Eigenvalues of the clamped plate

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Numbers

$\Omega$

$n$

$\Lambda_n(\Omega)$

unit disk

104.3631055588443069217226196733478504926323754398933759856840104698636921970114261209766249526384382

comment: The fundamental mode is axisymmetric.

unit disk

452.0045101331736988467916595258984454009644450918701964899603306426486157873653552620258016438930306

unit disk

452.0045101331736988467916595258984454009644450918701964899603306426486157873653552620258016438930306

equals: Eigenvalues_of_the_clamped_plate#unit-disk,2

unit disk

1216.407599710232222733545643005433062027903421270876486730021495747232191797182325545017814425922727

unit disk

1216.407599710232222733545643005433062027903421270876486730021495747232191797182325545017814425922727

equals: Eigenvalues_of_the_clamped_plate#unit-disk,4

unit disk

1581.744232046243252622929569151627795239025101346622184990568727436864020758041167764353601662834239

unit disk

2604.064521475449833922023312520128000382939700031631070394100347937308461246857159022825555230573092

unit disk

2604.064521475449833922023312520128000382939700031631070394100347937308461246857159022825555230573092

equals: Eigenvalues_of_the_clamped_plate#unit-disk,7

unit disk

3700.127315387753587399456988721996117461275902883879006442961301091061542516216634552455108566650566

unit disk

10:

3700.127315387753587399456988721996117461275902883879006442961301091061542516216634552455108566650566

equals: Eigenvalues_of_the_clamped_plate#unit-disk,9

unit disk

11:

4853.327966673681641730090020971294160865557679431324697448953296934731128046732383354758217364698962

unit disk

12:

4853.327966673681641730090020971294160865557679431324697448953296934731128046732383354758217364698962

equals: Eigenvalues_of_the_clamped_plate#unit-disk,11

unit disk

13:

7154.224605147623064464254736132426079661825688938347027412168785611778109393490229032054565216330035

unit disk

14:

7154.224605147623064464254736132426079661825688938347027412168785611778109393490229032054565216330035

equals: Eigenvalues_of_the_clamped_plate#unit-disk,13

unit disk

15:

7939.548473333865817263505628735069847072698724123947418655681977785980708441354409238343285804979083

unit disk

16:

8233.563637917670338834742657552718284779400855211964930681912320111057599648915877365573578074069928

unit disk

17:

8233.563637917670338834742657552718284779400855211964930681912320111057599648915877365573578074069928

equals: Eigenvalues_of_the_clamped_plate#unit-disk,16

unit disk

18:

12325.75389527305708772729914597506464536282597368167960904849694741148639424242518245335203370012153

unit disk

19:

12325.75389527305708772729914597506464536282597368167960904849694741148639424242518245335203370012153

equals: Eigenvalues_of_the_clamped_plate#unit-disk,18

unit disk

20:

13044.50009834193317120172848526954954576475241724709333926971040581170495678441861407478055782527525

unit disk

21:

13044.50009834193317120172848526954954576475241724709333926971040581170495678441861407478055782527525

equals: Eigenvalues_of_the_clamped_plate#unit-disk,20

unit disk

22:

14419.02305826541798619403621968627437952163357646889810197699514412844449880841829639206745686817896

unit disk

23:

14419.02305826541798619403621968627437952163357646889810197699514412844449880841829639206745686817896

equals: Eigenvalues_of_the_clamped_plate#unit-disk,22

unit disk

24:

19615.71264544360012231878098355244037349935659617387081940286914055743255098614047570321298039266981

unit disk

25:

19615.71264544360012231878098355244037349935659617387081940286914055743255098614047570321298039266981

equals: Eigenvalues_of_the_clamped_plate#unit-disk,24

unit disk

26:

19630.22471473453184808176671236668177880178727751759813632934930186426522635320988002313129506290336

unit disk

27:

19630.22471473453184808176671236668177880178727751759813632934930186426522635320988002313129506290336

equals: Eigenvalues_of_the_clamped_plate#unit-disk,26

unit disk

28:

23659.08034615677324428626217020222550809661709984221004242192679543681074424528221131616770093587113

unit disk

29:

23659.08034615677324428626217020222550809661709984221004242192679543681074424528221131616770093587113

equals: Eigenvalues_of_the_clamped_plate#unit-disk,28

unit disk

30:

25022.24592150141509641542516675749880207835593547178792028896015990404132175862638589771266536216870

Definition

The clamped-plate eigenvalues $\Lambda_n(\Omega)$ are the values of $\Lambda$ for which a nonzero function $u$ satisfies $\Delta^2u=\Lambda u$ in the planar domain $\Omega$, with $u=\partial_\nu u=0$ on its boundary, where $\nu$ is the outward unit normal [1].

Parameters

$\Omega$

— domain ($\Omega$ is one of the listed classical planar domains.)

$n$

— eigenvalue index ($n\geq1$)

Formulas

(1)

For $m\geq0$, let $\gamma_{m,k}$ be the $k$th positive zero of $W_m(x)=J_m(x)I_{m+1}(x)+J_{m+1}(x)I_m(x)$, where $J_m$ and $I_m$ are the ordinary and modified Bessel functions of the first kind. The disk spectrum is the sorted multiset of $\gamma_{m,k}^4$, with each $m=0$ root once and each $m>0$ root twice [1].

(2)

For $\gamma=\gamma_{m,k}$, the radial factor is $R_{m,k}(r)=I_m(\gamma)J_m(\gamma r)-J_m(\gamma)I_m(\gamma r)$. It satisfies $R_{m,k}(1)=R_{m,k}^{\prime}(1)=0$ [1].

(3)

If $j_{m,k}$ is the $k$th positive zero of $J_m$, then $j_{m,k}<\gamma_{m,k}<j_{m,k+1}$. For fixed $k$, the root $\gamma_{m,k}$ increases strictly with $m$ [2].

(4)

For a uniform thin isotropic plate on the unit disk with bending rigidity $D$, mass density $\rho$ and thickness $h$, the angular frequency satisfies $\omega_n=\sqrt{D/(\rho h)}\,\sqrt{\Lambda_n}$. The radial characteristic parameter is $\Lambda_n^{1/4}$ [4].

Comments

(5)

The unit disk is $B_1=\{(x,y):x^2+y^2<1\}$. Its radius agrees with the Dirichlet spectrum table; the domain is not rescaled to unit area.

(6)

The eigenvalues are positive, ordered in nondecreasing order and repeated according to multiplicity. The index $n$ orders the full spectrum, including all angular modes. The value stored is the eigenvalue $\Lambda_n$ of $\Delta^2$.

(7)

Let $m=0,1,2,\ldots$ be the angular order and $k\geq1$ the radial root index. Each radial root contributes one angular mode for $m=0$ and two, proportional to $\cos(m\theta)$ and $\sin(m\theta)$, for $m>0$. Distinct $(m,k)$ pairs have distinct eigenvalues throughout the range held here. This last statement is not asserted for the entire infinite spectrum; [1] discusses possible additional multiplicities.

(8)

Here $J_m$ is the Bessel function of the first kind and $I_m$ is the modified Bessel function of the first kind. [3] gives the separated clamped-disk eigenfunctions.

Programs

(P1)

Python

import mpmath as mp
mp.mp.dps = 110
def disk_mode(m, k):
    # One angular/radial mode; the returned quantity is the eigenvalue.
    a, b = mp.besseljzero(m, k), mp.besseljzero(m, k+1)
    def f(x):
        return (mp.besselj(m+1, x) + mp.besselj(m, x)
                * mp.besseli(m+1, x) / mp.besseli(m, x))
    g = mp.findroot(f, (a+(b-a)/5, a+2*(b-a)/5))
    return g**4
print(mp.nstr(disk_mode(0, 1), 100))

References

[1]

Dan Mangoubi and Daniel Rosenblatt, On multiplicity bounds for eigenvalues of the clamped round plate, 2025, Section 1.1. (arXiv)

[2]

Árpád Baricz, Saminathan Ponnusamy and Sanjeev Singh, Cross-product of Bessel functions: monotonicity patterns and functional inequalities, Proceedings of the Indian Academy of Sciences, Mathematical Sciences 128 (2018), article 36. (arXiv) (doi)

Links

[3]

Mangoubi and Rosenblatt: clamped round plate, Section 1.1

[4]

Steven L. Garrett: Membranes, Plates, and Microphones, Section 6.4.1

Similar tables

Dirichlet_eigenvalues_of_the_classical_planar_domains — the second-order membrane spectrum on the same domain geometries, with displacement zero at the boundary

T485 — the second-order membrane spectrum with zero normal derivative at the boundary

T486 — the one-dimensional fourth-order vibration problem, stored as characteristic roots

T487 — a different Bessel cross-product, involving J and Y, that determines annular membrane spectra

Zeros_of_Bessel_J_functions — the disk characteristic roots interlace with these zeros

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds the first 30 eigenvalues of the unit disk, including multiplicities and ending after a complete eigenspace. These low modes provide reference values for clamped circular plates and numerical eigenvalue solvers. The other domains in the Dirichlet spectrum table are omitted)

How they were obtained:

Every entry is stored to 100 significant digits. Arb at 513 bits certifies opposite signs at exact rational endpoints of each characteristic-root bracket, whose width is $2\cdot10^{-112}$. The derivative excludes zero in each bracket. Certified signs on a unit mesh, a proved zero-spacing bound for $J_m$ and the interlacing theorem [2] establish the radial indices. Monotonicity in angular order excludes all higher angular modes. The returned eigenvalue balls have a measured minimum accuracy of 371 bits, exceeding 111 decimal digits.

The attached checks independently construct the displacement-and-slope determinant from power-series solutions of the fourth-order radial differential equation. Geometric bounds enclose the series tails. At 768 bits this determinant has opposite signs at both exact endpoints of every written decimal interval. Its known value and first derivative at zero serve as controls. Roots computed independently with mpmath at 140 and 180 decimal digits agree to at least 125 relative places and lie inside the written intervals. Ordering, multiplicities and these numerical checks are repeated on the values read back from the draft. The Programs snippet computes approximations; the attached generator supplies the certified enclosures.