AP Physics 1: Wave Reflection and Transmission

Waves don't just travel forever in empty space; they inevitably encounter boundaries. Understanding what happens when a wave pulse or a continuous wave hits the edge of its medium or passes into a new one is crucial for explaining phenomena from the echo in a canyon to the clarity of an ultrasound image. This analysis of boundary behavior connects fundamental wave properties to powerful real-world applications in engineering, medicine, and communications.

The Wave-Medium Partnership and Boundary Interactions

To predict a wave's behavior at a boundary, you must first understand its relationship with the medium—the material through which it travels. A wave is a disturbance that transfers energy, and the medium's physical properties determine how easily this energy propagates. The key property here is the wave speed, $v$, which for a mechanical wave on a string, for example, depends on the string's tension ($T$) and its linear density ($\mu$): $v=\sqrt{T/\mu }$.

When a wave traveling in one medium (Medium 1) encounters a boundary with a second medium (Medium 2), the energy of the incident wave must be accounted for. It cannot simply vanish. The energy splits, resulting in two distinct waves: a reflected wave that travels back into the first medium, and a transmitted wave that continues forward into the second medium. This division of energy is why reflection and transmission are often partial, not total.

Analyzing Fixed and Free End Boundaries

The simplest boundaries to analyze are the extremes: a fixed end and a free end. These are special cases where the wave is completely reflected, but with a critical difference in the phase of the reflected pulse.

A fixed end is a boundary where the medium is rigidly clamped and cannot move. Imagine a rope tied tightly to a solid wall. When a pulse reaches this fixed point, the wall exerts an upward force to keep the end from moving. This reactive force generates a reflected pulse that is inverted—it is upside-down relative to the incident pulse. We say the pulse undergoes a phase change of 180° (or $\pi$ radians) upon reflection. A crest reflects as a trough.

A free end is a boundary where the medium is free to move. Picture a rope hanging vertically, with its end loose. When a pulse arrives, the end overshoots, and the inertia creates a reflected pulse that is upright. No phase change occurs; a crest reflects as a crest. This principle applies to any boundary where the medium is not fixed, such as the open end of a pipe in a sound wave.

Reflection and Transmission at a Medium Boundary

Most practical situations involve a wave moving from one distinct medium into another, like light passing from air into glass or a sound wave moving from water into air. Here, both partial reflection and partial transmission occur. The key factors are the wave speed ($v$) and the characteristic impedance of the media—a property related to the medium's density and stiffness that determines how much it "resists" wave motion.

The relative "heaviness" or "stiffness" of the two media governs what happens. We define Medium 2 as denser or slower if the wave speed in it ($v_2$) is less than the wave speed in Medium 1 ($v_1$).

• From Fast to Slow (Low to High Impedance): When a wave travels from a faster medium into a slower one (e.g., from a light string to a heavy string), the reflected pulse is inverted (180° phase change), just like at a fixed end. The transmitted pulse continues upright but with a slower speed and shorter wavelength.
• From Slow to Fast (High to Low Impedance): When a wave travels from a slower medium into a faster one (e.g., from a heavy string to a light string), the reflected pulse is upright (no phase change), analogous to a free end. The transmitted pulse is upright, with a faster speed and longer wavelength.

The amplitude of the transmitted and reflected waves depends on the impedance mismatch. A large difference (like sound hitting a solid wall) leads to strong reflection and weak transmission. A small difference leads to more energy transmission, which is why ultrasound gel is used to match impedances between the transducer and skin for better imaging.

Quantitative Relationships: Amplitude and Wave Speed

You can predict the amplitudes of the reflected and transmitted waves using the principle of superposition and boundary conditions (continuity of displacement and force). The results are often expressed in terms of the wave speeds in the two media. For a pulse on a string incident from Medium 1 ($v_1$) onto Medium 2 ($v_2$), the equations are:

• Transmitted Amplitude: $A_t=\left(\frac{2v_2}{v_2+v_1}\right)A_i$
• Reflected Amplitude: $A_r=\left(\frac{v_2-v_1}{v_2+v_1}\right)A_i$

Where $A_i$ is the incident amplitude. Let's work through an example: A wave pulse with amplitude 6.0 cm travels on a light string ($v_1=4.0$ m/s) attached to a heavier string ($v_2=2.0$ m/s). The transmitted amplitude is $A_t=\left(\frac{2(2.0)}{2.0+4.0}\right)(6.0)=\left(\frac{4.0}{6.0}\right)(6.0)=4.0$ cm. The reflected amplitude is $A_r=\left(\frac{2.0-4.0}{2.0+4.0}\right)(6.0)=\left(\frac{-2.0}{6.0}\right)(6.0)=-2.0$ cm. The negative sign on $A_r$ confirms the inversion (phase change) upon reflection into the faster medium.

The frequency of the wave does not change at the boundary—the source dictates the frequency. Since $v=f\lambda$, a change in wave speed ($v$) must be accompanied by a change in wavelength ($\lambda$). The transmitted wave in a new medium will have a different wavelength than the incident wave: ${\lambda }_2={\lambda }_1(v_2/v_1)$.

Common Pitfalls

1. Confusing "Fixed/Free" with "Fast-to-Slow/Slow-to-Fast": Students often memorize that "fixed end means inversion" but then misapply it to medium boundaries. Remember: Reflection at a true fixed end always inverts. Reflection at a medium boundary inverts only when going from a faster medium to a slower medium (low to high impedance). Going from slow to fast results in an upright reflection, like a free end.

1. Assuming Frequency Changes: A common mistake is to think the transmitted wave has a different frequency. The frequency is determined by the source and remains constant across the boundary. Only the wave speed and wavelength change in the new medium. The equation $v=f\lambda$ is your best tool to track these changes.

1. Misinterpreting the Amplitude Ratio Sign: In the reflected amplitude equation $A_r=\left(\frac{v_2-v_1}{v_2+v_1}\right)A_i$, the sign of the ratio is critical. A positive result means no phase change (upright reflection). A negative result means a 180° phase change (inverted reflection). Ignoring the sign loses half the physical information.

1. Overlooking Energy Conservation: The sum of the energies of the reflected and transmitted waves must equal the energy of the incident wave (assuming no absorption). Since wave energy is proportional to amplitude squared, a small amplitude difference can mean a large energy difference. A transmitted wave with half the incident amplitude carries only one-quarter of the energy.

Summary

• When a wave encounters a boundary, energy is conserved through partial reflection and transmission, except at perfectly fixed or free ends where reflection is total.
• A wave reflected from a fixed end inverts (180° phase change). A wave reflected from a free end reflects upright (no phase change).
• At a boundary between two media, the phase of the reflected wave depends on the relative wave speeds: reflection inverts when moving from a faster medium to a slower one and remains upright when moving from a slower medium to a faster one.
• The amplitude of transmitted and reflected waves depends on the wave speed ratio, with more energy transmitted when the media have similar impedances.
• While frequency remains constant across a boundary, the wave speed and wavelength of the transmitted wave change according to the properties of the new medium.