Introduction to Superposition and Standing Waves

Welcome to one of the most fascinating parts of Physics! In previous chapters, we looked at how individual waves behave. But what happens when two waves meet? Unlike solid objects (like two cars crashing), waves pass right through each other. As they overlap, they combine in a process called superposition. This simple idea explains everything from how noise-cancelling headphones work to why a guitar string makes music.

Don't worry if these terms feel a bit "maths-heavy" at first. We will break them down into simple rules and clear pictures.

Note: This chapter focuses on how waves combine. For basic wave definitions like frequency and wavelength, see the "Wave properties" chapter.


The Principle of Superposition

The Principle of Superposition states that when two or more waves meet at a point, the total displacement at that point is the vector sum of the displacements of the individual waves.

Imagine two "pulses" on a rope heading toward each other:

  • Constructive Interference: If both pulses are "up" (positive displacement), they combine to make a single, much larger "up" pulse at the moment they meet.
  • Destructive Interference: If one pulse is "up" and the other is "down" (negative displacement), they subtract from each other. If they are the same size, they will momentarily look like a flat line!

Quick Tip: Always remember the word vector. Because displacement has a direction, you must add positive and negative values together.


Coherence and Interference

To see a clear, steady pattern of interference (where waves continuously cancel out or add up), the wave sources must be coherent.

Coherence means the waves have:
1. A constant phase relationship (they stay "in step" with each other).
2. The same frequency.

If you use two different light bulbs, you won't see an interference pattern because the light waves are emitted in random bursts—they aren't coherent. This is why we often use a single laser split into two beams to ensure coherence.


Path Difference and Phase

Whether two waves interfere constructively or destructively depends on their phase difference at the point they meet. This is often caused by a path difference.

1. Path Difference \((\Delta x)\)

The path difference is simply the difference in the distance traveled by two waves from their sources to a specific point.

2. Phase Difference

Phase difference describes how far "out of sync" two waves are, measured in degrees or radians. One full cycle is \(360^\circ\) or \(2\pi\) radians.

The Golden Rules for Interference:

Let \(n\) be any whole number (\(0, 1, 2, ...\)):

  • Constructive Interference: Occurs when the path difference is a whole number of wavelengths.
    \(\text{Path Difference} = n\lambda\)
    The waves arrive in phase (phase difference = \(0, 2\pi, 4\pi...\)).
  • Destructive Interference: Occurs when the path difference is an odd number of half-wavelengths.
    \(\text{Path Difference} = (n + 0.5)\lambda\)
    The waves arrive out of phase or in "antiphase" (phase difference = \(\pi, 3\pi, 5\pi...\)).

The Relationship Formula:
\(\text{Phase Difference} = \frac{\text{Path Difference}}{\lambda} \times 2\pi\)


Standing (Stationary) Waves

A standing wave (also called a stationary wave) is a unique pattern that looks like it is vibrating in place rather than traveling along a medium. They are formed when two progressive waves with the same frequency and amplitude, traveling in opposite directions, superpose.

Key Features of Standing Waves:
  • Nodes: Points where the amplitude is always zero. Destructive interference is happening here constantly.
  • Antinodes: Points where the amplitude is at its maximum. Constructive interference is happening here.
  • Energy: Unlike progressive waves, standing waves do not transfer energy from one place to another. The energy is "trapped" in the vibrations.

Did you know? On a standing wave, all points between two adjacent nodes are in phase with each other, but they have different maximum amplitudes!


Transverse Waves on a String

When you pluck a guitar string, waves reflect off the fixed ends and travel back, creating a standing wave. The speed \(v\) of a transverse wave on a string depends on how tight the string is and how heavy it is.

The formula for the speed is:
\(v = \sqrt{\frac{T}{\mu}}\)

Where:
\(v\) = wave speed (\(\text{m s}^{-1}\))
\(T\) = tension in the string (\(\text{N}\))
\(\mu\) = mass per unit length of the string (\(\text{kg m}^{-1}\))

Combined with the wave equation \(v = f\lambda\), we can see that increasing the tension \(T\) increases the frequency \(f\) (making the note higher), while using a thicker string (higher \(\mu\)) decreases the frequency (making the note lower).


Core Practical 5: Investigating Vibrating Strings

In this practical, you investigate how the frequency \(f\) of the first harmonic (the simplest standing wave) is affected by the string's length, tension, and mass per unit length.

Standard Setup:

A vibration generator is connected to a signal generator. A string is stretched over a bridge and a pulley, with weights hanging off the end to provide tension.

Key Variables:
  • Length (\(L\)): As length increases, wavelength increases, so frequency decreases (\(f \propto \frac{1}{L}\)).
  • Tension (\(T\)): As tension increases, wave speed increases, so frequency increases (\(f \propto \sqrt{T}\)).
  • Mass per unit length (\(\mu\)): As the string gets heavier, wave speed decreases, so frequency decreases (\(f \propto \frac{1}{\sqrt{\mu}}\)).

Common Mistake: Students often forget that for the "first harmonic" (the basic loop), the length of the string \(L\) is only half a wavelength. So, \(\lambda = 2L\).


Quick Review Box

Superposition: Adding displacements of overlapping waves.

Coherence: Same frequency and constant phase relationship.

Constructive: Path difference = \(n\lambda\); waves are in phase.

Destructive: Path difference = \((n+0.5)\lambda\); waves are in antiphase.

Nodes: Zero amplitude in standing waves.

Antinodes: Max amplitude in standing waves.

String Speed: \(v = \sqrt{\frac{T}{\mu}}\)