Superposition of Displacements (6.6.2) | AP Physics 2: Algebra Notes

AP Syllabus focus: 'When waves overlap, the resulting displacement is determined by adding the individual displacements. This principle is called superposition.'

When multiple waves occupy the same region, the medium responds to all of them at once. The key AP Physics idea is to add their displacements carefully, point by point and instant by instant.

What superposition means

Superposition applies whenever two or more waves are present in the same place at the same time. The medium does not display separate, independent displacements; instead, it has one actual displacement from equilibrium at each location. That single displacement depends on the contribution made by every overlapping wave. Whether the disturbances are short pulses or smooth repeating waves, the same rule is used.

Superposition: When two or more waves overlap, the displacement of the medium at any point is the algebraic sum of the individual displacements at that point.

This principle is local and instantaneous. You do not combine whole waves by comparing only their amplitudes or by deciding which wave is “stronger.” You combine the displacements at one chosen position and one chosen instant. Then you repeat that process at neighboring positions if you want the full resulting shape. The same overlap can therefore produce different net displacements at different points.

Finding the resulting displacement

To apply superposition, treat the overlap as an algebra problem with signs. First choose a positive direction relative to equilibrium. Displacements on one side are positive, and displacements on the other side are negative. The net displacement is the algebraic sum of all individual contributions.

$y_{net} = y_1 + y_2 + \cdots + y_n$

$y_{net}$ = resulting displacement of the medium at one location and one instant, in meters

$y_1,\ y_2,\ \cdots,\ y_n$ = individual displacements from each overlapping wave at that same location and instant, in meters

If a diagram shows several waves, the procedure is systematic:

Two waves (red and blue) are graphed as displacement versus position, and the resultant (black) is constructed by adding their vertical displacements at the same horizontal position. The inset highlights specific x-values where the graph is read as $y_{net}=y_1+y_2$ , emphasizing that superposition is a local, point-by-point operation on snapshot graphs. Source

choose a single location on the medium
read the displacement from each wave at that location
keep the correct sign for each displacement
add the values
move to the next location and repeat

This process works for short pulses and for extended periodic waves.

Two triangular pulses are shown on a grid during overlap, with the net displacement drawn as the sum of the individual pulse displacements at each position. The picture makes it clear that only the region where both pulses are nonzero changes; outside the overlap, the resultant matches the single existing pulse. Source

The important idea is that the addition is done point by point, not by adding total amplitudes, widths, or wavelengths. If three or more waves overlap, the same method still applies: add every displacement present at that location.

Positive and negative contributions

When two overlapping displacements are on the same side of equilibrium, their magnitudes add and the resulting displacement is larger. When they are on opposite sides, one partially or completely cancels the other. If the magnitudes are equal and the signs are opposite, the net displacement at that point is zero.

A zero net displacement does not mean the individual waves stopped existing. It only means their contributions cancel exactly at that location and at that instant. At a nearby location or at a slightly different time, the net displacement may be positive, negative, or zero depending on the values being added.

Reading wave graphs correctly

Superposition questions often use graphs. The most common mistake is adding values taken from different positions or different times. Before adding anything, identify what the horizontal axis represents and make sure you are comparing matching points.

Snapshot graphs

If the graph is displacement versus position at one instant, add the vertical displacements of the waves at the same horizontal position. The resulting curve is built from those vertical sums. When sketching the combined wave, place each new point relative to equilibrium and connect the points smoothly if the original waves are smooth.

Time graphs

If the graph is displacement versus time for one position, add the displacements at the same time. In this case, the horizontal axis represents time rather than location, but the superposition rule is unchanged.

If only part of one wave overlaps another, apply superposition only in the region or time interval where both are present. Outside that overlap, the resultant displacement is simply the displacement of the single wave that is present.

What AP Physics 2 expects

On the AP exam, superposition is mainly a reasoning tool. You may need to identify the sign of each displacement, describe the combined shape, or decide whether the net displacement becomes larger, smaller, or zero at a particular point. Often the hardest step is recognizing that you must compare matching positions or matching times before adding.

Keep these ideas in mind:

Displacement is the quantity being added.
The addition is algebraic, so signs matter.
The rule can involve two waves or many waves.
Different wave shapes can still be combined with the same principle.
The resulting displacement can vary from point to point even within the same overlap region.

The exam usually rewards careful, point-by-point reasoning more than memorized wording.

Frequent errors to avoid

Common errors include:

adding amplitudes instead of the actual local displacements
ignoring whether a displacement is above or below equilibrium
combining values from different positions on a snapshot graph
combining values from different times on a time graph
assuming the same net displacement applies everywhere in an overlap region
forgetting that the result describes the medium’s displacement at one chosen instant

In superposition problems, the safest strategy is to ask: What is the displacement from each wave here, right now?

FAQ

Superposition works when the medium’s response to several disturbances equals the sum of the responses to each disturbance acting alone. That kind of behavior is called linear.

In AP Physics 2, waves are usually modeled this way. Real systems can deviate from perfect linearity at very large amplitudes, but the course treatment assumes linear behavior so algebraic addition of displacements is valid.

Yes. A complicated disturbance can often be represented as the sum of many simpler disturbances.

In more advanced physics, this idea is extended to combinations of sinusoidal waves. That is why superposition is so powerful: once the simpler components are known, the overall wave is found by adding their contributions.

Yes. Displacement tells you where the medium is relative to equilibrium, not whether it is moving at that moment.

A point can pass through equilibrium with nonzero speed, so two waves can produce a net displacement of $0$ while the particles of the medium are still moving. To know the full motion, displacement alone is not enough.

They generate a second sound wave designed to match the unwanted sound but with opposite phase at the ear.

When the pressure variations from the two sounds combine, the net pressure change is reduced. The cancellation is not perfect for every frequency or every location, which is why these systems work best for steady background sounds such as engine noise.

It can break down when the medium behaves nonlinearly, meaning one disturbance changes the medium so much that simple addition no longer describes the response.

Examples include very large-amplitude waves, shock waves, or systems where restoring forces are not proportional to displacement. Those cases are beyond the usual AP Physics 2 model, which assumes ordinary linear wave behavior.

Practice Questions

At a point on a rope, one wave produces a displacement of $+0.030\ m$ and a second wave produces a displacement of $-0.018\ m$ at the same instant. What is the resultant displacement, and what principle is being used? [2 marks]

1 mark: Calculates the resultant displacement as $+0.012\ m$ .
1 mark: Identifies the principle as superposition, or states that the displacements are added algebraically.

At one instant, Wave A on a rope has displacement $+2.0\ cm$ from $x=0$ to $x=3.0\ cm$ and zero elsewhere. Wave B has displacement $-1.0\ cm$ from $x=1.0\ cm$ to $x=4.0\ cm$ and zero elsewhere. Describe the resultant displacement in the regions $0 \le x < 1.0\ cm$ , $1.0\ cm \le x \le 3.0\ cm$ , and $3.0\ cm < x \le 4.0\ cm$ . State the principle you use. [5 marks]

1 mark: States that the resultant is found by superposition, adding displacements point by point.
1 mark: Gives resultant displacement $+2.0\ cm$ for $0 \le x < 1.0\ cm$ .
1 mark: Gives resultant displacement $+1.0\ cm$ for $1.0\ cm \le x \le 3.0\ cm$ .
1 mark: Gives resultant displacement $-1.0\ cm$ for $3.0\ cm < x \le 4.0\ cm$ .
1 mark: Gives a complete description by noting that the displacement is zero outside these intervals, or provides an equivalent correct piecewise result.

Try All Topic Practice Questions

Written by:

Dr Shubhi Khandelwal

Qualified Dentist and Expert Science Educator

Shubhi is a seasoned educational specialist with a sharp focus on IB, A-level, GCSE, AP, and MCAT sciences. With 6+ years of expertise, she excels in advanced curriculum guidance and creating precise educational resources, ensuring expert instruction and deep student comprehension of complex science concepts.