How is the classical wave equation derived from Newton's second law?

Conditions

• The string is ideal (knot-free and inextensible), so tension $F_T$ is uniform.
• The wave is transverse, so particles move only vertically.
• The string disturbance is very small, allowing the small-angle approximation.

Reasoning, step by step

1. Take an infinitesimal element $\Delta l$ on the rope with horizontal projection $\Delta x$.
2. Write the net vertical force as $F_T[\sin(\theta+\Delta\theta)-\sin\theta]$.
3. Express the mass of the element as $m = \mu\Delta x/\cos\theta$.
4. Express the vertical acceleration as $a = \partial^2 f(x,t)/\partial t^2$.
5. Apply Newton's second law: $F_T[\sin(\theta+\Delta\theta)-\sin\theta] = \mu(\Delta x/\cos\theta)\partial^2 f(x,t)/\partial t^2$.
6. Apply the small-angle approximation to replace $\sin$ with $\tan$ and $\cos\theta$ with 1.
7. Substitute $\tan\theta$ with the first partial derivative $\partial f/\partial x$.
8. Divide by $\Delta x$ and take the limit to obtain the second partial derivative $\partial^2 f/\partial x^2$.

Example

The screen step by step demonstrates the simplification process from Newton's second law equation to the final wave equation, yielding $\frac{\partial^2 f(x,t)}{\partial x^2} = \frac{\mu}{F_T} \cdot \frac{\partial^2 f(x,t)}{\partial t^2}$.

Common misconceptions

• Assuming the derivation holds for large-amplitude waves without the small-angle approximation.
• Confusing the wave propagation speed with the transverse velocity of the string particles.

Watch the explanation

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• How is the small-angle approximation applied in deriving the wave equation?
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