3. Consider an upright cylindrical container of radius R partially filled with water. A standing wave mode with azimuthal symmetry (no 0 dependence) has vertical displacement amplitude proportional to y(r) = Jo(ar/R), where Jo'(a) = 0. Consider the following question: Is the signed volume of the Bessel function measured from equilibrium (y = 0) equal to zero? Mathematically, the question is %3D %3D R R ar rdedr = 2n rJ. R ar dr = 0? R r= 0 0 = 0 r = 0 To determine whether or not the integral is zero, perform the following steps. First, use the change-of-variable x = ar/R to transform the integral, and note that this is a definite integral. Then, use a recursion relationship to perform the integral. Finally, evaluate the
3. Consider an upright cylindrical container of radius R partially filled with water. A standing wave mode with azimuthal symmetry (no 0 dependence) has vertical displacement amplitude proportional to y(r) = Jo(ar/R), where Jo'(a) = 0. Consider the following question: Is the signed volume of the Bessel function measured from equilibrium (y = 0) equal to zero? Mathematically, the question is %3D %3D R R ar rdedr = 2n rJ. R ar dr = 0? R r= 0 0 = 0 r = 0 To determine whether or not the integral is zero, perform the following steps. First, use the change-of-variable x = ar/R to transform the integral, and note that this is a definite integral. Then, use a recursion relationship to perform the integral. Finally, evaluate the
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:3. Consider an upright cylindrical container of radius R partially filled with water. A
standing wave mode with azimuthal symmetry (no 0 dependence) has vertical
displacement amplitude proportional to y(r) = Jo(ar/R), where Jo'(a) = 0. Consider the
following question: Is the signed volume of the Bessel function measured from
equilibrium (y = 0) equal to zero? Mathematically, the question is
%3D
R
R
ar
rdedr
ar
dr
rJ,
R
2n
0 ?
%3D
0.
r=0 e = 0
r= 0
To determine whether or not the integral is zero, perform the following steps. First, use
the change-of-variable x = ar/R to transform the integral, and note that this is a definite
integral. Then, use a recursion relationship to perform the
integral and use the fact J1(x) = -Jo'(x). Is the integral zero? Be sure to show all of your
work.
ral. Finally, evaluate the
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