From Oscillators to Waves: The Wave Equation

Lesson notes

Key points

• Treat a taut string as tiny masses joined by tension; each segment's acceleration depends on the curvature around it.
• Applying Newton's second law to one segment gives the one-dimensional wave equation, linking time and space derivatives.
• For a string the wave speed is the square root of tension over linear mass density, so tighter strings carry faster waves.
• Because the equation is linear, any two solutions add, which is the principle of superposition behind interference and standing waves.

Definitions

Curvature

The second spatial derivative ∂²y/∂x²: how sharply the string bends, which sets the net tension force on a segment.

Takeaway: Waves obey a linear equation, so superposition always applies.

Coupled oscillatorsWave equationWave speedSuperposition