Standing Waves

• Imagine a string of length $L$ that is rigidly fixed at $x = 0$ and $x = L$. This implies boundary conditions of $y(0, t) = 0$ and $y(L, t) = 0$.

• Making use of $y(0, t) = 0$ and Eq. (3), we find that $A = -C$ and $B = -D$, so
    $$y(x,t) = A [\sin (\omega t - kx) - \sin(\omega t + kx)] + B [\cos(\omega t - kx) - \cos (\omega t + kx)].$$ (4)

• Using sinusoidal sum and difference formulae, $\sin(x \pm y) = \sin(x)\cos(y) \pm \cos(x)\sin(y)$ and $\cos(x \pm y) = \cos(x)\cos(y) \mp \sin(x)\sin(y)$, we obtain
  $$\begin{eqnarray*} y(x,t) &=& 2 A \sin(k x)\cos(\omega t) - 2 B \sin(k x)\sin((\omega t) \\ &=& 2 [A \cos(\omega t) - B \sin(\omega t)] \sin(k x). \end{eqnarray*}$$

• The boundary condition $y(L, t) = 0$ requires that $\sin(k L) = 0$ or $\omega L / c = n \pi$, so that $\omega$ is restricted to values $\omega_n = n \pi c / L$ or $f_n = n c / 2 L$.

• Thus, the rigidly fixed string has normal modes of vibration given by
    $$y_n(x,t) = (A_n \sin(\omega_n t) + B_n \cos(\omega_n t)) \sin(\omega_n x / c),$$ (5)
  which are harmonic because each $f_n$ is a integer multiple $n$ times $f_1 = c / 2 L$. These correspond to the well known standing wave patterns on a stretched string.