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Superposition, Reflection, and Standing Waves

The Principle Of Superposition Of Waves ($ y = y_1 + y_2 $)

The Principle of Superposition of Waves is a fundamental principle that governs the behaviour of waves when two or more waves meet or overlap in the same region of space. It states that for linear media, the net disturbance at any point and at any time is the vector sum of the disturbances produced by each wave individually, as if the other waves were not present.

Statement of the Principle

When two or more waves traverse the same medium, the displacement of any particle in the medium at any instant is the vector sum of the displacements due to each wave acting independently at that instant.

If $\vec{y}_1(x,t)$ is the displacement due to wave 1 at position $x$ and time $t$, and $\vec{y}_2(x,t)$ is the displacement due to wave 2 at the same position and time, then the resultant displacement $\vec{y}(x,t)$ is:

$ \vec{y}(x,t) = \vec{y}_1(x,t) + \vec{y}_2(x,t) $

For waves in one dimension (e.g., transverse waves on a string, where displacement is vertical, or longitudinal waves in air, where displacement is horizontal), the superposition is a simple scalar sum:

$ y(x,t) = y_1(x,t) + y_2(x,t) $

This principle applies to all types of waves (mechanical waves, electromagnetic waves) as long as the medium is linear (meaning its properties do not change with the amplitude of the wave). Most media behave linearly for small wave amplitudes.

Consequences of Superposition

The principle of superposition is the basis for understanding phenomena like:

The principle allows us to analyse complex wave patterns by breaking them down into simpler components and summing their effects.

Diagram illustrating the superposition of two pulses on a string.

(Image Placeholder: Two diagrams showing the superposition of wave pulses on a string. One shows two pulses crest-up moving towards each other, overlapping to form a larger pulse, and then moving apart unchanged (constructive superposition). Another shows one pulse crest-up and one pulse crest-down moving towards each other, cancelling out at the point of overlap, and then moving apart unchanged (destructive superposition).)

During superposition, the waves pass through each other without being permanently altered, although the medium's displacement at the point of overlap is the sum of their individual displacements.


Reflection Of Waves

When a wave encounters a boundary between two different media or an obstacle, part or all of the wave is sent back into the original medium. This phenomenon is called reflection.

The reflection of waves follows the law of reflection: the angle of incidence equals the angle of reflection.

The behaviour of the reflected wave at the boundary depends on the nature of the boundary.

Reflection from a Fixed Boundary

When a wave in a medium reaches a boundary where the medium is rigidly fixed or more dense (e.g., a pulse on a string reflecting from a fixed end, a sound wave reflecting from a wall), the reflected wave is inverted relative to the incident wave. This means there is a phase change of $\pi$ radians (or 180°) upon reflection. For example, a crest arriving at a fixed boundary reflects back as a trough.

Diagram showing reflection of a wave pulse from a fixed end.

(Image Placeholder: A string fixed at one end. Show an incoming pulse travelling towards the fixed end (e.g., a crest). Show the reflected pulse travelling back, inverted (a trough).)

Reflection from a Free Boundary

When a wave reaches a boundary that is free to move or less dense (e.g., a pulse on a string reflecting from a free end where the string can move vertically, a sound wave reflecting from an open end of a tube), the reflected wave is not inverted relative to the incident wave. There is no phase change upon reflection (or a phase change of 0 or $2\pi$ radians). A crest arriving at a free boundary reflects back as a crest.

Diagram showing reflection of a wave pulse from a free end.

(Image Placeholder: A string with a free end (e.g., attached to a light ring sliding on a vertical rod). Show an incoming pulse (e.g., a crest) travelling towards the free end. Show the reflected pulse travelling back, not inverted (a crest).)

Applications of Reflection

Standing Waves And Normal Modes

When two identical progressive waves with the same amplitude and frequency travel in opposite directions in a bounded medium (e.g., a string fixed at both ends, an air column in a pipe), they interfere to produce standing waves (also called stationary waves).

Formation of Standing Waves

A standing wave appears stationary; it does not transfer energy from one end to the other. Points in the medium oscillate with varying amplitudes, but the pattern of displacement nodes (points of zero displacement) and antinodes (points of maximum displacement) is fixed in space.

Standing waves are formed by the superposition of the original wave and its reflection from the boundary. The interference between these waves creates points of constructive interference (antinodes) and destructive interference (nodes) that remain at fixed positions.

Let two identical waves travelling in opposite directions be described by:

$ y_1(x,t) = A \sin(kx - \omega t) $ (travelling in +x direction)

$ y_2(x,t) = A \sin(kx + \omega t + \delta) $ (travelling in -x direction, with phase difference $\delta$)

By superposition, the resultant displacement is $y(x,t) = y_1(x,t) + y_2(x,t)$. Using trigonometric identities:

$ y(x,t) = A [\sin(kx - \omega t) + \sin(kx + \omega t + \delta)] $

$ y(x,t) = A [2 \sin(\frac{(kx - \omega t) + (kx + \omega t + \delta)}{2}) \cos(\frac{(kx + \omega t + \delta) - (kx - \omega t)}{2})] $

$ y(x,t) = 2A \sin(kx + \delta/2) \cos(\omega t + \delta/2) $

This equation describes a standing wave. The term $2A \sin(kx + \delta/2)$ represents the amplitude of oscillation at position $x$. This amplitude varies with $x$. The term $\cos(\omega t + \delta/2)$ represents the time-varying part of the oscillation at position $x$, with angular frequency $\omega$.

Nodes and Antinodes

Nodes are separated by a distance of $\lambda/2$. Antinodes are also separated by a distance of $\lambda/2$. A node is always halfway between two adjacent antinodes, and vice versa.

Diagram illustrating a standing wave with nodes and antinodes.

(Image Placeholder: A diagram showing a standing wave on a string fixed at both ends. The string is shown vibrating in a pattern that has fixed points (nodes) and points of maximum vibration (antinodes). Label the nodes and antinodes. Show the shape of the string at different times, outlining the envelope of the standing wave.)

Boundary Conditions and Normal Modes

Standing waves can only form in a bounded medium if they satisfy the boundary conditions imposed by the ends of the medium (e.g., fixed end, free end). These boundary conditions restrict the possible wavelengths and frequencies of the standing waves that can exist in the medium.

These boundary conditions lead to a set of discrete possible wavelengths and corresponding frequencies for standing waves in a given system. These specific allowed modes of vibration are called normal modes or harmonics.

For a string of length $L$ fixed at both ends, the boundary condition is that there must be nodes at $x=0$ and $x=L$. The possible wavelengths are such that $L = n (\lambda/2)$, where $n = 1, 2, 3, ...$ (an integer representing the mode number). So, $\lambda_n = 2L/n$. The corresponding frequencies are $\nu_n = v/\lambda_n = v/(2L/n) = n(v/2L)$, where $v$ is the wave speed on the string. The lowest frequency ($n=1$) is the fundamental frequency or first harmonic ($\nu_1 = v/2L$). Higher frequencies are overtones or harmonics ($\nu_n = n\nu_1$).

Applications of Standing Waves: