Bring a good question into the classroom. — Bingkun Qi

Learning goal

Distinguish instantaneous displacement from amplitude, explain the effect of phase difference, and check an observation with an equation.

Define the model first

• We consider two sinusoidal waves obeying linear superposition. Their displacements, measured along the same direction at the same position and time, add algebraically.
• The waves have the same frequency and a fixed phase difference φ. This demonstration also uses the same wavelength and direction of travel, so φ is independent of position.
• The amplitude of wave 1 is fixed at A₁ = 1. A₂ can vary from 0 to 1. All displacements and amplitudes use the same relative units.

2 min

Predict: commit to an explanation

Before looking at the resultant-amplitude readout, set A₁ = A₂ = 1 in your model. Predict R for phase differences of 0°, 90° and 180°. Record a number or a range, with a reason.

• Will the crests of both waves occur at the same position?
• If the resultant displacement is zero at one instant, must its amplitude be zero?

3 min

Observe: add point by point

In the lab below, set the amplitude of wave 2 to 1, then move the phase slider to 0°, 90° and 180°. Pause the animation. At one horizontal coordinate, read the two displacements and check that the bright resultant curve equals their algebraic sum. Then check the resultant amplitude.

• At φ = 90°, the resultant displacement is 1 at a crest of wave 1. Is that also the maximum displacement of the resultant?
• Play the animation and watch one fixed position: instantaneous displacement changes with time, while the amplitude stays constant for the current settings.

3 min

Derive: express the pattern mathematically

Let θ = kx − ωt. Add the displacements, then expand the trigonometric expression:

y₁ = A₁ sin θ; y₂ = A₂ sin(θ + φ)

y = y₁ + y₂ = (A₁ + A₂ cos φ) sin θ + A₂ sin φ cos θ

y = R sin(θ + δ)

R² = (A₁ + A₂ cos φ)² + (A₂ sin φ)²

R = √(A₁² + A₂² + 2A₁A₂ cos φ)

• Match the coefficients of sin θ and cos θ with the expansion of R sin(θ + δ). Square and add to eliminate δ.
• The formula gives the amplitude R, not the displacement y at a particular position and time. When R = 0, the phase δ of the zero resultant need not be defined.

Check your predictions

2 min

Challenge: do opposite phases always cancel?

Keep A₁ = 1, set A₂ = 0.50 and φ = 180°. Calculate R before testing it with the sliders. Explain why the resultant is no longer a line of zero displacement.

Without testing every setting, find the minimum and maximum R as the phase varies with A₁ = 1 and A₂ = 0.50.

Check after completing the activity

The distinction to keep

Superposition adds signed displacements at the same position and time: y = y₁ + y₂. Amplitude is the maximum distance from equilibrium. In general, R is not A₁ + A₂; in this model that equality holds when the waves are in phase.

How this example is designed

Keep the original prediction and reasoning, then vary one parameter at a time. When observations disagree with a prediction, read all three displacements at the same horizontal coordinate before discussing amplitude. Use unequal amplitudes to test the claim that opposite phases always cancel, as well as the ability to substitute into a formula.

Teaching experience

• Physics, mathematics and advanced mathematics in an international school
• Undergraduate physics lecture and laboratory teaching at CUNY
• Teaching assistant experience at EPFL

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