{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Fourier perspective\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/fourier-perspective). 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 complicated wave shape can often be understood as a sum of simple sinusoidal patterns.\n", "Fourier analysis supplies the weights and phases of those patterns. This is more than a\n", "plotting trick: in a linear wave system, each sinusoidal mode can evolve independently,\n", "turning one complicated partial differential equation into many simple oscillator\n", "equations.\n", "\n", "## A periodic function as sines and cosines\n", "\n", "For a real function with period $L$, one\n", "common Fourier-series convention is\n", "\n", "\n", "$$\n", "f(x)=\\frac{a_0}{2}+\\sum_{n=1}^{\\infty}\\left[a_n\\cos\\left(\\frac{2\\pi n x}{L}\\right)+b_n\\sin\\left(\\frac{2\\pi n x}{L}\\right)\\right]\n", "$$\n", "\n", "\n", "The coefficients are projections onto mutually orthogonal basis functions:\n", "\n", "\n", "$$\n", "a_n=\\frac{2}{L}\\int_0^L f(x)\\cos\\left(\\frac{2\\pi n x}{L}\\right)dx\n", "$$\n", "\n", "\n", "\n", "$$\n", "b_n=\\frac{2}{L}\\int_0^L f(x)\\sin\\left(\\frac{2\\pi n x}{L}\\right)dx\n", "$$\n", "\n", "\n", "Orthogonality means different integer-frequency sines and cosines have zero inner product\n", "over a whole period. It lets a projection isolate one component even while all components\n", "are present in the waveform. The constant term represents the average:\n", "\n", "\n", "$$\n", "\\frac{a_0}{2}=\\frac{1}{L}\\int_0^L f(x)\\,dx\n", "$$\n", "\n", "\n", "Not every function has a pointwise convergent series everywhere. Under standard\n", "piecewise-smooth conditions, the series converges to the function at continuous points\n", "and to the midpoint of the jump at a discontinuity. Near a jump, truncated series show\n", "the Gibbs overshoot. Adding more terms narrows the affected region but does not eliminate\n", "the limiting overshoot height.\n", "\n", "## The complex form stores amplitude and phase\n", "\n", "Euler's relation combines sines and cosines into complex exponentials:\n", "\n", "\n", "$$\n", "f(x)=\\sum_{n=-\\infty}^{\\infty}c_n e^{i2\\pi n x/L},\\qquad c_n=\\frac{1}{L}\\int_0^L f(x)e^{-i2\\pi n x/L}dx\n", "$$\n", "\n", "\n", "For real $f(x)$, coefficients obey\n", "\n", "\n", "$$\n", "c_{-n}=c_n^*\n", "$$\n", "\n", "\n", "The negative-frequency coefficient is therefore not an accidental duplicate. Together,\n", "the pair encodes a real sinusoid's amplitude and phase. The magnitude\n", "$|c_n|$ measures the\n", "component strength under the chosen normalization, and\n", "$\\arg(c_n)$ records phase.\n", "\n", "Changing the sign in the forward exponential or moving the normalization factor between\n", "forward and inverse transforms is allowed. Mixing conventions is not. Always state the\n", "pair of definitions used.\n", "\n", "## Boundary conditions choose the spatial basis\n", "\n", "Periodic exponentials are not the only Fourier basis. A string fixed at\n", "$x=0$ and\n", "$x=L$ naturally uses sine modes:\n", "\n", "\n", "$$\n", "y(x,0)=\\sum_{n=1}^{\\infty}A_n\\sin\\left(\\frac{n\\pi x}{L}\\right)\n", "$$\n", "\n", "\n", "The sine functions already vanish at both endpoints. If the initial velocity is also\n", "specified, it supplies a second set of modal coefficients. The complete ideal-string\n", "motion can be written\n", "\n", "\n", "$$\n", "y(x,t)=\\sum_{n=1}^{\\infty}\\sin\\left(\\frac{n\\pi x}{L}\\right)\\left[C_n\\cos(\\omega_n t)+D_n\\sin(\\omega_n t)\\right]\n", "$$\n", "\n", "\n", "with\n", "$\\omega_n=vn\\pi/L$.\n", "This equation is the practical advantage of the Fourier viewpoint: project the initial\n", "shape and velocity once, evolve each coefficient like a harmonic oscillator, and add the\n", "modes to reconstruct the field." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "length, speed, time = 1.0, 1.0, 0.2\n", "x = np.linspace(0, length, 1001)\n", "initial = np.sin(np.pi * x) + 0.3 * np.sin(3 * np.pi * x)\n", "reconstructed = np.zeros_like(x)\n", "for mode in range(1, 6):\n", " shape = np.sin(mode * np.pi * x / length)\n", " coefficient = 2 / length * np.trapezoid(initial * shape, x)\n", " omega = speed * mode * np.pi / length\n", " reconstructed += coefficient * np.cos(omega * time) * shape\n", "print(\"Midpoint displacement:\", reconstructed[500])\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "The division by modal angular frequency converts an initial-velocity coefficient into the\n", "correct displacement amplitude. The zero-frequency mode, when a problem has one, must be\n", "handled separately rather than divided by zero.\n", "\n", "## The discrete Fourier transform\n", "\n", "Measured or computed signals contain finitely many samples. For\n", "$N$ samples\n", "$x_j$, one common discrete Fourier\n", "transform convention is\n", "\n", "\n", "$$\n", "X_k=\\sum_{j=0}^{N-1}x_j e^{-i2\\pi jk/N},\\qquad k=0,1,\\ldots,N-1\n", "$$\n", "\n", "\n", "The matching inverse is\n", "\n", "\n", "$$\n", "x_j=\\frac{1}{N}\\sum_{k=0}^{N-1}X_k e^{i2\\pi jk/N}\n", "$$\n", "\n", "\n", "If samples are separated by $\\Delta t$,\n", "the sample rate is $f_s=1/\\Delta t$ and bin\n", "spacing is\n", "\n", "\n", "$$\n", "\\Delta f=\\frac{f_s}{N}=\\frac{1}{N\\Delta t}\n", "$$\n", "\n", "\n", "Bins from $k=0$ through the Nyquist\n", "region represent nonnegative frequencies. Higher stored indices conventionally represent\n", "negative frequencies, with\n", "$f_k=(k-N)f_s/N$ for those\n", "indices. For a real signal, bin $k$\n", "is paired with bin $N-k$." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "samples = np.sin(2 * np.pi * np.arange(64) / 64)\n", "spectrum = np.fft.fft(samples)\n", "print(\"Dominant frequency bins:\", np.argsort(abs(spectrum))[-2:])\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "This direct calculation uses order\n", "$O(N^2)$\n", "operations. A fast Fourier transform computes the same DFT more efficiently, commonly in\n", "order $O(N\\log N)$,\n", "but does not change the meaning of the bins.\n", "\n", "## Sampling, aliasing, and leakage\n", "\n", "Sampling loses information unless the signal is sufficiently band-limited. Frequencies\n", "above half the sample rate can masquerade as lower frequencies. The Nyquist frequency is\n", "\n", "\n", "$$\n", "f_{\\mathrm{Nyquist}}=\\frac{f_s}{2}\n", "$$\n", "\n", "\n", "The phrase “sample at least twice the frequency” is a boundary, not a comfortable design\n", "rule. A practical system uses an analog low-pass filter and margin below the Nyquist\n", "frequency.\n", "\n", "Even a frequency below Nyquist can spread across many DFT bins when the finite record does\n", "not contain an integer number of cycles. This is spectral leakage: multiplying an\n", "infinite signal by a finite observation window changes its spectrum. A tapering window can\n", "reduce distant leakage but broadens peaks and changes amplitude calibration. Name the\n", "window and correct its gain when quantitative amplitudes matter.\n", "\n", "Frequency resolution and sample rate are separate. Increasing\n", "$N$ at fixed sample rate lengthens the record\n", "and narrows bin spacing. Increasing sample rate at fixed\n", "$N$ raises the Nyquist frequency but\n", "shortens the record and widens bin spacing.\n", "\n", "## Parseval's relation and physical energy\n", "\n", "With the DFT convention above, Parseval's relation is\n", "\n", "\n", "$$\n", "\\sum_{j=0}^{N-1}|x_j|^2=\\frac{1}{N}\\sum_{k=0}^{N-1}|X_k|^2\n", "$$\n", "\n", "\n", "It is a powerful round-trip and normalization check. However, squared sample magnitude is\n", "not automatically physical energy. A string's energy also depends on velocity, tension,\n", "linear density, spatial derivatives, and grid measure. Parseval tells us how a quadratic\n", "norm is distributed between representations; the physical model tells us which norm\n", "corresponds to energy.\n", "\n", "## Tests for a trustworthy transform\n", "\n", "Transform a constant array. Only the zero-frequency bin should be nonzero, apart from\n", "roundoff. Transform a pure sine whose frequency falls exactly on bin\n", "$m$. For a real signal, the dominant\n", "coefficients should appear at $m$ and\n", "$N-m$ with conjugate symmetry. Then\n", "shift the sine phase and verify that magnitudes stay fixed while coefficient phases change.\n", "\n", "Apply the inverse transform and calculate the largest sample error:" ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "recovered = np.fft.ifft(spectrum).real\n", "assert np.allclose(recovered, samples)\n", "print(\"Round-trip error:\", np.max(abs(recovered - samples)))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "Finally, try a non-bin-aligned sine and observe leakage, a frequency above Nyquist and\n", "observe aliasing, and an impulse and observe its flat-magnitude spectrum. These fixtures\n", "distinguish a correct transform from a spectrum that merely looks plausible.\n", "\n", "Fourier representations assume linear addition and a chosen basis. Nonlinear systems\n", "couple modes and create harmonics or combination frequencies. Time-varying signals may\n", "need short-time or wavelet methods because one spectrum hides when each component occurs.\n", "Even then, the central discipline remains: state the basis, sampling, normalization, and\n", "information discarded by the representation.\n", "\n", "## A worked Python notebook\n", "\n", "Reconstruct a string shape from its first and third sine modes. The question for this calculation is precise: how does **two-mode displacement**\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", "The sine basis is chosen for fixed endpoints; other boundary conditions require another basis. 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", "two-mode displacement, 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", "$$\n", "f(q) = \\sin{\\left(\\pi q \\right)} + 0.3 \\sin{\\left(3 \\pi q \\right)}\n", "$$\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 0.25 m, the expression gives 0.919239 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)+0.3*sin(3*pi*q)\", locals={\"q\": q})\n", "reference_input = 0.25\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m\")\n", "print(\"Predicted two-mode displacement:\", 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", "0.7 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 two-mode displacement. 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(\"Two-mode displacement (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", "two-mode displacement change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 0.25 m is\n", "0.222144 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", "$$\n", "f'(q) = \\pi \\cos{\\left(\\pi q \\right)} + 0.9 \\pi \\cos{\\left(3 \\pi q \\right)}\n", "$$\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 = 0.255\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 0.255 m. The tangent estimate is\n", "0.92035 m, while the full expression gives\n", "0.92003 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 two-mode displacement 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: The sine basis is chosen for fixed endpoints; other boundary conditions require another basis. 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*, Fourier analysis and normal-mode discussions](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, “Music and Noise”](https://physics.info/music/).\n", "- [Physics LibreTexts, *The Physics of Waves*, Chapter 10, “Signals and Fourier Analysis”](https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/The_Physics_of_Waves_%28Goergi%29/10%3A_Signals_and_Fourier_Analysis)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "fourier-perspective.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/fourier-perspective", "source_sha256": "c0124d7c0c39493d9f044e1d8be5e0ee1b565b1e51cca9d877cb68d1890d1805" } }, "nbformat": 4, "nbformat_minor": 5 }