Boundary conditions

A wave equation describes how a field changes in the interior of a region, but it does not by itself determine a unique motion. We must also specify initial conditions and what happens at every boundary. An endpoint may be fixed, free, driven, joined to another medium, repeated periodically, or treated as approximately open. Each choice represents different physics and produces different reflections.

Initial conditions and boundary conditions do different jobs

For the one-dimensional wave equation

\[\frac{\partial^2 y}{\partial t^2}=v^2\frac{\partial^2 y}{\partial x^2}\]

two initial functions are normally required because the equation is second order in time:

\[y(x,0)=f(x),\qquad \left.\frac{\partial y}{\partial t}\right|_{t=0}=g(x)\]

Boundary conditions specify behavior at the spatial edges for later times. The initial data must be compatible with them. If an endpoint is fixed at zero for all time, then \(f\) must be zero there and, for smooth compatible data, the endpoint's initial velocity must also agree with the constraint. An inconsistent initialization often launches an unintended high-frequency pulse on the first step.

Fixed endpoint: displacement is prescribed

At a fixed endpoint at \(x=0\), the string displacement is constrained:

\[y(0,t)=0\]

This is a homogeneous Dirichlet condition. A nonzero prescribed displacement, \(y(0,t)=h(t)\), is also Dirichlet, but it acts as a driver and can inject or remove energy.

Suppose a right-moving incident pulse reaches a fixed endpoint. The reflected displacement pulse is inverted. Incident and reflected displacements must add to zero at the endpoint at every instant. “Inverted” refers to displacement: a crest returns as a trough. Which velocity or force component changes sign must be derived rather than inferred from the picture.

import numpy as np

x = np.linspace(0, 1, 101)
values = np.sin(np.pi * x)
values[0] = values[-1] = 0.0
assert values[0] == values[-1] == 0.0
print("Fixed endpoint values:", values[[0, -1]])

Setting only the displayed endpoint to zero is not enough if the hidden solver state still uses a different value. The constraint must participate consistently in the numerical update.

Free endpoint: transverse force vanishes

At an ideal free string end, no support supplies a transverse force. Under the small-slope string model, this means the spatial slope is zero:

\[\left.\frac{\partial y}{\partial x}\right|_{\mathrm{end}}=0\]

This is a homogeneous Neumann condition. The reflected displacement pulse is not inverted: a crest returns as a crest. A centered finite-difference approximation at the right edge can use a ghost value satisfying

\[\frac{y_{N+1}-y_{N-1}}{2\Delta x}=0\quad\Longrightarrow\quad y_{N+1}=y_{N-1}\]

The exact indexing depends on whether the physical boundary lies on a grid point or a cell face. Mixing those layouts changes accuracy and can move the effective boundary by half a cell.

An interface between two strings

At a junction, displacement is continuous and transverse forces balance when the junction has negligible mass. Reflection is controlled by mechanical wave impedance. For a string, define

\[Z=\mu v=\sqrt{T\mu}\]

where \(\mu\), \(T\), and \(v=\sqrt{T/\mu}\) describe the medium. For a displacement wave incident from medium 1 onto medium 2, using amplitudes at the interface and the same positive transverse direction, the reflection and transmission amplitude ratios are

\[r=\frac{A_r}{A_i}=\frac{Z_1-Z_2}{Z_1+Z_2},\qquad t=\frac{A_t}{A_i}=\frac{2Z_1}{Z_1+Z_2}\]

These signs depend on the stated displacement convention. The limits are informative: \(Z_2\to\infty\) gives \(r\to-1\), while \(Z_2\to0\) gives \(r\to+1\).

Amplitude coefficients are not energy fractions. For lossless strings under these assumptions,

\[\mathcal R=r^2,\qquad \mathcal T=\frac{Z_2}{Z_1}t^2=\frac{4Z_1Z_2}{(Z_1+Z_2)^2},\qquad \mathcal R+\mathcal T=1\]

The impedance factor in \(\mathcal T\) is essential. Squaring the transmitted displacement ratio alone generally does not give the transmitted power fraction.

Periodic boundaries

A periodic domain identifies opposite edges. For a domain of length \(L\),

\[y(x+L,t)=y(x,t)\]

A wave leaving the right re-enters from the left. This describes a ring or an endlessly repeated computational cell, not an open laboratory. A periodic grid must wrap neighbor indices exactly:

periodic_values = np.sin(2 * np.pi * np.arange(100) / 100)
left_neighbors = np.roll(periodic_values, 1)
right_neighbors = np.roll(periodic_values, -1)
print("Neighbors of the first point:", left_neighbors[0], right_neighbors[0])

Use distinct edge values in tests. A symmetric or constant field can conceal an off-by-one error because the wrong neighbor happens to have the same value.

Driven boundaries and boundary work

A prescribed endpoint motion can generate waves. If \(y(0,t)=A_d\sin(\omega_d t)\), the boundary has displacement amplitude \(A_d\) and angular frequency \(\omega_d\). The domain response depends on reflections, damping, and proximity to its normal-mode frequencies.

Prescribing motion is not the same as prescribing force. A force boundary is a Neumann or traction-type condition, while a displacement boundary is Dirichlet. The source power is force times endpoint velocity, so it may be positive or negative during a cycle. Near resonance, its average can be positive even though energy flows back toward the driver during part of each cycle.

Approximately absorbing boundaries

A finite numerical domain often stands in for a much larger region. Simply fixing the far edge creates a reflection. For the uniform one-dimensional wave equation, a right-going wave \(y=F(x-vt)\) satisfies

\[\frac{\partial y}{\partial t}+v\frac{\partial y}{\partial x}=0\]

At a left boundary, an outgoing left-moving wave satisfies

\[\frac{\partial y}{\partial t}-v\frac{\partial y}{\partial x}=0\]

These Sommerfeld-type conditions are exact for one-directional waves of the assumed one-dimensional, constant-speed model. Their discrete versions are approximate and may reflect high-frequency content. In multiple dimensions, oblique incidence makes a simple one-way condition less effective. Absorbing layers can reduce reflections over a broader range but introduce parameters that also require tests.

# Test a first-order absorbing right boundary for a travelling Gaussian.
speed, dx, dt = 1.0, 0.01, 0.005
courant = speed * dt / dx
x = np.arange(201) * dx
pulse = lambda time: np.exp(-(((x - 1.8 - speed * time) / 0.1) ** 2))
old = pulse(0.0)
new_interior = pulse(dt)
ratio = (courant - 1) / (courant + 1)
predicted_boundary = old[-2] + ratio * (new_interior[-2] - old[-1])
print("Boundary displacement error:", predicted_boundary - new_interior[-1])
# A full reflected-energy test needs a time history after the pulse exits.

Calling a boundary “absorbing” is not a verification. Measure its reflected amplitude or energy over the frequencies and angles the application actually uses.

Boundary order and numerical consistency

An interior stencil may be second-order accurate while a crude boundary closure is only first order. The global solution can then converge more slowly than expected. Ghost cells, one-sided derivatives, penalty methods, or characteristic conditions must match the PDE, grid placement, and desired order.

Apply boundary rules at a clearly defined point in the update. For constrained endpoints, tests should inspect both current and next state. For an interface, check displacement and force residuals. For a periodic boundary, test every wrapped neighbor direction. For an open boundary, track outgoing energy and reflected energy separately.

Boundary terminology is field-dependent. A rigid end is a displacement node for a string, but in acoustics a rigid wall is an air-displacement node and a pressure antinode. Electric and magnetic fields have their own interface conditions, and quantum wavefunctions obey conditions tied to the potential and probability current. Transfer the mathematical idea, not an unqualified string rule.

A worked Python notebook

Use a sinusoidal shape that obeys zero displacement at both boundaries. The question for this calculation is precise: how does fixed-end displacement shape 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.

A compatible initial shape does not excuse the time-stepping method from preserving the boundary values. 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 0 to 1 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 fixed-end displacement shape, 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.

\[f(q) = \sin{\left(\pi q \right)}\]

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 0.4 m, the expression gives 0.951057 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("sin(pi*q)", locals={"q": q})
reference_input = 0.4
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted fixed-end displacement shape:", 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 m. The midpoint, at 0.5 m, gives 1 m. The upper endpoint gives 0 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([0.0, 0.5, 1.0])
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 fixed-end displacement shape. 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(0.0, 1.0, 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("Fixed-end displacement shape (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 fixed-end displacement shape change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.4 m is 0.970806 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = \pi \cos{\left(\pi q \right)}\]

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 = 0.405
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 0.405 m. The tangent estimate is 0.955911 m, while the full expression gives 0.955793 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 = 1.0
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 fixed-end displacement shape 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: A compatible initial shape does not excuse the time-stepping method from preserving the boundary values. 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(1.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 1.0, 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

Share to: