Understanding Wave Reflection

To find the reflected wave, you determine its characteristics (amplitude, frequency, wavelength, and phase) based on the properties of the incident wave and the nature of the boundary it encounters.

Understanding Wave Reflection

When a wave traveling through a medium reaches a boundary where the medium changes or ends, part of the wave is reflected back into the original medium. The properties of this reflected wave depend significantly on how the boundary behaves.

There are primarily two types of boundaries for transverse waves (like waves on a string or light waves) and similar concepts apply to longitudinal waves (like sound waves):

Fixed Boundary: The end of the medium is fixed and cannot move (e.g., a string tied to a wall).
Free Boundary: The end of the medium is free to move (e.g., a string tied to a loose ring on a pole).

Finding the Reflected Wave at a Free Boundary

Finding the equation for a reflected wave involves understanding how the boundary condition affects the wave's phase and amplitude.

For a free boundary, the wave is reflected without a phase reversal. This means the reflected wave's displacement is in the same direction as the incident wave's displacement at the boundary.

As stated in the provided reference on the Mathematical Equation of Reflection of Waves from Free End:
Since the phase difference in reflection from a free boundary is zero. The equation of the reflected wave would be yr(x,t)=asin(kx+ωt).

Here's a breakdown of the equation and its components:

yr(x,t): Represents the displacement of the reflected wave at a position x and time t.
a: Is the amplitude of the wave. For an ideal reflection from a free boundary in the same medium, the amplitude of the reflected wave is often equal to the amplitude of the incident wave.
k: Is the wave number, related to the wavelength ($\lambda$) by $k = 2\pi/\lambda$.
x: Is the position along the wave's path.
ω: Is the angular frequency, related to the frequency ($f$) by $\omega = 2\pi f$.
t: Is time.
asin(kx+ωt): This sinusoidal function describes the wave's oscillation. The + sign before ωt indicates that the reflected wave is traveling in the negative x direction (assuming the incident wave yi(x,t) = asin(kx-ωt) travels in the positive x direction). The key point from the reference is the zero phase difference, which means there is no extra phase shift added to the wave function upon reflection at a free end, unlike a fixed end.

Key Characteristics at a Free Boundary

Amplitude: Typically the same as the incident wave, assuming no energy loss.
Frequency and Wavelength: Remain unchanged as the wave is in the same medium.
Speed: Remains unchanged.
Phase: The most crucial aspect at the boundary – there is no phase change upon reflection at a free end. The wave effectively reflects as if it reached the end and continued traveling, just in the opposite direction.

Contrasting with a Fixed Boundary

For context, understanding reflection from a fixed boundary helps highlight the uniqueness of the free boundary case:

At a fixed boundary, the wave is reflected with a 180-degree (or $\pi$ radians) phase difference. The wave is inverted upon reflection.
The equation for a reflected wave from a fixed end (if the incident wave is $yi(x,t)=asin(kx-ωt)$) would typically involve this phase shift, often appearing as a sign change: $yr(x,t)=-asin(kx+ωt)$ or $asin(kx+ωt + \pi)$.

Practical Examples

Waves on a String: If a string is tied to a loose ring that can slide freely on a pole (a free end), a wave pulse arriving at the ring will reflect back on the same side of the string it arrived on, with no inversion.
Sound Waves: When sound waves in an air pipe reach an open end, it approximates a free boundary. A compression pulse reflects as a rarefaction pulse (and vice-versa), which involves a phase change when considering pressure, but reflects without a phase change when considering displacement.

Summary: Steps to Find the Reflected Wave (Free End)

Identify the incident wave's properties (amplitude a, wave number k, angular frequency ω).
Confirm the boundary is a free boundary.
Recognize that for a free boundary, the phase difference upon reflection is zero.
Determine the direction of the reflected wave (opposite to the incident wave). If incident is $asin(kx-\omega t)$, reflected is $asin(kx+\omega t)$.
Use the general form, incorporating the identified properties and the zero phase shift. For a free end, the equation is directly given as $yr(x,t)=asin(kx+ωt)$, assuming the incident wave was $asin(kx-ωt)$.