A guitar string of length \(l\) stretched along the \(x\)-axis is plucked in the middle. The initial deflection of the string has the shape \[ u(x, t = 0) = u_0(x) = \begin{cases} \frac{2 A_0 x}l & x < \frac l2 \\\\ \frac{2 A_0 (l - x)}l & x \geq \frac l2, \end{cases} \] where \(A_0\) is in the middle (see diagram). The string is then released and swings freely for time \(t > 0,\) and the resulting string vibration can be written as a superposition of standing waves \[u(x, t) = \sum_{n = 1}^\infty A_n \cos(2 \pi f_n t) \sin\left( \frac{n\pi}{l} x \right),\] with eigenfrequencies \(f_n = n \cdot \nu\) and fundamental frequency \(\nu\).

What is the relative amount of vibrational energy stored in the fundamental mode \(f = \nu?\) To do this, you'll have to determine the amplitudes \(A_n\).

Hint:

• The set of sine functions has the property \[ \int_0^l \sin\left( \frac{n\pi}{l} x \right) \sin\left( \frac{m\pi}{l} x \right) dx = \begin{cases}\frac l2 & n = m \\ 0 & n \not= m, \quad n, m \in \mathbb{N}. \end{cases} \]
• The kinetic energy of a small section of the string at position \(x\) of length \(dx\) is given by \(\frac12 \overbrace{\left(\frac{m}{l}dx\right)}^\textrm{mass} \left(\frac{du(x,t)}{dt}\right)^2,\) where \(m\) is the mass of the string. Assume that this holds along the entire string.