Energy Distribution in String Vibration via Fourier Series

Problem Statement

In string instruments such as guitars, plucking (picking) a string at a specific location gives the string a particular initial displacement, exciting an infinite number of normal modes. In this problem, we rigorously analyze this phenomenon using the 1D wave equation and Fourier series expansion.

Take the horizontal direction as the $x$x-axis. A flexible, uniform string of linear mass density $\rho$ρ and tension $T$T is fixed at positions $x = 0$x=0 and $x = L$x=L. Let the displacement of the string be $y(x, t)$y(x,t), which obeys the 1D wave equation $\rho\frac{\partial^2 y}{\partial t^2} = T\frac{\partial^2 y}{\partial x^2}$ρ∂t2∂2y​=T∂x2∂2y​ for small vibrations. Damping due to air resistance or internal friction is neglected.

At time $t = 0$t=0, the point $x = \frac{L}{3}$x=3L​ on the string is pulled perpendicular to the string by distance $h$h and gently released (initial velocity is zero everywhere). The initial shape consists of two line segments: one from $x = 0$x=0 to $x = \frac{L}{3}$x=3L​, and another from $x = \frac{L}{3}$x=3L​ to $x = L$x=L, forming a triangle.

The string vibration $y(x, t)$y(x,t) can be expressed as a superposition of normal modes (Fourier sine series):

$$y(x, t) = \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi x}{L}\right)\cos(\omega_n t)$$y(x,t)=n=1∑∞​An​sin(Lnπx​)cos(ωn​t)

where $y_n(x,t)$yn​(x,t) is the displacement of the $n$n-th harmonic (fundamental at $n = 1$n=1), $\omega_n$ωn​ is its angular frequency, and $A_n$An​ is the Fourier coefficient determining the amplitude of each mode.

Define the mechanical energy (sum of kinetic and elastic energy) of the $n$n-th harmonic alone as $E_n$En​.

Under the given constraints, compute the mechanical energy $E_1$E1​ of the fundamental mode ($n = 1$n=1).

Constraints

• $L = 0.90 \text{ [m]}$L=0.90 [m]
• $\rho = 5.0 \times 10^{-3} \text{ [kg/m]}$ρ=5.0×10−3 [kg/m]
• $T = 160 \text{ [N]}$T=160 [N]
• $h = 0.03 \text{ [m]}$h=0.03 [m]

Input Format

The calculated energy $E_1 \text{ [J]}$E1​ [J] is expressed as:

$$E_1 = \frac{A}{B \pi^C}$$E1​=BπCA​

where $A$A and $B$B are coprime positive integers and $C$C is a positive integer.

Give the value of $A + B + C$A+B+C as a positive integer.