Bessel's equation and Bessel functions (intro, physical relevance): Question bank

Before we start, one reminder of what each symbol is, so no line uses an undefined term:

$u (r, θ)$ = the physical field on the disc (drum displacement, temperature, ...); $u_{r}$ means $\partial u / \partial r$ , its slope in the radial direction.
$r$ = the actual radius (distance from the centre); $a$ = the rim radius of the disc, where the edge is clamped.
$k$ = the wave-number, a fixed positive constant setting the wave's scale; it packages the frequency into the radial equation via the substitution $x = k r$ .
$x$ = the independent variable (a stretched radius, $x = k r$ ). It moves.
$ν$ (Greek "nu") = the order, a fixed number you choose once. It does not move.
$J_{ν} (x)$ = Bessel function of the first kind — finite at $x = 0$ , the "sine for round things."
$Y_{ν} (x)$ = Bessel function of the second kind — blows up at $x = 0$ .
$α_{ν, k}$ = the $k$ -th positive zero of $J_{ν}$ , i.e. the $k$ -th value of $x$ (counting outward, $k = 1, 2, 3, \dots$ ) where $J_{ν} (x) = 0$ . Here $k$ is a counting index, unrelated to the wave-number $k$ above — the letter is overloaded by tradition, so read it from context.
$c_{1}, c_{2}$ = arbitrary constants of integration: the two free constants any second-order linear ODE's general solution carries, fixed later by boundary conditions.
The drum problem: a circular membrane of rim radius $a$ , clamped at the edge, so the boundary condition is $u = 0$ (equivalently $R (a) = 0$ ) at $r = a$ , while $u$ stays finite at the centre $r = 0$ .

True or false — justify

TF1. Bessel's equation is a linear ODE.

True —

y

y^{'}

y^{''}

each appear to the first power with coefficients depending only on

x

; no

y^{2}

y y^{'}

ever appears, so solutions superpose.

TF2. Because

J_{ν} (x)

oscillates and crosses zero, it is periodic like

cos x

False — its zeros are not equally spaced and its amplitude shrinks like

1/ x

. It is a stretched, damped cosine, never exactly periodic.

TF3. On a solid disc the general solution still keeps both

J_{ν}

and

Y_{ν}

False — a solid disc includes

r = 0

where

Y_{ν} \to \pm \infty

; physical displacement is finite there, so its coefficient is forced to

0

. Both survive only on an annulus (hole in the middle).

TF4. In

x^{2} y^{''} + x y^{'} + (x^{2} - ν^{2}) y = 0

, the term

- ν^{2}

is a constant, not a function of

x

True —

ν

is the fixed order, so

- ν^{2}

is just a number subtracted from

x^{2}

; reading it as variable is the classic slip.

TF5. Every

J_{n}

starts at height

1

when

x = 0

False — only

J_{0} (0) = 1

; for

n \geq 1

J_{n} (0) = 0

because the series for

J_{ν}

begins with the power

(x /2)^{ν}

, which vanishes at

0

unless

ν = 0

TF6. Point

x = 0

is an ordinary point, so a plain Taylor series solves Bessel's equation.

False — dividing by

x^{2}

makes the coefficients blow up, so

x = 0

is a regular singular point; you need the Frobenius method with the extra power

x^{s}

TF7. The two indicial roots are always

+ ν

and

- ν

regardless of whether

ν

is an integer.

True — the indicial equation is

s^{2} - ν^{2} = 0

for every

ν

; but when

ν

is an integer the two roots differ by an integer, which is why the second solution needs a

ln x

term.

TF8. Half-integer orders like

ν = \frac{1}{2}

give genuinely new "special" functions with no closed form.

False — half-integer orders collapse to elementary functions, e.g.

J_{1/2} (x) = 2/ (π x) sin x

. They are a favourite sanity check precisely because they are elementary.

TF9. A drum's overtones are integer multiples of its fundamental, like a guitar string.

False — the frequencies scale with the zeros

α_{0, k}

(

2.405, 5.520, 8.654, \dots

), whose ratios are irrational. That inharmonicity is why a drum sounds like a drum, not a string.

TF10. The

x y^{'}

term would still be there if we set up the wave equation on a straight string instead of a disc.

False — the

x y^{'}

traces back to the

\frac{1}{r} u_{r}

(radial slope) in the polar Laplacian, which exists because area grows like

r

. A straight string has no such geometric spreading, so no first-derivative term and a clean sine appears.

Why questions

WH1. Why does separating variables on a disc force

ν

to be an integer, while on an annulus it need not be?

On a full disc the angle wraps all the way around, so

Θ (θ + 2 π) = Θ (θ)

(single-valued); only integer

ν

satisfies this. See Separation of variables.

WH2. Why does the

+ x^{2}

term (not

- x^{2}

) make the solution oscillate rather than grow?

+ x^{2}

acts like a restoring "

+ k^{2} y

" spring term, so solutions curve back toward zero and wave; a

- x^{2}

would push

y

away from zero, giving exponential-type growth (that is the modified Bessel case).

WH3. Why does the amplitude of

J_{ν}

decay like

1/ x

instead of staying constant?

Energy in a circular wave spreads over a ring of circumference

\propto x

, so energy density falls like

1/ x

and amplitude (its square root) like

1/ x

— visible in the asymptotic

2/ (π x) cos (\dots)

WH4. Why is

x = 0

a regular singular point and not an irregular one?

In standard form the coefficients are

\frac{1}{x}

and

\frac{x ^{2} - ν ^{2}}{x ^{2}}

; multiplying by

x

and

x^{2}

respectively gives analytic (finite, well-behaved) functions at

0

. That "mild" blow-up is the definition of regular — see Regular singular points.

WH5. Why does the recurrence link

a_{m}

a_{m - 2}

(a jump of two) rather than

a_{m - 1}

The oscillating term

+ x^{2} y

raises the power of every series term by

2

, so it ties each coefficient to the one two steps behind. This also kills all odd coefficients since

a_{1} = 0

WH6. Why does the Gamma function appear in the normalization of

J_{ν}

For non-integer

ν

the denominators are factorials of non-integers; the Gamma function is the continuous extension of the factorial that lets

(m + ν)!

make sense.

WH7. Why do zeros of

J_{ν}

— and not the value of

J_{ν}

— set a drum's frequencies?

The clamped edge forces

R (a) = 0

, i.e.

J_{ν} (k a) = 0

; only special products

k a = α_{ν, k}

satisfy this, and each such

k

selects one allowed vibration frequency.

WH8. Why do the two solutions merge (needing a log term) exactly when

ν

is an integer?

The indicial roots

\pm ν

then differ by an integer, so the smaller-root Frobenius series collides with the larger one; Frobenius method then guarantees the second solution carries a

ln x

factor — that is

Y_{n}

Edge cases

EC1. What happens at

ν = 0

that does not happen for

ν > 0

The two indicial roots

\pm ν

coincide at

0

, giving a repeated root; the second solution

Y_{0}

then necessarily contains

ln x

and diverges logarithmically at the origin.

EC2. What is the behaviour of

J_{ν} (x)

x \to 0^{+}

for

ν > 0

It approaches

0

(x /2)^{ν} /Γ (ν + 1)

— the leading power of the series — so it is finite and vanishing, with a flatter start for larger

ν

EC3. What is the limiting shape of

J_{ν} (x)

x \to \infty

It looks like

2/ (π x) cos (x - \frac{ν π}{2} - \frac{π}{4})

: an ever-slower-shrinking cosine whose zeros become almost evenly spaced (spacing

\to π

EC4. On an annular region

b \leq r \leq a

(a washer), how many Bessel functions survive and why?

Both

J_{ν}

and

Y_{ν}

survive, because the centre

r = 0

is excluded — nothing is unbounded on the domain, so both constants

c_{1}, c_{2}

stay and two boundary conditions fix them.

EC5. If a problem gives

x^{2} y^{''} + x y^{'} + (x^{2} + ν^{2}) y = 0

(a plus

ν^{2}

), is it Bessel's equation?

Not the standard one — a

+ ν^{2}

would still reduce to order

i ν

, but more commonly the sign flip to be careful about is

x^{2} \to - x^{2}

, which gives the modified Bessel equation with non-oscillating solutions

I_{ν}, K_{ν}

EC6. What does

J_{1/2}

tell you about the "half-integer" edge of the family?

That the family connects smoothly to elementary trigonometry:

J_{1/2} (x) = 2/ (π x) sin x

is an exact closed form, confirming the

1/ x

decay explicitly.

Recall One-line survival guide

" $x$ moves, $ν$ is fixed; $J$ stays (finite at $0$ ), $Y$ flies (infinite at $0$ ); it's a damped, unevenly-spaced cosine, not a periodic one." Every trap above is one of these three facts in disguise.

Connections

Bessel's equation and Bessel functions (intro, physical relevance) — the parent this bank drills.
Frobenius method — why the series (and the log in $Y_{n}$ ) appear.
Regular singular points — the reason Taylor fails at $x = 0$ .
Separation of variables — where the integer- $ν$ constraint is born.
Gamma function — the factorial extension inside $J_{ν}$ .

Question bank — Bessel's equation and Bessel functions (intro, physical relevance)