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Bessel y

BESSELY Function

The BESSELY function in Excel computes the Bessel function of the second kind, denoted as Yₙ(x).
This mathematical function arises in solving differential equations with cylindrical symmetry in various scientific
and engineering applications, such as acoustic wave propagation, electromagnetics, and vibration analysis.

Key Features of BESSELY:

  • Computes the value of the Bessel function of the second kind for a given order (n).
  • Specifically computes Yₙ(x):
    • x is the input value (must be greater than zero).
    • n is the order of the function, representing the degree of the Bessel function.
  • It is often used to model problems involving cylindrical symmetry where boundary conditions
    require a second independent solution (compared to the Bessel function of the first kind).

Syntax:

BESSELY(x, n)
  • x: The input value at which the Bessel function is evaluated.
    • Must be a numeric value greater than 0.
  • n: The order of the Bessel function.
    • Must be a numeric value. If a non-integer value is provided, Excel truncates it to the nearest integer.

Examples:

  1. Basic Example:
    =BESSELY(2, 0)
    Computes the Bessel function of the second kind for x = 2 and n = 0.
    Result: -0.5103757 (approximate value).

  2. Higher Order Calculation:
    =BESSELY(3, 2)
    Computes the Bessel function of the second kind for x = 3 and n = 2.
    Result: -0.1289432 (approximate value).

  3. Smaller Value of x:
    =BESSELY(0.5, 1)
    Evaluates the Bessel function for x = 0.5 and n = 1.
    Result: -1.4714724 (approximate value).

Notes:

  • Behavior of BESSELY:
    • The Bessel function of the second kind, Yₙ(x), is singular (undefined) at x = 0.
    • Larger values of n cause the function to oscillate more rapidly.
    • Used as a companion to the Bessel function of the first kind (BESSELJ), forming a complete set of linearly independent solutions to Bessel's differential equation.
  • Error values:
    • #VALUE!: Occurs if x or n is non-numeric.
    • #NUM!: Occurs if x is less than or equal to zero (invalid input for this function).
  • Truncation of n:
    • The order n must be a whole number; Excel truncates non-integer values automatically.

Applications:

  • Use Case: The BESSELY function is widely used in wave equations and vibration analysis for cylindrical structures,
    including calculations in structural mechanics, acoustics, and fluid dynamics.

  • Complementary to BESSELJ: The BESSELY function complements the Bessel function of the first kind (Jₙ(x)),
    particularly when a second independent solution is needed for modeling wave behavior in physics.