Reflection of Waves

Reflection of Waves

When a wave travelling through a medium encounters a boundary — a point where the medium changes — part or all of the wave is reflected. This phenomenon, reflection of waves, is fundamental to understanding how waves behave at interfaces. The key question is: what happens to the wave's displacement, velocity, and phase when it bounces back?

Consider a wave pulse travelling along a string. The string is attached at one end to a rigid wall. When the pulse reaches the wall, it cannot continue — the fixed end cannot move. The pulse must reflect. But the reflected pulse is not identical to the incident one; it is inverted. A crest becomes a trough, and a trough becomes a crest.

Why does this inversion happen? At the fixed end, the displacement must always be zero. The incident wave alone would produce a non-zero displacement at the wall. To keep the net displacement zero, the reflected wave must arrive with opposite displacement — exactly cancelling the incident wave's effect at the boundary. This is the physical origin of the phase change of $\pi$ radians (180°) upon reflection from a rigid boundary.

Now, let us examine the mathematics. Suppose an incident wave travelling along the positive $x$-direction is described by

At a rigid boundary at $x=0$, the reflected wave travels in the negative $x$-direction. Because of the inversion, we write

The negative sign represents the phase reversal. The total displacement at any point is the sum of the incident and reflected waves:

Using the trigonometric identity $\sin P - \sin Q = 2 \cos\frac{P+Q}{2} \sin\frac{P-Q}{2}$, we get

This is the equation of a standing wave. At the boundary $x=0$, $y(0,t) = 0$ for all $t$, as required.

For a free boundary — say, a string tied to a massless ring that slides on a frictionless rod — the end is free to move. The boundary condition is that the slope at the free end is zero (no transverse force). In this case, the reflected wave is not inverted:

The total displacement becomes

At the free end $x=0$, $y(0,t) = 2A \sin(\omega t)$ — the displacement is maximum, not zero.

Properties of Reflection

The textbook lists three key properties of wave reflection. Each is derived from the boundary conditions and the principle of superposition.

Property 1: Reflection from a rigid boundary produces a phase change of $\pi$

When a wave reflects from a rigid boundary (a fixed end), the displacement at the boundary must remain zero at all times. The incident wave alone would produce a non-zero displacement there. The reflected wave must therefore arrive with opposite displacement — a phase shift of $\pi$ radians — so that the two cancel exactly at the boundary.

Proof: Let the incident wave be $y_i = A \sin(\omega t - kx)$ and the reflected wave be $y_r = A' \sin(\omega t + kx + \phi)$, where $\phi$ is the phase change upon reflection. At the boundary $x=0$, the total displacement must be zero:

For this to hold for all $t$, we need $A' = A$ (amplitude unchanged for perfect reflection) and $\sin(\omega t) + \sin(\omega t + \phi) = 0$. Using the identity $\sin P + \sin Q = 2 \sin\frac{P+Q}{2} \cos\frac{P-Q}{2}$, we get

For this to be zero for all $t$, we require $\cos(\phi/2) = 0$, which gives $\phi/2 = \pi/2$, so $\phi = \pi$. The reflected wave is $y_r = A \sin(\omega t + kx + \pi) = -A \sin(\omega t + kx)$.

Property 2: Reflection from a free boundary produces no phase change

When a wave reflects from a free boundary, the end is free to move. The boundary condition is that the slope at the free end is zero — no transverse force acts at the boundary.

Proof: For a free end at $x=0$, the slope $\partial y/\partial x$ must be zero at all times. Let $y_i = A \sin(\omega t - kx)$ and $y_r = A' \sin(\omega t + kx + \phi)$. The total displacement is

The slope at $x=0$ is

For this to hold for all $t$, we need $A' = A$ and $\cos(\omega t) = \cos(\omega t + \phi)$. This requires $\phi = 0$ (or $2\pi$, which is equivalent). The reflected wave is $y_r = A \sin(\omega t + kx)$ — no phase change.

Property 3: The reflected wave travels with the same speed as the incident wave

The speed of a wave on a string depends only on the tension and linear mass density of the string: $v = \sqrt{T/\mu}$. Since the reflected wave travels in the same medium (the same string), both $T$ and $\mu$ are unchanged. Therefore, the speed of the reflected wave is identical to that of the incident wave. The wavelength and frequency also remain the same, as $v = f\lambda$.

The Principle of Superposition at Boundaries

The reflected wave is not created independently. It arises because the incident wave and the reflected wave must together satisfy the boundary condition at all times. This is a direct consequence of the superposition principle: the net displacement at any point is the sum of the displacements of all waves present. …