The derivation of the wave equation in terms of a string’s kinetic and potential energy uses Lagrangian mechanics, a reformulation of classical mechanics. Here we focus on the potential energy term \(V\), especially why it contains the square of the spatial derivative, \(\left(\partial \psi / \partial x\right)^2\).
In simple terms, the potential energy \(V\) is associated with the stretching of the string. This is easiest to understand by looking at a small segment of the string after it has been displaced.
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Intuitive Understanding
Imagine taking a tiny section of the string between \(x\) and \(x + \mathrm{d}x\). If this segment is displaced vertically by a small amount \(\psi(x,t)\), it is not the absolute height that creates stretching. A segment lifted uniformly remains almost the same length. Stretching appears when neighboring points are displaced by different amounts.
The derivative
measures how quickly the displacement \(\psi(x,t)\) changes along the string. In other words, it measures the local slope of the string. A larger slope means that a small piece of the string is longer than it would be in equilibrium, so more energy is stored in the tension of the string.
Squaring the derivative,
makes the energy depend on the size of the slope rather than on its sign. A segment sloping upward and a segment sloping downward store the same amount of potential energy if the magnitude of the slope is the same.
Mathematical Interpretation
The potential energy term is
This is an integral over all points \(x\) along the string, from \(0\) to \(\ell\). Here, \(\mathcal{T}\) is the tension in the string. The tension acts as a restoring influence: when the string is stretched away from its equilibrium shape, energy is stored in that deformation.
The quantity
is the potential energy density per unit length, in the small-slope approximation. The integral then sums these local contributions across the whole string to give the total potential energy.
Thus the appearance of \(\left(\partial \psi / \partial x\right)^2\) reflects a simple physical fact: the string stores potential energy when its shape changes from point to point, not merely when it is displaced vertically. This term encodes the string’s resistance to changes in shape due to the tension \(\mathcal{T}\), and it is essential in deriving the wave equation.