Waves Can Be Represented as Superpositions of Sine Waves.
The shape of a plucked string is represented as a sum of sine waves. The convergence as more terms are added is then compared for a continuous shape and a shape with jumps in height.
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項数を20から1000へ増やすと、行きすぎは跳びの高さの何%に近づくか？
幅0.02 mでつなぐと、20項から1000項で行きすぎはどう変わるか？
The equations explain the phenomenon
1A point on a circle can be represented by 1 number.eiθ=cosθ+i sinθ……
2Differentiation can be represented by multiplication.ddt eiωt=iω eiωt……
3The string has a set of normal modes.yn=sin(nπxL) cos(ωn t)……
4Superposition produces the desired shape.SN(x)=Σn=1N bn sin(nπxL)……
5Sine waves of different orders are orthogonal.∫0Lsin(nπxL)sin(mπxL)dx=(L2) δnm……
6The coefficient bn can be isolated.bn=(2L) ∫0Lf(x) sin(nπxL) dx……
The calculations from this point use the dimensionless coordinate ξ = x/L, obtained by dividing position by the length L.The equations in Steps 7–10 use ξ. Since L = 1 m, the value of ξ equals the numerical value of x measured in m. The horizontal axis of each graph remains x [m].
7The coefficients for a plucked string can be calculated.bn=8h sin(nπ/2)n²π²……
The following sections consider functions with jumps in height, rather than the string itself (displacement is continuous in the small-slope string model). Only the triangle in Step 7 is referred to as a string shape.
8The coefficients for a shape with jumps can be calculated.bn=(2nπ)[cos(nπ/4)− cos(3nπ/4)]……
9The peak height near a jump can be calculated.For large NPeak height of SN≒12+Si(π)π≒1.0895……
10An error bound can be found for the shape with straight ramps replacing the jumps.|SN − f|≦8aπ² N……
Further study: the time evolution of a string and the Fourier transform
12A short signal can be represented as a sum over continuous frequencies.Δt · Δω=1 (Gaussian width)……
What this chapter has shown
The shape of a plucked string can be expressed as a sum of sine waves. For a shape with a discontinuity, adding more terms makes the peak narrower and brings it closer to the discontinuity, but the relative overshoot approaches about 9%.
8.98%Overshoot relative to the height of the discontinuity
When the discontinuity is replaced by a straight segment of width 0.02 m, the overshoot is 0.17% at N=1000.
In everyday lifeHeavy compression of a photograph can produce fine ripple patterns next to sharp edges.
Detailed summary
Using eiθ turns differentiation of oscillations into multiplication, reducing differential equations to algebraic equations. The shape of a string fixed at both ends can be expressed as a sum of sine functions corresponding to its normal modes, and the coefficients can be found using orthogonality. For a continuous shape, even one with a sharp corner like a plucked string, the sum converges to the original shape at every point. For a shape with a discontinuity, an overshoot of about 9% remains near the jump. Each component oscillates at its own frequency, so adding the components gives the complete motion.