Grade 11

Advanced

7 min

Lesson 2 of 12

The Principle of Superposition

Not Started

Exploring how waves interact when they occupy the same space simultaneously through vector addition of displacements.

Have you ever wondered how noise-canceling headphones can turn a roaring jet engine into a whisper of silence? It isn't magic—it's the mathematical power of waves colliding and canceling each other out with surgical precision.

Learning Objectives

1.State the principle of superposition for both longitudinal and transverse waves
2.Calculate the resultant displacement of overlapping wave pulses using vector addition
3.Distinguish between constructive and destructive interference in complex wave patterns

The Core Principle

The Principle of Superposition is the fundamental rule governing how waves interact. Unlike solid objects, two waves can occupy the exact same space at the same time. When they do, they don't bounce off each other; instead, they pass through one another. At the moment of overlap, the resultant displacement is the algebraic (vector) sum of the individual displacements. If

y_1

is the displacement of the first wave and

y_2

is the displacement of the second, the total displacement

y_{total}

is given by:

y_{total} = y_1 + y_2

This applies to both transverse waves (like light or strings) and longitudinal waves (like sound).

Two pulses travel toward each other on a string. Pulse A has an amplitude of $+3\text{ cm}$ and Pulse B has an amplitude of $+5\text{ cm}$ .

1. Identify the displacements: $y_A = +3$ and $y_B = +5$ . 2. Apply the superposition formula: $y_{total} = y_A + y_B$ . 3. Calculate: $3 + 5 = 8\text{ cm}$ . 4. Result: At the instant they overlap perfectly, the string reaches a height of $+8\text{ cm}$ .

Quick Check

If a pulse with displacement $+4\text{ m}$ meets a pulse with displacement $-4\text{ m}$ , what is the resultant displacement at that point?

Remember to treat the displacements as positive and negative values.

Answer

0 meters (Complete destructive interference).

Constructive vs. Destructive Interference

Interference is the phenomenon that occurs during superposition. We categorize it into two types. Constructive Interference occurs when waves have displacements in the same direction (both positive or both negative), resulting in a larger amplitude. Destructive Interference occurs when waves have displacements in opposite directions. If the waves have equal but opposite amplitudes, they can momentarily produce a displacement of zero, known as total destructive interference. Crucially, after the waves pass through each other, they continue on their path with their original shapes and speeds unchanged.

A square wave pulse of height $+10\text{ units}$ and width $2\text{ m}$ moves right. A triangular pulse of peak height $-6\text{ units}$ and width $2\text{ m}$ moves left.

1. When they overlap perfectly, we add the displacements at every point along the $2\text{ m}$ width. 2. At the peak of the triangle: $y_{total} = (+10) + (-6) = +4\text{ units}$ . 3. The resulting shape will look like a square pulse with a 'dip' or 'notch' taken out of it, reaching a minimum height of $+4$ .

Quick Check

True or False: After two waves interfere destructively and cancel out, they are destroyed and stop moving.

Think of waves as energy passing through a medium, not as solid colliding objects.

Answer

False. Waves pass through each other and regain their original shape and velocity after the overlap.

Graphical Analysis of Complex Waves

To analyze complex or non-symmetrical waves, we use graphical addition. This involves picking specific points along the horizontal axis (

x

), measuring the displacement of each wave at that

x

-coordinate, and adding them together. For a continuous wave function, the total displacement at any point

x

and time

t

is:

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

This method is essential for understanding standing waves and beats, where the resulting pattern looks significantly different from the individual component waves.

Consider two sine waves $y_1 = A\sin(kx)$ and $y_2 = A\sin(kx + \phi)$ .

1. If the phase difference $\phi = 0$ , the waves are 'in phase'. $y_{total} = A\sin(kx) + A\sin(kx) = 2A\sin(kx)$ . This is maximum constructive interference. 2. If $\phi = \pi$ (180 degrees), the waves are 'out of phase'. Since $\sin(kx + \pi) = -\sin(kx)$ , the sum is $y_{total} = A\sin(kx) - A\sin(kx) = 0$ . 3. This mathematical cancellation is exactly how noise-canceling technology works: it generates a 'secondary' sound wave with a phase shift of $\pi$ to cancel the ambient noise.

Key Takeaways

1The Principle of Superposition states that the total displacement is the vector sum of individual wave displacements: $y_{res} = \sum y_i$ .
2Constructive interference increases amplitude (same sign), while destructive interference decreases it (opposite signs).
3Waves are 'ghost-like'—they pass through each other without losing energy or changing their original properties.
4Noise cancellation is a practical application of total destructive interference using phase-shifted waves.

Quick Check

Multiple Choice

Two pulses with amplitudes of $+2\text{ cm}$ and $-5\text{ cm}$ overlap. What is the resultant amplitude?

Multiple Choice

What happens to the shape of two wave pulses after they have finished overlapping?

True/False

The principle of superposition only applies to transverse waves, not longitudinal sound waves.

What's Next?

Review Tomorrow

In 24 hours, try to sketch two different shaped pulses (a triangle and a semi-circle) and draw what they would look like at the exact moment their centers overlap.

Practice Activity

Look up 'PhET Wave Interference' simulation online and experiment with overlapping two pulses to see constructive and destructive interference in real-time.

Browse

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The Principle of Superposition

Learning Objectives

The Core Principle

Constructive vs. Destructive Interference

Graphical Analysis of Complex Waves

Key Takeaways

Quick Check

What's Next?

The Principle of Superposition

Learning Objectives

The Core Principle

Constructive vs. Destructive Interference

Graphical Analysis of Complex Waves

Key Takeaways

Quick Check

What's Next?