Two-mode fixed-end string: exact solution, d'Alembert cross-check and the O(N^{-4}) tail bound at N=1

Let $L>0$ be a length, $c>0$ a speed, $A_{\ast }>0$ a displacement scale with $[A_{\ast }]=\mathrm{\ell }$, and $\beta \in \mathbb{R}\setminus \{0\}$ dimensionless. Let $[x]=\mathrm{\ell }$ and $[t]=\tau$, and consider the fixed-end string model of The string model: d'Alembert solutions, fixed-end modes and uniqueness for the adopted one-dimensional continuum with initial displacement and velocity

$F(x):=A_{\ast }[\sin (\frac{\pi x}{L})+\beta \sin (\frac{3\pi x}{L}\left)\right],G(x):=0.$

Then $F$ is odd and $2L$-periodic of class $C^{\mathrm{\infty }}$ and $G$ is odd and $2L$-periodic of class $C^4$, with $M_4=\sup \mathrm{\mid }{G}^{(4)}\mathrm{\mid }=0$ and $M_5=\sup \mathrm{\mid }{F}^{(5)}\mathrm{\mid }\le A_{\ast }(\pi \mathrm{/}L{)}^5(1+243\mathrm{\mid }\beta \mathrm{\mid })=:{\bar{M}}_5$ (Sine coefficients, sine series and the odd periodic extension). Put ${\omega }_n:=n\pi c\mathrm{/}L$ and

$u(t,x):=A_{\ast }[\cos ({\omega }_{1}t)\sin (\frac{\pi x}{L})+\beta \cos ({\omega }_{3}t)\sin (\frac{3\pi x}{L}\left)\right]\mathrm{.}$

Then:

1. the sine coefficients of $F$ are $b_1(F)=A_{\ast }$, $b_3(F)=\beta A_{\ast }$ and $b_n(F)=0$ otherwise, and the coefficients of $G$ all vanish; hence the series solution of The classical $C^2$ fixed-end string: the sine series solves the wave equation with the given data, uniquely, with an explicit tail bound is exactly the displayed finite sum $u$;
2. $u$ is $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^2$, satisfies $u_{tt}=c^2{u}_{xx}$, vanishes at $x=0$ and $x=L$ for every $t$, and has $u(0,\cdot )=F$, $u_t(0,\cdot )=0$; by the uniqueness clause of the theorem it is the unique $C^2$ fixed-end solution with these data;
3. the d'Alembert formula of d'Alembert's formula solves the wave equation, is the unique $C^2$ solution, and the fixed-end sine modes solve it exactly with data $(F,0)$ gives the same field: $\frac{1}{2}[F(x-ct)+F(x+ct)]=u(t,x)$ by the product-to-sum identity (The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine);
4. the tail bound at $N=1$ is verified: with $S_{1}u=A_{\ast }\cos ({\omega }_{1}t)\sin (\pi x\mathrm{/}L)$ one has $\mathrm{\mid }u-S_{1}u\mathrm{\mid }=A_{\ast }\mathrm{\mid }\beta \mathrm{\mid }\mathrm{\mid }\cos ({\omega }_{3}t)\mathrm{\mid }\mathrm{\mid }\sin (3\pi x\mathrm{/}L)\mathrm{\mid }\le A_{\ast }\mathrm{\mid }\beta \mathrm{\mid }$, while the theorem gives $\mathrm{\mid }u-S_{N}u\mathrm{\mid }\le C(N+1{)}^{-4}$ with $C\le \bar{C}:=4A_{\ast }(1+243\mathrm{\mid }\beta \mathrm{\mid })$, so the explicit upper estimate is $A_{\ast }(1+243\mathrm{\mid }\beta \mathrm{\mid })\mathrm{/}4$ at $N=1$, which also dominates the direct bound $A_{\ast }\mathrm{\mid }\beta \mathrm{\mid }$ for every $\beta$;
5. the mode frequencies satisfy $[{\omega }_n]={\tau }^{-1}$ under $[c]=\mathrm{\ell }{\tau }^{-1}$ and $[L]=\mathrm{\ell }$ (Continuum body data, the fixed inertial frame and the dimensions of the fields).

• The two-mode data make every series finite, so the check isolates the theorem's content — the mode frequencies, the boundary and initial conditions, uniqueness, the d'Alembert correspondence and the tail constant — without requiring convergence arguments beyond the supplier theorem.

• Modes with larger $n$ oscillate faster and the displayed bound is deliberately conservative at small $N$; the $O(N^{-4})$ decay becomes informative for large mode numbers and smooth data, as the theorem states. No measured frequency, tension or density appears.