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Boundary Conditions

Level 1 - Physics topic page in Mechanical Waves.

Principle

Reflection at a boundary is set by the condition the medium must satisfy at that boundary.

The inversion or non-inversion of the reflected wave is not a separate rule to memorize; it comes from enforcing the correct boundary condition.

Notation

\(y_i\)
incident-wave displacement
\(\mathrm{m}\)
\(y_r\)
reflected-wave displacement
\(\mathrm{m}\)
\(y\)
total displacement
\(\mathrm{m}\)
\(x=L\)
boundary position
\(\mathrm{m}\)

Method

Derivation 1: Apply the fixed-end condition

At a fixed end, the medium cannot move. The total displacement must therefore be zero at the boundary for all times.

Fixed end
\[y(L,t)=0\]
Add incident and reflected parts
\[y_i(L,t)+y_r(L,t)=0\]
Reflection rule
\[y_r(L,t)=-y_i(L,t)\]
The reflected pulse is inverted at the boundary.

The fixed-end reflection below shows that the reflected pulse has opposite sign so the total displacement vanishes at the wall.

0246810-1.2-0.600.61.2xyincidentreflectedfixed

Derivation 2: Apply the free-end condition

At a free end, the boundary cannot supply a transverse force. For a string, that means the slope must vanish there.

Free end
\[\frac{\partial y}{\partial x}(L,t)=0\]
Slope cancellation
\[\frac{\partial y_i}{\partial x}(L,t)+\frac{\partial y_r}{\partial x}(L,t)=0\]
Reflection rule
\[y_r(L,t)=y_i(L,t)\]
The reflected pulse is not inverted.

Derivation 3: Identify what does and does not change on reflection

The reflection changes phase according to the boundary condition, but the wave speed in the same medium is unchanged.

Same medium
\[v_{r}=v_i\]
Fixed-end phase change
\[\Delta\phi=\pi\]
Free-end phase change
\[\Delta\phi=0\]

Rules

These are the compact results from the method above.

Fixed-end condition
\[y(L,t)=0\]
Fixed-end reflection
\[y_r(L,t)=-y_i(L,t)\]
Free-end condition
\[\frac{\partial y}{\partial x}(L,t)=0\]
Free-end reflection
\[y_r(L,t)=y_i(L,t)\]

Examples

Question
A crest pulse on a string reaches a rigid wall. What shape returns, and what happens to the wave speed?
Answer
The reflected pulse returns as a trough because a fixed end requires zero displacement at the wall. The speed is unchanged because the string medium is unchanged.

Checks

  • Inversion comes from the boundary condition, not from the direction of travel alone.
  • A free end reflects without inversion.
  • Reflection does not require the wave speed to change in the same medium.
  • At a fixed end, the total displacement at the boundary is always zero.

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