📐 Bessel Function Zeros 計算機

This tool finds the s-th positive zero of the Bessel functions of the first kind J_v(x) and second kind Y_v(x) (the Neumann or Weber function) for a real order v. These zeros, written j_{v,s} and y_{v,s}, appear throughout physics and engineering: vibrating circular membranes (drumheads), heat conduction in cylinders, electromagnetic waveguides, and Fourier-Bessel series. It is a pure-mathematics special-function tool and is universal, with no regional or unit dependence.

Enter the Order v (a real number, typically 0 to 200) and the Zero index s (a positive integer 1, 2, 3, …). The calculator returns j_{v,s}, the s-th positive root of J_v(x), and y_{v,s}, the s-th positive root of Y_v(x). For example, v = 0, s = 1 gives the fundamental drumhead mode.

Both functions solve Bessel's equation x^2 y'' + x y' + (x^2 - v^2)y = 0. J_v is evaluated from its convergent power series; Y_v uses

with the integer-order case handled as a numerical limit. To locate the s-th zero we seed with the McMahon asymptotic estimate, bracket the root, then refine by bisection until convergence. Because Y_v(x) \to -\infty at x = 0, the search begins at a small positive x to avoid the logarithmic singularity.

For v = 0, s = 1: the first positive zero of J_0(x) is 2.4048255577, and the first positive zero of Y_0(x) is 0.8935769663. For v = 1, s = 1: j_{1,1} = 3.8317059702 and y_{1,1} = 2.1971413260.

Bessel-function zeros are not an abstract curiosity — they are the discrete eigenvalues that emerge whenever a wave, heat, or potential problem is posed on a circular or cylindrical domain. The boundary condition forces the radial part of the solution to vanish (or its derivative to vanish) at the boundary, and that condition is satisfied only at the zeros j_{v,s}.

Vibrating circular membrane (drumhead)

For an ideal circular drum of radius a with a clamped edge, the displacement separates into modes labelled by (v,s), where v counts angular nodal diameters and s counts radial nodal circles. The allowed eigenfrequencies are f_{v,s}=((c) ÷ (2π a)) j_{v,s}, where c is the wave speed. The fundamental tone uses j_{0,1}=2.4048256; higher s and higher v both raise the pitch, and because the j_{v,s} are not integer multiples of one another a drum's overtones are inharmonic.

Cylindrical heat conduction

When solving the heat equation in a long cylinder with a fixed-temperature surface, the radial eigenfunctions are J_0(\lambda_s r/a) with \lambda_s=j_{0,s}. Each mode decays in time as \exp\!≤ft(-\alpha (j_{0,s}/a)^2 t\right), so the smallest zero j_{0,1} governs the slowest-decaying, longest-lived temperature profile. Larger s gives larger eigenvalues and therefore faster decay.

Waveguide cutoff frequencies

In a hollow circular metallic waveguide of radius a, transverse-magnetic (TM) modes cut off at frequencies set by j_{v,s} and transverse-electric (TE) modes by the zeros of the derivative J_v'. For TM modes the cutoff is f_{c}=((c j_{v,s}) ÷ (2π a)); only above this frequency does the mode propagate. The lowest TM mode (TM_{01}) again uses j_{0,1}.

Fourier–Bessel series

Any reasonable function on a disk can be expanded as f(r)=\sum_{s=1}^{\infty} c_s J_v\!≤ft(j_{v,s} r/a\right). The scaled zeros j_{v,s}/a act exactly like the wavenumbers nπ/L of an ordinary Fourier sine series, and the orthogonality of the J_v(j_{v,s}r/a) over the disk (with weight r) lets the coefficients be computed by integration, c_s=((2) ÷ (a^2 J_{v+1}^2(j_{v,s})))\int_0^a f(r) J_v\!≤ft(j_{v,s}r/a\right) r dr.

How the zeros shift

For a fixed order, increasing the index s increases j_{v,s} in steps that approach π (the oscillation becomes nearly periodic far from the origin). For a fixed index, increasing the order v pushes the first zero outward roughly like j_{v,1}≈ v + 1.8557 v^{1/3}+·s for large v, reflecting how the centrifugal v^2/x^2 term in Bessel's equation delays the onset of oscillation. The practical takeaway: higher angular complexity (v) and more radial nodes (s) both correspond to larger eigenvalues and therefore higher frequencies or faster decay.

For a concrete drum example with c=100\ m/s and a=0.30\ m, the fundamental frequency is f_{0,1}=((100) ÷ (2π(0.30))) (2.4048256)≈ 127.6\ Hz. The first overtone uses j_{1,1}=3.8317060, giving ≈ 203\ Hz, so the overtone-to-fundamental ratio is j_{1,1}/j_{0,1}≈ 1.593 — audibly not an octave or a fifth, which is why drums sound non-pitched compared with strings.