fcce4e80-a14e-4bed-bb9a-31c5d30b506f 38 4 5 decimal

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Axisymmetric vibrations of a circular membrane

contour radius a 1 :

wave speed c 1 :

Approximation of the Bessel function J0(x) by trigonometric functions x P.0 19128 x 2^* / - 367532768 x 4^* / + 24012454194304 x 6^* / - 130439058752147483648 x 8^* / + : x Q.0 18 x * / - 751024 x 3^* / + 59535262144 x 5^* / - 5797291533554432 x 7^* / + : x J.0maxX 2 π x * / sqrt x π 4 / - cos x P.0 * x π 4 / - sin x Q.0 * - (* (: x J.0minX 1 x 2 / (2^- 122^/ x 2 / (4^(* + 162^/ x 2 / (6^(* - : Approximation of the Bessel function J0(x) x J.0 x J.0minX x 2 ≤ (x 2 - ≥ (& (* x J.0maxX x 2 > ((* + x - J.0maxX x 2 - ≤ (* + : 61 line

График функции J0(x)

x J.0

JZeros 2.4048 5.5201 8.6537 31 mat : (Zeros of the Bessel funktion ≡ k λ0 1 a 2^/ JZeros k el (2^* : (Frequency and periods of oscillations of standing waves ≡ k T0 2 π * c k λ0 sqrt * / : k tmax 60 k T0 * 1 T0 / eval : (Number of animation frames ≡ 41 line

u v t F1 u v cos * u v sin * 1 λ0 sqrt u * J.0 2 π * t * 1 tmax / cos * 31 mat :

u v t F2 u v cos * u v sin * 2 λ0 sqrt u * J.0 2 π * t * 2 tmax / cos * 31 mat :

u v t F3 u v cos * u v sin * 3 λ0 sqrt u * J.0 2 π * t * 3 tmax / cos * 31 mat :

t w1 u v t F1 a - a 0 π 1112 CreateMesh γ * 9 * eval :

t w2 u v t F2 a - a 0 π 1112 CreateMesh γ * 9 * eval :

t w3 u v t F3 a - a 0 π 1112 CreateMesh γ * 9 * eval :

γ 0.8232 - 0.4194 - 0.3827 0.5677 0.6187 - 0.543 0.009 0.6643 0.7474 33 mat :

τ1 01 tmax range :

τ2 02 tmax range :

τ3 03 tmax range :

Mode1 with JZeros1=2.4048

1 λ0 sqrt x * J.0 2 π * t * 1 tmax / cos * ( x a ≤ ( x a - ≥ ( & ( * 1 1 sys

Mode2 with JZeros1=5.5201

2 λ0 sqrt x * J.0 2 π * t * 2 tmax / cos * ( x a ≤ ( x a - ≥ ( & ( * 1 1 sys

Mode3 with JZeros1=8.6537

3 λ0 sqrt x * J.0 2 π * t * 3 tmax / cos * ( x a ≤ ( x a - ≥ ( & ( * 1 1 sys

Mode 1 with JZeros1 = 2.40483

t w1 2 1 sys

Mode 2 with JZeros1 = 5.52008

t w2 2 1 sys

Mode 3 with JZeros1= 8.65373

t w3 2 1 sys