AP Physics 1: Wave Reflection and Transmission

Understanding what happens when a wave meets a barrier or passes from one material into another is crucial for explaining phenomena from earthquake safety to the design of musical instruments and fiber optic cables. This analysis of wave behavior at boundaries connects the abstract principles of wave mechanics to tangible, real-world engineering and scientific applications.

The Incident, Reflected, and Transmitted Wave

When a traveling wave encounters a boundary—the interface between two different media—it does not simply stop. Its energy is partitioned. The original wave approaching the boundary is called the incident wave. Upon reaching the boundary, part of its energy is sent back into the original medium, creating a reflected wave. The remaining energy is carried forward into the new medium, creating a transmitted wave (sometimes called the refracted wave).

The division of energy, and thus the amplitudes of the reflected and transmitted waves, depends on the properties of the two media. If the media are very similar, most of the energy transmits, and the reflected wave is weak. If they are very different, most of the energy reflects. A key principle here is the conservation of energy: the energy carried by the incident wave is equal to the sum of the energies carried by the reflected and transmitted waves (ignoring absorption). For a wave on a string, the relevant property is the linear mass density ($\mu$, mass per unit length) and the tension ($F_T$). The wave speed on a string is given by $v=\sqrt{F_T/\mu }$.

Phase Changes Upon Reflection: Fixed vs. Free Ends

The phase of the reflected wave—whether its pulse is upright or inverted compared to the incident pulse—is determined by the nature of the boundary. This is a critical concept for understanding standing waves.

At a fixed end (or rigid boundary), the boundary condition is that the medium cannot move. Imagine a rope tied tightly to a wall. When a crest pulse arrives, it exerts an upward force on the wall. By Newton's third law, the wall exerts a downward force back on the rope. This downward force generates a reflected pulse that is inverted (a trough). We say the wave undergoes a 180° phase change upon reflection from a fixed end.

At a free end (or loose boundary), the medium is free to move vertically. Imagine a rope attached to a vertical ring that can slide without friction. When a crest arrives, it pulls the ring up, and the ring's inertia carries it past equilibrium, creating a new crest pulse traveling back. The reflected pulse from a free end is upright and experiences no phase change.

The Role of Medium Properties: Impedance Mismatch

The characteristic that determines how much energy reflects and transmits is the impedance of the medium. For a mechanical wave on a string, impedance ($Z$) is defined as $Z=\mu v=\sqrt{F_T\mu }$. It represents how much the medium "resists" the wave's motion. The greater the difference in impedance between two media—the greater the impedance mismatch—the greater the fraction of energy reflected.

Let's analyze two strings tied together, with the same tension but different linear densities. The wave speed is higher in the lighter string ($v=\sqrt{F_T/\mu }$). When a wave travels from a light string (low $\mu$, low $Z$) to a heavy string (high $\mu$, high $Z$), it is like moving from a "fast" medium to a "slow" medium. In this case, the reflected wave is inverted (a 180° phase change), similar to a fixed end. The transmitted wave continues upright but with reduced amplitude and slower speed.

Conversely, when a wave travels from a heavy string (high $Z$) to a light string (low $Z$), it moves from "slow" to "fast." The reflected wave is upright (no phase change), similar to a free end. The transmitted wave is upright, with a higher speed but reduced amplitude.

The exact amplitudes can be derived from boundary conditions requiring continuity of the wave function and its derivative at the knot. The results show that complete transmission ($A_{transmitted}=A_{incident}$) only occurs when the impedances are matched ($Z_1=Z_2$).

Common Pitfalls

1. Confusing end conditions with medium changes. A fixed end always causes an inversion, and a free end never does. However, when going between two different strings, the phase of the reflection depends on the direction of travel. Traveling from low impedance to high impedance causes an inverted reflection (like a fixed end), while high to low causes an upright reflection (like a free end). Mixing up this directional rule is a frequent error.
2. Assuming amplitude determines speed. The speed of a wave is determined solely by the properties of the medium ($v=\sqrt{F_T/\mu }$ for a string). The amplitude of the transmitted wave may be larger or smaller than the incident wave, but its speed in the new medium is fixed and unrelated to that amplitude. Do not think a bigger pulse travels faster.
3. Forgetting that energy is proportional to amplitude squared. When comparing energies, remember that a wave's energy is proportional to the square of its amplitude ($E\propto A^2$). If the transmitted wave has half the amplitude of the incident wave, it carries only one-quarter of the energy, not half.
4. Overlooking conservation principles. In a closed system, the energy of the incident wave must be accounted for. It doesn't disappear. If a problem gives you amplitudes, you can often check your work by verifying that energy (amplitude squared) is conserved across the boundary, considering the different wave speeds and densities if necessary.

Summary

• When a wave meets a boundary between two media, its energy is split into a reflected wave (back into the first medium) and a transmitted wave (into the second medium).
• Reflection from a fixed end results in a 180° phase change (inversion), while reflection from a free end results in no phase change.
• For two connected strings, the phase of the reflection depends on the impedance mismatch: low-to-high impedance reflection is inverted, while high-to-low impedance reflection is upright.
• The wave speed in a medium is an intrinsic property ($v=\sqrt{F_T/\mu }$) and does not change with amplitude.
• The amplitude of the transmitted and reflected waves depends on the relative impedance of the two media, with maximum transmission occurring when impedances are matched.
• The total energy is conserved, with the incident wave's energy distributed between the reflected and transmitted waves.