# Graphical Representation of Superposition ## Representing Superposition Graphically - Superposition can be represented using graphs to show how waves combine - When two waves meet at a point, the resultant wave is formed by adding their displacements at each position - This graphical approach visually demonstrates the principle of superposition

The resultant wave is the algebraic sum of the displacements of individual waves at each point

## Key Features of Graphical Superposition - The black line (or resultant wave) shows the combined effect of both interfering waves - Complete constructive or destructive interference appears most clearly when: - Both waves have the same speed - Both waves have the same frequency - Both waves have similar amplitudes - Any two waves can superpose, whether: - Both are longitudinal or both are transverse - They have different properties ## Different Types of Interference Patterns ### Waves With Same Properties (In Phase) - When waves with identical amplitude, wavelength, and speed travel in the same direction and are in phase: - Peaks align with peaks - Troughs align with troughs - The resultant amplitude doubles (constructive interference) **Example: Constructive Interference** The graph below shows two waves (amplitude = 1.0 units each) that are perfectly in phase. Notice how the resultant wave has amplitude = 2.0 units:

### Waves in Anti-Phase - When waves are perfectly out of phase (180° or $\pi$ radians): - Peaks of one wave align with troughs of the other - The waves cancel completely - Resultant wave has zero amplitude (destructive interference) **Example: Destructive Interference** The graph below shows two waves (amplitude = 1.0 units each) that are in anti-phase. The resultant wave has zero amplitude at all points:

### Decreasing Phase Difference - As the phase difference between two waves gradually changes: - The amount of constructive and destructive interference varies - The resultant amplitude changes between maximum (double amplitude) and minimum (zero) - Intermediate phase differences produce intermediate amplitudes **Example: Partial Interference** The graph below shows two waves with a phase difference of 90° (quarter wavelength). The resultant amplitude is between zero and double amplitude:

### Waves in Phase (Constructive Interference) - When two identical waves are perfectly in phase: - Path difference = $n\lambda$ (where $n = 0, 1, 2, 3...$) - Phase difference = $2n\pi$ or even multiples of $\pi$ - Resultant amplitude = $2A$ (where $A$ is individual wave amplitude) - Maximum energy transfer occurs

Count the wavelengths along each wave's path to determine if constructive or destructive interference will occur

## Worked Example: Sketching Resultant Waves **Scenario:** Two overlapping waves of the same type travel in the same direction. The variation with $x$ and $y$ displacement of the waves is shown. **Task:** Use the principle of superposition to sketch the resultant wave. ### Solution Method **Step 1:** Identify key points where displacement values are easy to read - Look for peaks (maximum positive displacement) - Look for troughs (maximum negative displacement) - Look for zero crossings **Step 2:** At each position along the $x$-axis, add the displacements algebraically - If both waves have positive displacement, add them: $y_{\text{resultant}} = y_1 + y_2$ - If one is positive and one is negative, subtract: $y_{\text{resultant}} = y_1 - y_2$ - If both are negative, add them (more negative): $y_{\text{resultant}} = (-y_1) + (-y_2)$ **Step 3:** Find maximum and minimum amplitudes - Maximum amplitude occurs where both waves have maximum displacement in the same direction - Minimum amplitude (possibly zero) occurs where waves have equal but opposite displacements **Example calculation:** - Point A: Wave 1 has displacement $+0.7$, Wave 2 has displacement $+0.7$ - Resultant = $0.7 + 0.7 = 1.4$ units - Point B: Wave 1 has displacement $+0.5$, Wave 2 has displacement $-0.5$ - Resultant = $0.5 + (-0.5) = 0$ units - Point C: Wave 1 has displacement $-1.0$, Wave 2 has displacement $-0.5$ - Resultant = $-1.0 + (-0.5) = -1.5$ units **Step 4:** Join the calculated points smoothly - The resultant wave should maintain the same wavelength as the original waves (if they have the same frequency) - The resultant curve should be smooth and continuous - Check that the shape makes physical sense

Where waves intersect on a graph determines how much constructive or destructive interference occurs at that position

## Practical Tips for Graphical Superposition **Finding maximum amplitude:** 1. Look for points where both waves have their peak in the same direction 2. Add the amplitudes: $A_{\text{max}} = A_1 + A_2$ **Finding minimum amplitude:** 1. Look for points where peaks of one wave align with troughs of the other 2. Subtract the amplitudes: $A_{\text{min}} = |A_1 - A_2|$ **Plotting the resultant:** 1. Choose regular intervals along the horizontal axis (e.g., every $\lambda/4$) 2. At each interval, read off both displacements from the graph 3. Add them algebraically 4. Plot the new point 5. Connect all points with a smooth curve

Always check the vertical scale—amplitudes may differ between the two interfering waves

## Key Principles to Remember - Superposition is **instantaneous** at each point - The resultant displacement exists only while the waves overlap - Each wave continues unchanged after passing through the overlap region - The resultant wave shows: - **Constructive interference** where amplitudes add (larger resultant) - **Destructive interference** where amplitudes subtract (smaller or zero resultant) - **Partial interference** at intermediate positions - Graphical representation helps visualize: - How energy is redistributed (not created or destroyed) - Why interference patterns form in space - The relationship between phase difference and resultant amplitude ## Common Graph Features | Feature | Description | Indicates | |---------|-------------|-----------| | Resultant amplitude = Sum of individual amplitudes | Both waves displace in same direction | Constructive interference | | Resultant amplitude = Difference of amplitudes | Waves displace in opposite directions | Destructive interference | | Resultant amplitude = Zero | Equal but opposite displacements | Complete destructive interference | | Resultant amplitude between zero and sum | Partial overlap of peaks and troughs | Partial interference | ## Mathematical Approach For two waves with displacements $y_1$ and $y_2$ at position $x$ and time $t$: $$y_1 = A_1 \sin(kx - \omega t)$$ $$y_2 = A_2 \sin(kx - \omega t + \phi)$$ Where $\phi$ is the phase difference. The resultant displacement: $$y_{\text{resultant}} = y_1 + y_2$$ For waves with the same amplitude $A$ and wavelength: - When $\phi = 0$ (in phase): $y_{\text{resultant}} = 2A \sin(kx - \omega t)$ - When $\phi = \pi$ (anti-phase): $y_{\text{resultant}} = 0$