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Waves & Optics · Physics

The wave equation explains why shapes travel at a definite speed.

Connect the mathematical wave equation to physical properties of a medium—especially the speed of waves on a stretched string.

This topic

Wave Equation and Wave Speed

Learn what the 1D wave equation says, how speed depends on the medium, and how waves transport energy.

Core
One-dimensional wave equation
The 1D wave equation captures “shape travels” behavior. The key is interpreting what the equation implies, not memorizing the symbol pattern.
  • What the equation describes physically
  • Why second derivatives appear
  • Traveling wave solutions (conceptual)
  • What the wave speed parameter means
Start nano-lesson Interpret the equation before solving it.
String
Wave speed on a string
A stretched string supports transverse waves. Speed is not set by how hard you wiggle the string, but by tension and mass distribution.
  • What “wave speed” refers to
  • Speed vs particle motion speed
  • Qualitative dependence on tension
  • Why heavier strings are slower
Start nano-lesson Do not confuse wave speed with amplitude.
Dependence
Dependence on tension and mass density
The speed on a string depends on tension and linear mass density. Dimensional analysis helps you check the form and avoid algebra mistakes.
  • Tension increases speed
  • Higher linear density decreases speed
  • Dimensional analysis consistency check
  • Interpreting changes: ratios and scaling
Start nano-lesson Scaling arguments save time.
Energy
Energy transport in waves
Waves carry energy across space. Power depends on how strongly the medium oscillates and how fast energy flows through the medium.
  • Energy stored in the medium
  • Power as energy per time
  • Amplitude dependence (conceptual)
  • What changes when the medium changes
Start nano-lesson Energy flow is the point of “waves.”
Practice
Practice & Exercises
Practice interpreting the wave equation, computing string wave speeds, and using scaling reasoning without over-memorization.
  • Interpretation questions about the wave equation
  • Speed changes under tension/density changes
  • Dimensional analysis checks
  • Concept checks: amplitude vs speed
  • Exam-style string-wave sets
Open practice Always label the medium parameters.