reality forever. Real strings have damping, which caps the amplitude at a large-but-finite value, and at large amplitude the string stops being linear, so the clean sine modes blur. Resonance is genuine and important — it is how every musical instrument and radio tuner works — but the literal blow-up is the model telling you it has left the regime where it can be trusted.
Energy: the bookkeeping that says it cannot fade
Why do the free modes ring forever, while heat always fades? The deepest answer is conservation of energy. Define the string's total energy as the integral over its length of (1/2) u_t^2 (kinetic, from how fast points move) plus (1/2) c^2 u_x^2 (potential, from how stretched the string is). Differentiate this in time, use u_tt = c^2 u_xx and integrate by parts, and on a clamped string the boundary terms vanish — leaving exactly zero. The total energy never changes.
- Define the energy. Let E(t) = integral over [0,L] of (1/2) u_t^2 + (1/2) c^2 u_x^2 dx — kinetic plus potential, both non-negative.
- Differentiate. dE/dt = integral of u_t u_tt + c^2 u_x u_xt dx, just by differentiating under the integral sign.
- Use the equation and integrate by parts. Replace u_tt by c^2 u_xx; integrating c^2 u_x u_xt by parts produces a boundary term plus a piece that cancels the first.
- Kill the boundary. On a clamped string u_t = 0 at both ends, so the boundary term is zero and dE/dt = 0: energy is conserved exactly.
Conservation is the formula behind the contrast you have been feeling all rung. The heat equation dissipates — its energy steadily decreases, which is why high modes decay and rough shapes smooth out and you cannot run it backward. The wave equation conserves — its energy is frozen, which is why modes ring on undimmed, why information is not lost, and why running it backward is perfectly well-posed (just play the vibration in reverse). It also explains resonance honestly: a driving force at omega_n keeps doing positive work on the string, feeding energy in with nothing to remove it, so E grows without bound in the lossless model — the energy ledger and the runaway amplitude are two views of the same fact.