Superposition principle

In a linear system, overlapping disturbances add point by point. Each wave continues to obey the same linear equation while the measurable displacement or field is their sum. Interference is this addition, not a separate force that switches on when waves meet.

Linearity of the wave equation

Write the one-dimensional wave equation as an operator equation

\[\mathcal L[y]=\frac{\partial^2y}{\partial t^2}-c^2\frac{\partial^2y}{\partial x^2}=0\]

If \(y_1\) and \(y_2\) satisfy it, then for constants \(a\) and \(b\),

\[\mathcal L[ay_1+by_2]=a\mathcal L[y_1]+b\mathcal L[y_2]=0\]

Therefore

\[y_{\mathrm{total}}(x,t)=\sum_i y_i(x,t)\]

is also a solution under compatible linear boundary and source conditions. The addition must occur at the same position and time.

import numpy as np

x = np.linspace(-5, 5, 501)
first = np.exp(-((x + 1) ** 2))
second = -0.5 * np.exp(-((x - 1) ** 2))
combined = first + second
assert combined.shape == x.shape
print("Combined displacement at x=0:", combined[250])

Source order should not change the mathematical result, apart from small floating-point addition-order differences when many values are summed.

Constructive and destructive interference

When two displacements have the same sign at a point, their sum has larger magnitude: this is constructive interference. Opposite signs partially or completely cancel: this is destructive interference. Cancellation means the net displacement is zero at that instant and position; the individual waves and their energy need not have vanished.

For equal sinusoids with phase difference \(\Delta\phi\),

\[y_1=A\cos\theta\qquad y_2=A\cos(\theta+\Delta\phi)\]

the sum is

\[y_{\mathrm{total}}=2A\cos\left(\frac{\Delta\phi}{2}\right)\cos\left(\theta+\frac{\Delta\phi}{2}\right)\]

The resultant amplitude is the magnitude of

\[A_{\mathrm{result}}=2A\left|\cos\left(\frac{\Delta\phi}{2}\right)\right|\]

At zero phase difference, amplitude is \(2A\). At phase difference \(\pi\), aligned samples cancel.

Unequal coherent amplitudes

For amplitudes \(A_1\) and \(A_2\) at the same frequency, phasor addition gives

\[A_{\mathrm{result}}^2=A_1^2+A_2^2+2A_1A_2\cos(\Delta\phi)\]

Complete cancellation is possible only when amplitudes are equal and phases are opposite. Otherwise destructive interference leaves a residual amplitude equal to their difference in the fully opposite case.

Use a quadrant-aware phase calculation if the resultant phase is needed. Adding peak amplitudes as ordinary scalars ignores phase and is wrong except for exactly aligned waves.

Pulses pass through in a linear medium

When two pulses overlap on an ideal linear string, total displacement is their pointwise sum. Afterward each continues with its original shape and direction if the medium is nondispersive and lossless. During overlap, a large crest or momentary cancellation does not imply the pulses collided like material objects.

for time in [0.0, 1.0, 2.0]:
    right = np.exp(-((x + 2 - time) ** 2))
    left = -np.exp(-((x - 2 + time) ** 2))
    combined = right + left
    print(time, "largest absolute displacement:", np.max(abs(combined)))

Retaining individual contributions makes the pass-through property visible and prevents the combined field from becoming an irreversible edit.

Standing waves from opposite traveling waves

Two equal waves traveling in opposite directions combine as

\[A\cos(kx-\omega t)+A\cos(kx+\omega t)=2A\cos(kx)\cos(\omega t)\]

The result has fixed nodes where the spatial cosine is zero and antinodes where its magnitude is maximal. Although the standing pattern has no net propagation in either direction, it consists of traveling contributions and stores energy that exchanges between kinetic and elastic forms.

Beats from nearby frequencies

Two equal waves at one location with angular frequencies \(\omega_1\) and \(\omega_2\) add to

\[A\cos(\omega_1t)+A\cos(\omega_2t)=2A\cos\left(\frac{\omega_1-\omega_2}{2}t\right)\cos\left(\frac{\omega_1+\omega_2}{2}t\right)\]

The rapid oscillation is modulated by a slow envelope. The number of loudness or amplitude maxima per second is the beat frequency

\[f_{\mathrm{beat}}=|f_1-f_2|\]

The envelope cosine changes sign, but perceived or measured amplitude magnitude produces two maxima per envelope cosine cycle, yielding the frequency difference.

Energy is quadratic, not simply additive

Displacements add linearly, but wave energy often depends on squared amplitude or squared derivatives. For a string,

\[u=\frac12\mu\left(\frac{\partial y}{\partial t}\right)^2+\frac12T\left(\frac{\partial y}{\partial x}\right)^2\]

Substituting \(y_1+y_2\) produces cross terms. Local energy can redistribute during interference. It is generally wrong to say that destructive displacement interference destroys energy.

For coherent waves, average intensity contains an interference term. For unrelated phases averaged over enough time, the cross term may average to zero and intensities add. State the coherence and averaging assumptions.

Samples must be aligned

Pointwise array addition is physical only when indices represent the same coordinates and times. Equal-looking arrays sampled with different origins or spacing are not aligned. One source shifted by one cell can leave a large residual where exact analytic cancellation was expected.

first_grid, second_grid = x.copy(), x.copy()
first_unit, second_unit = "m", "m"
assert first_unit == second_unit
assert np.array_equal(first_grid, second_grid)
combined = first + second
print("Aligned samples added:", len(combined))

Resampling introduces interpolation error and possibly phase change. Record the target grid and method, and compare against analytic values when possible.

A worked Python notebook

Add two equal pulses with opposite signs at a fixed instant. The question for this calculation is precise: how does combined pulse displacement change when we change position, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

Pointwise addition applies in a linear medium and on a common position grid. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from -4 to 4 m. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for position, expressed in m. The output is combined pulse displacement, expressed in m. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = e^{- \left(q + 1\right)^{2}} - e^{- \left(q - 1\right)^{2}}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 1 m, the expression gives -0.981684 m. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("exp(-(q+1)**2)-exp(-(q-1)**2)", locals={"q": q})
reference_input = 1.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted combined pulse displacement:", reference_output, "m")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 0.00012341 m. The midpoint, at 0 m, gives 0 m. The upper endpoint gives -0.00012341 m. Read these together with the assumptions: they describe different choices of position within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([-4.0, 0.0, 4.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} m -> {output_value:.6g} m")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows position; the vertical axis shows combined pulse displacement. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(-4.0, 4.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Position (m)")
ax.set_ylabel("Combined pulse displacement (m)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does combined pulse displacement change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1 m is -0.0732626 in those ratio units. Locally, the output decreases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = - \left(2 - 2 q\right) e^{- \left(q - 1\right)^{2}} + \left(- 2 q - 2\right) e^{- \left(q + 1\right)^{2}}\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 1.04
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "m")
print("Full expression:", full_prediction, "m")
print("Approximation difference:", full_prediction - linear_prediction, "m")

Here the changed input is 1.04 m. The tangent estimate is -0.984615 m, while the full expression gives -0.982819 m. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 8.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of position inside the stated interval, predict whether combined pulse displacement will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: Pointwise addition applies in a linear medium and on a common position grid. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of m: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(4.0) - evaluate(-4.0))
for points in [101, 201, 401]:
    grid = np.linspace(-4.0, 4.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources and boundary conditions

Superposing already-computed free waves is different from solving a forced boundary-value problem. In a linear equation, source responses can be added when they use the same linear operator and homogeneous boundary structure. If each solution assumes a different fixed endpoint or material, their sum may violate both problems.

At a boundary, incident and reflected waves superpose. Reflection phase depends on the boundary condition. The total field must satisfy the boundary at every time, which is a strong check on signs and timing.

When superposition fails

Large-amplitude string motion, nonlinear elasticity, wave breaking, shocks, saturation, contact, and nonlinear optical media can couple contributions. Then the response to two sources is not the sum of separate responses. Harmonics and combination frequencies may appear.

Clipping a summed signal is also nonlinear. Normalizing every displayed frame by its own maximum preserves appearance but destroys amplitude comparison and should never feed back into state. A renderer can display a clipped view only if the underlying physical values remain available.

Verification tests

Disable every source but one and require exact recovery of that source. Add identical arrays and expect double values. Add exact opposites and expect zero within floating-point tolerance. Scale all sources by one constant and expect the sum to scale by the same constant. Reorder sources and compare output.

Then test analytic interference: equal coherent sinusoids at zero and opposite phase, opposite traveling waves forming nodes, and nearby frequencies producing the predicted beat rate. Refine space and time sampling so cancellation or phase behavior is not a grid artifact.

The superposition principle is powerful precisely because it supplies these algebraic tests. Its use is justified only while the governing system, boundaries, and measurement operation remain linear.

Sources

• OpenStax, University Physics Volume 1, Section 16.5, "Interference of Waves".
• OpenStax, University Physics Volume 1, Section 16.6, "Standing Waves and Resonance".
• Benjamin Crowell, Light and Matter, superposition and wave-interference discussions.
• The Physics Hypertextbook, "Interference".
• Physics LibreTexts, "Interference of Waves".