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Superposition Amplitude Calculator

Calculate resultant amplitude from two harmonic waves with phase difference

Parameters

ⓘ
ⓘ
°ⓘ
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Controls

xⓘ

Calculated Values

Resultant Amplitude:
5.00;5.00;

Examples

Perpendicular phasors

A₁=3, A₂=4, φ=90° → A_R=5.

    Destructive

    A₁=5, A₂=5, φ=180° → A_R=0.

      Visualization

      Superposition — Resultant Amplitude of Two Harmonic Waves

      The principle of superposition states that when two or more waves overlap, the resultant displacement is the sum of individual displacements. For two harmonic waves of the same frequency ω, the sum is still harmonic at frequency ω, but with a new amplitude and phase.

      Let y₁ = A₁ sin(ωt) and y₂ = A₂ sin(ωt + φ), where φ is the constant phase difference between them. The resultant can be written y_R = A_R sin(ωt + φ_R) with A_R = √(A₁² + A₂² + 2A₁A₂ cos φ).

      Special cases: φ = 0 (in phase) → A_R = A₁ + A₂ (maximum constructive). φ = π (180° out of phase) → A_R = |A₁ − A₂| (maximum destructive). φ = π/2 (90°) → A_R = √(A₁² + A₂²) (quadrature — Pythagoras).

      Phasor picture: draw vector length A₁ along reference axis, vector length A₂ at angle φ; A_R is the vector sum magnitude. This is the same mathematics as AC circuits with phasors.

      Intensity I ∝ A_R². So constructive interference (φ = 0) gives I_max = (√I₁ + √I₂)² if amplitudes add; for equal A, I quadruples when φ = 0 vs random phase average.

      If frequencies differ, superposition gives beats (slow amplitude modulation) instead of constant A_R — the formula above requires same ω.

      Class 12 superposition and interference; JEE phasor addition, slit problems, and comparison with vector components.

      Key Concepts

      • A_R = √(A₁² + A₂² + 2A₁A₂ cos φ)
      • φ = phase difference (rad or deg)
      • Constructive: φ = 0, 2π, …
      • Destructive: φ = π, 3π, …
      • Phasor vector addition
      • I ∝ A_R²; same ω required

      Real-World Applications

      • Double-slit and multi-slit interference
      • Two-speaker loudness patterns
      • Phasor analysis in AC circuits (analogy)
      • Noise cancellation (destructive superposition)
      • Class 12–JEE superposition and phasor problems

      Explore Further

      More waves tools

      • Wave Properties

        Calculate wavelength, frequency, amplitude, and wave velocity.

      • Doppler Effect

        Analyze frequency shifts due to relative motion of source and observer.

      • Wave Interference

        Analyze constructive and destructive interference patterns between two waves with interactive visualization.

      • Standing Waves

        Analyze standing wave patterns, nodes, antinodes, and resonance frequencies.

      • Wave Reflection

        Calculate reflection coefficients, phase shifts, and energy transfer at boundaries.

      • Wave Transmission

        Analyze wave transmission between different media and wavelength changes.

      Physics Equations

      Resultant:
      AR=A12+A22+2A1A2cos⁡ϕA_R = \sqrt{A_1^2+A_2^2+2A_1A_2\cos\phi}

      Step-by-Step Solution

      See how the main results are calculated.

      1

      Step 1: Resultant Amplitude

      Equation:

      AR=A12+A22+2A1A2cos⁡ϕA_R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos\phi}

      Explanation:

      φ = phase difference between two harmonic waves of same frequency.

      2

      Step 2: Given

      Result:

      A1=3,A2=4,φ=90°A₁ = 3, A₂ = 4, φ = 90°
      3

      Step 3: Cosine Term

      Calculation:

      2A1A2cos⁡ϕ=2×3×4×cos⁡(90°)=0.00002A_1A_2\cos\phi = 2 \times 3 \times 4 \times \cos(90°) = 0.0000
      4

      Step 4: Sum Under Root

      Calculation:

      A12+A22+2A1A2cos⁡ϕ=25.0000A_1^2 + A_2^2 + 2A_1A_2\cos\phi = 25.0000
      5

      Step 5: Result

      Calculation:

      AR=25.0000A_R = \sqrt{25.0000}

      Result:

      AR=5.0000A_R = 5.0000
      6

      Step 6: Special Cases

      General superposition

      Explanation:

      Used in interference and phasor addition.

      Frequently Asked Questions (FAQ)

      Different frequencies?

      Use beat analysis; A_R formula applies only when both waves share the same ω.

      Complex exponential method?

      Write A₁e^(iφ₁) + A₂e^(iφ₂); magnitude gives A_R. Powerful for many sources.

      More than two waves?

      Add phasors sequentially or use complex sum; symmetry can simplify (e.g. three sources 120° apart).

      φ from path difference?

      φ = 2πΔx/λ plus any π shifts from reflection.

      Can A_R exceed A₁ + A₂?

      No for real sinusoids with this formula; maximum is A₁ + A₂ at φ = 0.

      Practice MCQs

      1. A_R formula:
      2. φ=0:
      3. φ=180°:
      4. φ=90°:
      5. Equal A, φ=0:
      6. Intensity ∝

      Related Calculators

      These tools connect to the same physics concepts used in this calculator.

      Beat FrequencyCalculate beat frequency from two superposing waves: f_beat = |f₁ − f₂|.Sound Intensity LevelConvert sound intensity to decibels using β = 10 log₁₀(I/I₀).String Fundamental FrequencyFind fundamental frequency of a string fixed at both ends: f₁ = v/(2L).Open Pipe HarmonicsResonant frequencies of an open organ pipe: fₙ = nv/(2L).Sound Wave PropertiesCalculate wavelength, period, and other properties of sound waves with interactive visualization.Related Acoustics topic.Snell's LawCalculate refraction angles and analyze light behavior at media boundaries.Related Optics topic.Error PropagationCombine measurement uncertainties for sums, products, quotients, and powers in experimental formulas.Related Measurement & Uncertainty topic.Wave PropertiesCalculate wavelength, frequency, amplitude, and wave velocity.Part of the Class 11 — Waves learning path.