{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Boundary conditions\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/boundary-conditions). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise." ] }, { "cell_type": "code", "id": "code-001", "metadata": {}, "source": [ "import importlib.metadata\n", "import subprocess\n", "import sys\n", "\n", "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n", "to_install = []\n", "for package, minimum in requirements.items():\n", " try:\n", " installed = importlib.metadata.version(package)\n", " version = tuple(int(part) for part in installed.split(\".\")[:2])\n", " except importlib.metadata.PackageNotFoundError:\n", " version = (0, 0)\n", " if version < minimum:\n", " to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n", "if to_install:\n", " subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "\n", "print(\"Python\", sys.version.split()[0])\n", "print(\"NumPy\", np.__version__)\n", "print(\"SymPy\", sp.__version__)\n", "print(\"Matplotlib\", matplotlib.__version__)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-002", "metadata": {}, "source": [ "A wave equation describes how a field changes in the interior of a region, but it does not\n", "by itself determine a unique motion. We must also specify initial conditions and what\n", "happens at every boundary. An endpoint may be fixed, free, driven, joined to another\n", "medium, repeated periodically, or treated as approximately open. Each choice represents\n", "different physics and produces different reflections.\n", "\n", "## Initial conditions and boundary conditions do different jobs\n", "\n", "For the one-dimensional wave equation\n", "\n", "\n", "$$\n", "\\frac{\\partial^2 y}{\\partial t^2}=v^2\\frac{\\partial^2 y}{\\partial x^2}\n", "$$\n", "\n", "\n", "two initial functions are normally required because the equation is second order in time:\n", "\n", "\n", "$$\n", "y(x,0)=f(x),\\qquad \\left.\\frac{\\partial y}{\\partial t}\\right|_{t=0}=g(x)\n", "$$\n", "\n", "\n", "Boundary conditions specify behavior at the spatial edges for later times. The initial\n", "data must be compatible with them. If an endpoint is fixed at zero for all time, then\n", "$f$ must be zero there and,\n", "for smooth compatible data, the endpoint's initial velocity must also agree with the\n", "constraint. An inconsistent initialization often launches an unintended high-frequency\n", "pulse on the first step.\n", "\n", "## Fixed endpoint: displacement is prescribed\n", "\n", "At a fixed endpoint at $x=0$, the\n", "string displacement is constrained:\n", "\n", "\n", "$$\n", "y(0,t)=0\n", "$$\n", "\n", "\n", "This is a homogeneous Dirichlet condition. A nonzero prescribed displacement,\n", "$y(0,t)=h(t)$, is also\n", "Dirichlet, but it acts as a driver and can inject or remove energy.\n", "\n", "Suppose a right-moving incident pulse reaches a fixed endpoint. The reflected displacement\n", "pulse is inverted. Incident and reflected displacements must add to zero at the endpoint\n", "at every instant. “Inverted” refers to displacement: a crest returns as a trough. Which\n", "velocity or force component changes sign must be derived rather than inferred from the\n", "picture." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "x = np.linspace(0, 1, 101)\n", "values = np.sin(np.pi * x)\n", "values[0] = values[-1] = 0.0\n", "assert values[0] == values[-1] == 0.0\n", "print(\"Fixed endpoint values:\", values[[0, -1]])\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "Setting only the displayed endpoint to zero is not enough if the hidden solver state still\n", "uses a different value. The constraint must participate consistently in the numerical\n", "update.\n", "\n", "## Free endpoint: transverse force vanishes\n", "\n", "At an ideal free string end, no support supplies a transverse force. Under the small-slope\n", "string model, this means the spatial slope is zero:\n", "\n", "\n", "$$\n", "\\left.\\frac{\\partial y}{\\partial x}\\right|_{\\mathrm{end}}=0\n", "$$\n", "\n", "\n", "This is a homogeneous Neumann condition. The reflected displacement pulse is not inverted:\n", "a crest returns as a crest. A centered finite-difference approximation at the right edge\n", "can use a ghost value satisfying\n", "\n", "\n", "$$\n", "\\frac{y_{N+1}-y_{N-1}}{2\\Delta x}=0\\quad\\Longrightarrow\\quad y_{N+1}=y_{N-1}\n", "$$\n", "\n", "\n", "The exact indexing depends on whether the physical boundary lies on a grid point or a cell\n", "face. Mixing those layouts changes accuracy and can move the effective boundary by half a\n", "cell.\n", "\n", "## An interface between two strings\n", "\n", "At a junction, displacement is continuous and transverse forces balance when the junction\n", "has negligible mass. Reflection is controlled by mechanical wave impedance. For a string,\n", "define\n", "\n", "\n", "$$\n", "Z=\\mu v=\\sqrt{T\\mu}\n", "$$\n", "\n", "\n", "where $\\mu$,\n", "$T$, and\n", "$v=\\sqrt{T/\\mu}$ describe\n", "the medium. For a displacement wave incident from medium 1 onto medium 2, using amplitudes\n", "at the interface and the same positive transverse direction, the reflection and\n", "transmission amplitude ratios are\n", "\n", "\n", "$$\n", "r=\\frac{A_r}{A_i}=\\frac{Z_1-Z_2}{Z_1+Z_2},\\qquad t=\\frac{A_t}{A_i}=\\frac{2Z_1}{Z_1+Z_2}\n", "$$\n", "\n", "\n", "These signs depend on the stated displacement convention. The limits are informative:\n", "$Z_2\\to\\infty$ gives\n", "$r\\to-1$, while\n", "$Z_2\\to0$ gives\n", "$r\\to+1$.\n", "\n", "Amplitude coefficients are not energy fractions. For lossless strings under these\n", "assumptions,\n", "\n", "\n", "$$\n", "\\mathcal R=r^2,\\qquad \\mathcal T=\\frac{Z_2}{Z_1}t^2=\\frac{4Z_1Z_2}{(Z_1+Z_2)^2},\\qquad \\mathcal R+\\mathcal T=1\n", "$$\n", "\n", "\n", "The impedance factor in $\\mathcal T$\n", "is essential. Squaring the transmitted displacement ratio alone generally does not give\n", "the transmitted power fraction.\n", "\n", "## Periodic boundaries\n", "\n", "A periodic domain identifies opposite edges. For a domain of length\n", "$L$,\n", "\n", "\n", "$$\n", "y(x+L,t)=y(x,t)\n", "$$\n", "\n", "\n", "A wave leaving the right re-enters from the left. This describes a ring or an endlessly\n", "repeated computational cell, not an open laboratory. A periodic grid must wrap neighbor\n", "indices exactly:" ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "periodic_values = np.sin(2 * np.pi * np.arange(100) / 100)\n", "left_neighbors = np.roll(periodic_values, 1)\n", "right_neighbors = np.roll(periodic_values, -1)\n", "print(\"Neighbors of the first point:\", left_neighbors[0], right_neighbors[0])\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "Use distinct edge values in tests. A symmetric or constant field can conceal an off-by-one\n", "error because the wrong neighbor happens to have the same value.\n", "\n", "## Driven boundaries and boundary work\n", "\n", "A prescribed endpoint motion can generate waves. If\n", "$y(0,t)=A_d\\sin(\\omega_d t)$,\n", "the boundary has displacement amplitude\n", "$A_d$ and angular frequency\n", "$\\omega_d$. The domain\n", "response depends on reflections, damping, and proximity to its normal-mode frequencies.\n", "\n", "Prescribing motion is not the same as prescribing force. A force boundary is a Neumann or\n", "traction-type condition, while a displacement boundary is Dirichlet. The source power is\n", "force times endpoint velocity, so it may be positive or negative during a cycle. Near\n", "resonance, its average can be positive even though energy flows back toward the driver\n", "during part of each cycle.\n", "\n", "## Approximately absorbing boundaries\n", "\n", "A finite numerical domain often stands in for a much larger region. Simply fixing the far\n", "edge creates a reflection. For the uniform one-dimensional wave equation, a right-going\n", "wave $y=F(x-vt)$ satisfies\n", "\n", "\n", "$$\n", "\\frac{\\partial y}{\\partial t}+v\\frac{\\partial y}{\\partial x}=0\n", "$$\n", "\n", "\n", "At a left boundary, an outgoing left-moving wave satisfies\n", "\n", "\n", "$$\n", "\\frac{\\partial y}{\\partial t}-v\\frac{\\partial y}{\\partial x}=0\n", "$$\n", "\n", "\n", "These Sommerfeld-type conditions are exact for one-directional waves of the assumed\n", "one-dimensional, constant-speed model. Their discrete versions are approximate and may\n", "reflect high-frequency content. In multiple dimensions, oblique incidence makes a simple\n", "one-way condition less effective. Absorbing layers can reduce reflections over a broader\n", "range but introduce parameters that also require tests." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "# Test a first-order absorbing right boundary for a travelling Gaussian.\n", "speed, dx, dt = 1.0, 0.01, 0.005\n", "courant = speed * dt / dx\n", "x = np.arange(201) * dx\n", "pulse = lambda time: np.exp(-(((x - 1.8 - speed * time) / 0.1) ** 2))\n", "old = pulse(0.0)\n", "new_interior = pulse(dt)\n", "ratio = (courant - 1) / (courant + 1)\n", "predicted_boundary = old[-2] + ratio * (new_interior[-2] - old[-1])\n", "print(\"Boundary displacement error:\", predicted_boundary - new_interior[-1])\n", "# A full reflected-energy test needs a time history after the pulse exits.\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "Calling a boundary “absorbing” is not a verification. Measure its reflected amplitude or\n", "energy over the frequencies and angles the application actually uses.\n", "\n", "## Boundary order and numerical consistency\n", "\n", "An interior stencil may be second-order accurate while a crude boundary closure is only\n", "first order. The global solution can then converge more slowly than expected. Ghost cells,\n", "one-sided derivatives, penalty methods, or characteristic conditions must match the PDE,\n", "grid placement, and desired order.\n", "\n", "Apply boundary rules at a clearly defined point in the update. For constrained endpoints,\n", "tests should inspect both current and next state. For an interface, check displacement and\n", "force residuals. For a periodic boundary, test every wrapped neighbor direction. For an\n", "open boundary, track outgoing energy and reflected energy separately.\n", "\n", "Boundary terminology is field-dependent. A rigid end is a displacement node for a string,\n", "but in acoustics a rigid wall is an air-displacement node and a pressure antinode. Electric\n", "and magnetic fields have their own interface conditions, and quantum wavefunctions obey\n", "conditions tied to the potential and probability current. Transfer the mathematical idea,\n", "not an unqualified string rule.\n", "\n", "## A worked Python notebook\n", "\n", "Use a sinusoidal shape that obeys zero displacement at both boundaries. The question for this calculation is precise: how does **fixed-end displacement shape**\n", "change when we change **position**, while keeping the other choices fixed?\n", "The earlier sections explain the physical relationship. Here we turn one part of that\n", "relationship into a calculation that you can run, inspect, and change in Google Colab.\n", "\n", "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\n", "its useful range, and a plotting library will usually draw that number without asking\n", "whether the physical assumptions still hold. The selected interval is from\n", "0 to 1 m. It gives us a specific domain to discuss\n", "instead of leaving the graph's range to a software default.\n", "\n", "Use the notebook link above the lesson, run the setup cell, and then run the cells in\n", "order. The code below is also included there. NumPy handles the arrays of numerical\n", "values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these\n", "libraries or notebook cells are unfamiliar, start with\n", "[Your first physics notebook](https://tirta.co/a-byte-of-physics/your-first-physics-notebook).\n", "\n", "### State the relationship before choosing a graph\n", "\n", "In this small example, `q` stands for position, expressed in m. The output is\n", "fixed-end displacement shape, expressed in m. The equation below uses the numerical values in those\n", "units. Any fixed coefficients therefore carry the units needed to make the relation\n", "consistent. A value supplied in another unit must be converted before it enters this\n", "expression.\n", "\n", "\n", "$$\n", "f(q) = \\sin{\\left(\\pi q \\right)}\n", "$$\n", "\n", "\n", "This is a function of one changing input. It is not a statement that the physical\n", "system has only one variable. We deliberately hold the other inputs fixed so that one\n", "cause of variation is visible. Changing several inputs at once would make it harder\n", "to tell which change produced the output difference. Write down the held-fixed choices\n", "before trying a different experiment.\n", "\n", "At 0.4 m, the expression gives 0.951057 m. This is our\n", "reference calculation. It is a prediction of the stated example, not a new measurement.\n", "Its displayed digits help us compare implementations. They should not be read as a\n", "claim that physical inputs are known to six significant figures." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "import numpy as np\n", "import sympy as sp\n", "import matplotlib.pyplot as plt\n", "\n", "q = sp.Symbol(\"q\", real=True)\n", "expression = sp.sympify(\"sin(pi*q)\", locals={\"q\": q})\n", "reference_input = 0.4\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m\")\n", "print(\"Predicted fixed-end displacement shape:\", reference_output, \"m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "`sp.Symbol` declares a mathematical symbol. It does not assign a numerical measurement\n", "to that symbol. `subs` then replaces the symbol with our chosen input, and `float`\n", "converts the resulting symbolic number into a standard Python numerical value.\n", "Keeping these steps separate lets us inspect the equation before numerical substitution.\n", "It also prevents a later plot from becoming the only surviving record of the model.\n", "\n", "### Compare a few cases you can explain\n", "\n", "Before drawing hundreds of points, calculate three. The lower endpoint gives\n", "0 m. The midpoint, at 0.5 m, gives\n", "1 m. The upper endpoint gives 0 m.\n", "Read these together with the assumptions: they describe different choices of position\n", "within the same example, with the other inputs held fixed.\n", "\n", "These three values are a useful first check, but they do not establish the shape between\n", "them. Two endpoints can be equal even when the curve bends or oscillates in between.\n", "Likewise, a large change between endpoints does not tell us whether the relationship is\n", "linear. We need the equation and intermediate samples to answer those questions." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.0, 0.5, 1.0])\n", "outputs = np.asarray(evaluate(inputs), dtype=float)\n", "outputs = np.broadcast_to(outputs, inputs.shape)\n", "for input_value, output_value in zip(inputs, outputs):\n", " print(f\"{input_value:.6g} m -> {output_value:.6g} m\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "`lambdify` turns the symbolic expression into a numerical function. We explicitly select\n", "NumPy as its numerical backend so it can accept an array. The same expression now has\n", "two useful forms: one for algebra and one for calculating many cases. The assertion\n", "checks that the chosen inputs produce finite outputs. It would catch an infinity or an\n", "undefined numerical result, but a finite value can still be physically inappropriate.\n", "\n", "### Make a graph that answers the question\n", "\n", "The horizontal axis shows position; the vertical axis shows fixed-end displacement shape. Every plotted\n", "point is a prediction of the same expression. We mark the reference case separately\n", "so that a reader can connect the earlier printed value to its place on the curve.\n", "The axis labels include units because a shape without a scale cannot tell us the size\n", "of a physical effect." ] }, { "cell_type": "code", "id": "code-013", "metadata": {}, "source": [ "input_grid = np.linspace(0.0, 1.0, 301)\n", "output_grid = np.asarray(evaluate(input_grid), dtype=float)\n", "output_grid = np.broadcast_to(output_grid, input_grid.shape)\n", "fig, ax = plt.subplots(figsize=(7, 4))\n", "ax.plot(input_grid, output_grid, label=\"Stated model\")\n", "ax.scatter([reference_input], [reference_output], label=\"Reference case\", zorder=3)\n", "ax.set_xlabel(\"Position (m)\")\n", "ax.set_ylabel(\"Fixed-end displacement shape (m)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "Look for a sign change, a turning point, or a region where the curve becomes steep.\n", "Connect each feature to the expression rather than to the colour of the line. If the\n", "curve has a minimum or maximum, changing the input in opposite directions may have\n", "similar effects on the output near that point. If it is nearly straight over a short\n", "interval, a local linear approximation may be useful even when the complete expression\n", "is nonlinear.\n", "\n", "Increasing the number of samples makes the line smoother and may reveal features that\n", "a coarse grid missed. It does not improve the physical inputs or enlarge the model's\n", "validity range. For quantities that can take only discrete values, use the line as a\n", "guide to the relationship and interpret only the allowed input values. A plotting\n", "function does not enforce such restrictions for you.\n", "\n", "### Measure the local sensitivity\n", "\n", "A derivative asks a more specific question than the whole graph: how quickly does\n", "fixed-end displacement shape change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 0.4 m is\n", "0.970806 in those ratio units. Locally, the output increases as the\n", "input increases. This statement is local; the sign or size of the slope can be different\n", "elsewhere on the curve.\n", "\n", "\n", "$$\n", "f'(q) = \\pi \\cos{\\left(\\pi q \\right)}\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 0.405\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"m\")\n", "print(\"Full expression:\", full_prediction, \"m\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "Here the changed input is 0.405 m. The tangent estimate is\n", "0.955911 m, while the full expression gives\n", "0.955793 m. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-017", "metadata": {}, "source": [ "input_span = 1.0\n", "for fraction in [1e-2, 1e-3, 1e-4]:\n", " step = input_span * fraction\n", " upper = float(evaluate(reference_input + step))\n", " lower = float(evaluate(reference_input - step))\n", " numerical_slope = (upper - lower) / (2 * step)\n", " difference = numerical_slope - slope_at_reference\n", " print(f\"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "For a smooth function, reducing the step often reduces the approximation error at first.\n", "Very small steps can eventually make subtraction of nearly equal floating-point values\n", "less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare\n", "the observed behaviour over a useful range of steps instead of choosing the smallest\n", "number the computer can represent.\n", "\n", "An absolute slope difference is useful when the expected slope is near zero. A relative\n", "difference would divide by a very small reference and could look enormous even when the\n", "absolute discrepancy is tiny. Away from zero, a relative difference can help compare\n", "quantities with very different scales. The tolerance should reflect the question and\n", "the numerical method, rather than a universal decimal threshold copied into every test.\n", "\n", "### Change the example and preserve the reasoning\n", "\n", "Make one copy of the reference calculation before editing it. Choose a second value of\n", "position inside the stated interval, predict whether fixed-end displacement shape will rise or fall,\n", "and then evaluate the expression. Place the prediction and the result in a text cell.\n", "If they disagree, identify whether the mistake was in the sign, the unit conversion,\n", "the algebra, or the assumption about how the curve behaves.\n", "\n", "Next choose a shorter plotting interval around the reference point. Compare the local\n", "curve with its tangent approximation. Then widen the interval again. The difference\n", "shows why a useful local estimate need not describe the whole range. Keep the axis\n", "labels and numerical scale visible when comparing the plots; automatic axis limits can\n", "make two different effects appear equally large.\n", "\n", "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\n", "to the setup that would violate that limit. State what new input, equation, or observation\n", "would be needed before trusting an answer in the changed situation. This last step\n", "connects the Python experiment to the physics instead of treating successful execution\n", "as the only criterion for a useful result.\n", "\n", "Before saving, restart the runtime and run all cells in order. A result that depends on\n", "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n", "unrounded values for later calculations, round only the reported output, and label any\n", "new data as measured, assumed, or generated. The graph and printed values should remain\n", "traceable to the same expression and the same choices of inputs.\n", "\n", "### Recover the output change by integrating the slope\n", "\n", "There is another way to check that the expression and its derivative describe the same\n", "relationship. Add up the small output changes predicted by the slope across the chosen\n", "input interval. In the continuous limit, this integral equals the output at the upper\n", "endpoint minus the output at the lower endpoint. The result has units of m:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-019", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(1.0) - evaluate(0.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.0, 1.0, points)\n", " slopes = np.asarray(slope_function(grid), dtype=float)\n", " slopes = np.broadcast_to(slopes, grid.shape)\n", " integrated_change = np.trapezoid(slopes, grid)\n", " print(\n", " points,\n", " \"integrated change:\",\n", " integrated_change,\n", " \"endpoint change:\",\n", " endpoint_change,\n", " )\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-020", "metadata": {}, "source": [ "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n", "their signed areas. A negative contribution remains negative; replacing every slope\n", "by its magnitude would calculate a different quantity. Increasing the sample count\n", "provides a practical refinement check. If the values do not settle toward the endpoint\n", "difference, first inspect the interval, singularities, and array ordering.\n", "\n", "Do not conclude that two agreeing calculations prove the model is physically complete.\n", "Both use the same selected relationship, so an omitted effect would be omitted from\n", "both. Their agreement is evidence that the numerical tools implement that relationship\n", "consistently. Comparing with an independent experiment or a more appropriate physical\n", "model is a separate test.\n", "\n", "Keep this comparison in the notebook when you change the expression. It is particularly\n", "useful after adding a coefficient, changing a sign convention, or converting an input\n", "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n", "calculation of the same output change gives you another place to notice them.\n", "\n", "## Sources\n", "\n", "- [OpenStax, *University Physics Volume 1*, Section 16.5, “Interference of Waves”](https://openstax.org/books/university-physics-volume-1/pages/16-5-interference-of-waves).\n", "- [OpenStax, *University Physics Volume 1*, Section 16.6, “Standing Waves and Resonance”](https://openstax.org/books/university-physics-volume-1/pages/16-6-standing-waves-and-resonance).\n", "- [Benjamin Crowell, *Light and Matter*, reflection, transmission, and standing-wave discussions](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, “Interfaces and Barriers”](https://physics.info/interface/).\n", "- [Physics LibreTexts, “Boundary Conditions: What Happens at the Ends?”](https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/The_Physics_of_Waves_%28Goergi%29/03%3A_Coupled_Oscillators/3.02%3A_Boundary_Conditions-_What_Happens_at_the_Ends)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "boundary-conditions.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/boundary-conditions", "source_sha256": "b9a382c150f7ede6caffb3bcfe05bf4bda80d41db00277a475acdf1e9aeb665b" } }, "nbformat": 4, "nbformat_minor": 5 }