Chemistry 352

Consider a string of length \( L \) stretched tightly between two fixed supports. If the string is displaced slightly and released, waves propagate along the string. The vertical displacement of the string is described by a function \( y(x,t) \), where \( x \) is the position along the string and \( t \) is time.

The motion of the string is governed by the one-dimensional wave equation:

\[ \frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}, \]

where \( v \) is the wave speed on the string.

Boundary conditions: fixed ends

Because the ends of the string are fixed, the displacement must be zero at \( x=0 \) and \( x=L \) for all times:

\[ y(0,t) = 0, \qquad y(L,t) = 0. \]

These boundary conditions strongly restrict the allowed motions of the string. Only certain spatial patterns—called standing waves—are permitted.

Separation of spatial and time variables

To solve the wave equation, we assume that the solution can be written as a product of a purely spatial function and a purely time-dependent function:

\[ y(x,t) = X(x)\,T(t). \]

Substituting this form into the wave equation and separating variables gives two ordinary differential equations:

\[ \frac{1}{T}\frac{d^2 T}{dt^2} = v^2 \frac{1}{X}\frac{d^2 X}{dx^2} = -\omega^2. \]

The separation constant \( -\omega^2 \) is chosen so that the solutions describe oscillatory motion.

Spatial solutions and allowed modes

The spatial equation becomes

\[ \frac{d^2 X}{dx^2} + k^2 X = 0, \qquad \text{with } k = \frac{\omega}{v}. \]

Applying the boundary conditions \( X(0)=0 \) and \( X(L)=0 \) leads to the allowed spatial solutions

\[ X_n(x) = \sin\!\left(\frac{n\pi x}{L}\right), \qquad n = 1,2,3,\ldots \]

Each integer \( n \) corresponds to a distinct standing wave with \( n-1 \) interior nodes. The associated angular frequencies are

\[ \omega_n = \frac{n\pi v}{L}. \]

Time dependence

For each spatial mode, the time-dependent function satisfies

\[ \frac{d^2 T}{dt^2} + \omega_n^2 T = 0. \]

The general solution is oscillatory:

\[ T_n(t) = A_n \cos(\omega_n t) + B_n \sin(\omega_n t). \]

Each spatial mode oscillates independently at its own characteristic frequency.

Superposition principle

The wave equation is linear. As a result, any linear combination of solutions is also a solution. This leads to the most general motion of the string:

\[ y(x,t) = \sum_{n=1}^{\infty} \left[ A_n \cos(\omega_n t) + B_n \sin(\omega_n t) \right] \sin\!\left(\frac{n\pi x}{L}\right). \]

This expression shows that the motion of the string can be written as a superposition of spatial eigenfunctions. The coefficients \( A_n \) and \( B_n \) are determined by the initial shape and initial velocity of the string.

Any physically allowed displacement of the string can be constructed by adding together these standing-wave spatial functions.

Key points (one glance)

• The wave equation separates into spatial and time-dependent parts.
• Fixed-end boundary conditions restrict allowed spatial functions.
• Only sine functions satisfy both boundary conditions.
• Each spatial mode oscillates with its own frequency.
• General motion is a superposition of standing-wave modes.
• This structure closely parallels quantum mechanical eigenstates.