Quick answer

Typical examples evaluate Jn at zeros that define modes, compare neighboring orders using recurrence, and read amplitudes from plots. The calculator supplies decimals; you supply the physical scaling.

Formula

  • Disk mode: Jn(jn,k r/a) with jn,k a zero of Jn
  • Cutoff: Jn(x) = 0 defines allowed k or frequency
  • Check: evaluate Jn(x) at tabulated zeros

Introduction

Examples turn symbols into intuition. Here we work through values you can repeat in under a minute on the Bessel Function Calculator, tied to problems from vibrations and waves.

Review formulas if recurrence notation is unfamiliar, and Jv of the first kind for deeper background on the family.

Each example states the order, the argument, and why that point matters: a boundary, a zero, or a check between neighbors. Copy the pattern onto your own homework sheet.

Mode problems add geometry: radius a, wavenumber k, and an index for the zero used. The Bessel piece is still Jn(k r) with k fixed by Jn(k a) = 0.

Keep the calculator plot open while you read. Seeing a near-zero value on the graph is stronger memory than a lone decimal in a table.

What makes a good example?

A good example states v, x, and why that point matters: a boundary, a zero, or a comparison between orders. It ends with a number you can look up independently.

Mode problems add geometry: radius a, wavenumber k, and an index for the zero used. The Bessel piece is still Jn(k r) with k fixed by Jn(k a) = 0.

Examples with Yn or In belong to other articles; this page stays with Jn in real arguments unless noted.

Dimensionless x is non-negotiable. If your radius is in centimeters but your wavenumber in inverse meters, convert before you type into the calculator.

When an example gives a frequency, trace back to β or k and show Jn(β a) = 0 explicitly. That line is what graders look for before they care about amplitude.

Formulas behind the numbers

  • J0(x) = 0 → x = j0,k (k-th zero)
  • J1(x) first zero ≈ 3.8317
  • Recurrence check at x = 2: J0(2) + J2(2) = (2/2) J1(2) for n = 1

Zeros are transcendental but stable in tables. The calculator evaluates the function at those locations; it does not replace a root finder for new parameters you invent.

When an example uses small x, you can also sum a few series terms from the formula article and compare. Agreement at three decimals is enough for classroom work.

Superposition examples add coefficients in front of each Jn mode. Orthogonality integrals pick those coefficients; the calculator only checks the shape Jn at a point.

Work an example end to end

  1. State the physical setup Write disk radius, boundary condition, and which mode index you need.
  2. Find or quote the zero Use a handbook value for Jn(j a) = 0, or scan a plot from the calculator near the expected crossing.
  3. Evaluate amplitudes Compute Jn at interior points, such as r = a/2, once k is fixed.
  4. Cross-check identities Verify one recurrence at your x to catch wrong n.
  5. Summarize in one line End with mode index, zero used, and one interior amplitude so you can find the example again during revision.

Example trio: J0, J1, recurrence

At x = 2: J0(2) ≈ 0.2239, J1(2) ≈ 0.5767, J2(2) ≈ 0.3528. For n = 1 the recurrence gives J0(2) + J2(2) = (2/2) J1(2), and 0.2239 + 0.3528 matches 0.5767.

For a waveguide-style cutoff, if J0(k a) = 0 and a is known, the first root gives k a ≈ 2.4048. Evaluating J0(2.4) on the calculator shows a value near zero, confirming you are close to the root.

For angular dependence n = 2 at x = 3, J2(3) ≈ 0.0584, much smaller than J0(3) ≈ 0.2600, illustrating how higher orders start smaller before their own oscillation dominates.