05 / CHAPTER 10
Reflection, transmission, and impedance
Predict what happens when a wave meets a boundary.
ANCHOR EQUATION
Impedance reflection
In symbols: R = (Z₂ − Z₁)/(Z₂ + Z₁)
An impedance mismatch controls how much wave amplitude reflects.
Impedance decides reflection
A wave meeting a boundary must satisfy continuity conditions, so a mismatch between media generally creates reflected and transmitted waves. Impedance measures how strongly a medium resists the motion associated with a traveling wave.
Match force and velocity at the boundary
For a string, impedance is Z = √(Tμ). A wave traveling from Z₁ to Z₂ has amplitude reflection coefficient R = (Z₂ − Z₁)/(Z₂ + Z₁). If Z₂ = Z₁, R = 0; if the second medium is much heavier, |R| approaches 1. The sign tells whether the reflected displacement flips.
If Z₁ = 2 and Z₂ = 6 in consistent units, R = 4/8 = 0.50, so half the displacement amplitude reflects. The reflected intensity fraction is R² = 0.25 in the simple lossless model.
Amplitude and energy coefficients differ
A 50% amplitude reflection is not a 50% energy reflection; energy scales with squared amplitude and the two media carry different impedances. Track the quantity being compared.
Work it: Compute amplitude and intensity reflection fractions for Z₁ = 4 and Z₂ = 1. Decide whether the reflected displacement flips sign.