standing waves continuous systems
In the last lesson, N masses meant N modes. Stretch those masses without end until they become one continuous string, and the modes become endless too. Each mode is a smooth standing wave with both ends held fixed, and its frequency is a whole-number multiple of the fundamental. That is why a string becomes music. The moment the masses crowd into a curve, vibration starts to speak the language of waves.
Raise N. With only a few masses the mode shape is still the jagged one from the last lesson, but as the count grows it gathers into one smooth curve, a continuous string. A continuous system is a multi-degree system whose masses never run out. When the gaps between the points disappear, the shape is read as a curve.
The simplest mode is the fundamental. The whole string rises and falls as one arch, with both ends held as nodes. Half a wavelength equals the length of the string, so λ₁ = 2L, and the frequency is the lowest one, f₁. The middle moves the most, and the ends do not move.
Choose a higher mode. The nth mode has n arches, and between them there are n−1 nodes. The frequency is fₙ = n·f₁, a whole-number multiple of the fundamental. That ladder of whole multiples is the harmonic series, and it is why a string sounds musical. The more nodes there are, the higher the pitch.
The superposition of A5 spreads across endless modes. Pluck the string at one point and that triangle is a sum of standing-wave modes. Add more harmonics and the corner grows sharp, gathering toward the plucked shape. Which mix you chose is the tone color. Fewer harmonics leave a round corner, and more of them make the plucked point distinct.
What sets the pitch? f₁ = (12L)√(Tμ), so the tension T, the length L, and the linear density μ decide it. Move the three. Tighter (T↑), shorter (L↓, which is fretting), or thinner (μ↓) raises the pitch. A bass string is thick for the same reason, and bridge cables and power lines hum by the same formula. Tighter, shorter, and lighter means a higher pitch.