• Superposition is what happens when two or more waves meet — their displacements add together to give one resultant wave.
• Constructive interference = waves add up to a bigger amplitude. Destructive interference = waves cancel to a smaller (or zero) amplitude.
• Interference is only visible if the sources are coherent — same frequency, constant phase difference.
• Phase difference is about how "out of step" two wave cycles are (measured in degrees/radians). Path difference is about how much extra distance one wave has travelled (measured in wavelengths).
• Constructive interference happens when path difference = nλ (whole number of wavelengths). Destructive happens when path difference = (n + ½)λ.
• A stationary wave forms when two waves of the same frequency and amplitude travel in opposite directions and superpose — usually a wave and its own reflection.
• Stationary waves have fixed nodes (zero displacement, always) and antinodes (maximum displacement, oscillating up and down).
• Wave speed on a stretched string: v = √(T/μ). Fundamental frequency: f₀ = (1/2L)√(T/μ).
• Core Practical 5 investigates how the frequency of the first harmonic depends on string length, tension, or mass per unit length.

Picture two ripples on a pond heading toward each other. What happens the instant they cross paths? They don't bounce off each other or get destroyed — instead, at every point where they overlap, the water surface height is simply the sum of what each wave would have done on its own. Once they've fully passed through each other, both ripples carry on completely unchanged, as if the meeting never happened.

This "just add the displacements together" rule is called the principle of superposition, and the process itself is called interference. It applies to any type of wave — water, sound, light, waves on a string — as long as they're travelling through the same medium at the same time.

Constructive vs Destructive Interference

There are two extreme outcomes of superposition, depending on how the waves line up:

Constructive interference happens when the two waves meet "in phase" — crest lines up with crest, trough lines up with trough. The resultant amplitude is larger than either individual wave (if both waves have amplitude A, the resultant can be as big as 2A).

Destructive interference happens when the waves meet "in antiphase" — crest lines up with trough. The resultant amplitude is smaller than the individual waves, and if both amplitudes are equal, it cancels to exactly zero.

Coherence — the Essential Requirement

Here's the catch: you only get a stable, observable interference pattern if the two sources are coherent. Coherent waves must have:

• The same frequency
• A constant phase difference (it doesn't have to be zero, it just can't keep randomly changing)

Why does this matter so much? Imagine two sources whose phase relationship keeps jumping around randomly. One instant they might add constructively, the next instant destructively — the interference pattern would flicker so fast (and randomly) that you'd never actually see fringes or a fixed pattern. It would just look like an average, blurred-out mess.

This is exactly why a laser is coherent (it emits monochromatic light — a single frequency — with waves locked in step), while a filament lamp is not coherent (it emits a jumble of different frequencies and random phases from millions of independent atoms).

These two terms sound alike, and students mix them up constantly on exams — but they describe two genuinely different things. Let's separate them clearly.

Phase Difference

Two points are "in phase" if they are at the exact same stage of their wave cycle at the same moment — same displacement, moving in the same direction. Think of a wave cycle as a full circle (0° to 360°): phase difference is simply the angle between where two points sit on that circle.

So if one wave is at its peak (phase = 90°) and another is at zero going upward (phase = 0°), their phase difference is 90°, or a quarter of a cycle.

Path Difference

Path difference is a completely different idea. It's about distance travelled, not angle. Formally:

Why does path difference matter? Because if one wave has travelled exactly one whole wavelength further than the other, it will have completed one extra full cycle — meaning it arrives back "in step" with the other wave, and you get constructive interference again. But if it's travelled half a wavelength further, it arrives exactly out of step, giving destructive interference.

Worked Example — Two Sources

Two coherent sources S₁ and S₂ emit identical waves. At point P₁, the wave from S₁ has travelled 6λ, and the wave from S₂ has travelled 6.5λ. At point P₂, S₁'s wave has travelled 7λ and S₂'s has travelled 6λ.

This matches the general rule perfectly: a path difference that's a whole number of wavelengths → constructive; a path difference that's a whole number plus a half → destructive.

Now for the main event of this chapter. A stationary wave (also called a standing wave) is a special, very particular result of superposition: it happens when two waves of the same frequency and the same amplitude travel in opposite directions and overlap continuously.

In practice, this is almost always created by taking a single travelling wave and reflecting it back on itself — for example, a wave sent down a string that's fixed at the far end. The original outward wave and its reflected wave then superpose continuously, and something remarkable happens: the resulting pattern's peaks and troughs stop moving along the string. They just oscillate up and down in place. That's why it's called "stationary" — even though the two waves that make it up are very much still travelling (in opposite directions), the interference pattern itself appears frozen in space.

Where You See This In Real Life

This chapter covers three classic demonstrations, and it's genuinely useful to picture each one:

These all work on the same underlying physics: a driven wave reflects off some kind of boundary and interferes with itself.

Nodes and Antinodes

Every stationary wave is built from two key kinds of points:

One more subtle but important fact: between two adjacent nodes, every point on the stationary wave is in phase with every other point in that same "loop" (they all reach max displacement together, and all pass through zero together) — they just have different amplitudes depending on how close they are to the antinode. But cross over a node into the next loop, and everything flips into antiphase (exactly opposite direction) compared to the loop before it.

How fast does a wave actually travel along a taut string, like a guitar string? It turns out this depends on just two physical properties: how tightly the string is stretched (tension), and how "heavy" the string is per unit length. Makes intuitive sense — a tighter string snaps back faster (higher wave speed), and a heavier/thicker string is more sluggish to move (lower wave speed).

Getting to the Fundamental Frequency

Now combine this with what we know about stationary waves. At the fundamental frequency (also called the first harmonic) of a string of length L fixed at both ends, we established that λ = 2L (one loop, nodes at both ends, antinode in the middle).

The universal wave equation says v = fλ, so at the fundamental:

Harmonics — Higher Modes of Vibration

The fundamental (first harmonic) isn't the only possible standing wave pattern a string can support. If you drive the string at higher frequencies, you can get 2, 3, or more loops — these are called the second harmonic, third harmonic, and so on.

Notice the pattern: each successive harmonic's frequency is just a whole-number multiple of the fundamental frequency. This is exactly why musical instruments have such a rich, characteristic sound — the string vibrates in a mix of all these harmonics at once, not just the fundamental.

Worked Example

A guitar string of mass 3.2 g and length 90 cm is fixed onto a guitar. It's tightened to a tension of 65 N between two bridges 75 cm apart. Find (a) the wave speed, and (b) the fundamental frequency.

This is a required practical, so exam boards love asking about the method, the apparatus, the graph, and — especially — the sources of error. Let's build the whole picture.

Aim & Variables

The overall aim is to measure how the frequency of the first harmonic depends on one of three things (you only vary one at a time, keeping the others constant):

Dependent variable in every case: the frequency of the first harmonic, f.

Apparatus & Its Purpose

Method (varying length, as an example)

1. Attach one end of the string to the vibration generator, pass the other end over the pulley, and secure it to the mass hanger.
2. Position the wooden bridge so the length L (from vibration generator to bridge) can be measured with a metre ruler.
3. Turn on the signal generator to set the string oscillating.
4. Increase the frequency until the first harmonic is observed (nodes at both ends, single antinode in the middle) — read off this frequency.
5. Repeat with different values of L (a good spread, e.g. 0.2 m intervals over at least 1.0–1.6 m range).
6. Repeat each frequency reading at least twice more and average.
7. Calculate tension using T = mg (m = mass on hanger, g = 9.81 N kg⁻¹).
8. Find μ by weighing a known length of string (1 m is ideal) on the balance: μ = mass ÷ length.

Analysing the Results — the Linear Graph Trick

Here's the elegant bit. We know f = v/(2L), which can be rewritten as:

Compare this to the straight-line equation y = mx:

So if you plot f against 1/L, you should get a straight line through the origin. The gradient of that line, multiplied by 2, gives you the wave speed v — which you can then cross-check against v = √(T/μ) calculated independently. This is a really elegant way to test the theory using real data rather than a single calculation.

Evaluating: Errors & How to Reduce Them

Uncertainty Calculation Example

A student measures a wire: mass = 0.16 g, length weighed = 1.0 m, distance between fixed ends L = 0.4 m. Resolution: metre ruler = 1 mm (but real set-up errors up to 1 cm), top-pan balance = 0.005 g.

Key Terms

Formulas

• I can explain the principle of superposition and how it leads to interference.
• I can distinguish constructive and destructive interference by comparing peaks/troughs.
• I can explain what coherence means and why it's needed to observe an interference pattern.
• I can explain the difference between phase difference and path difference.
• I can apply the nλ and (n+½)λ conditions to determine the type of interference at a point.
• I can describe how a stationary wave forms from two waves travelling in opposite directions.
• I can describe the stretched string, microwave, and air column demonstrations of stationary waves.
• I can identify nodes and antinodes and explain the "NO Disturbance" memory trick.
• I know the distance between adjacent nodes is always λ/2.
• I can use v = √(T/μ) to calculate wave speed on a stretched string.
• I can derive and use f₀ = (1/2L)√(T/μ) for the fundamental frequency.
• I understand the wavelength patterns for the 1st, 2nd, and 3rd harmonics.
• I can describe the full method for Core Practical 5, including variables and control variables.
• I know why plotting f against 1/L gives a straight line, and what its gradient represents.
• I can identify systematic and random errors in the stationary wave practical and how to reduce them.
• I know that percentage uncertainty is halved for quantities under a square root.