Cylindrical wave

A cylindrical wave is a model of the wave process , a wave radially diverging from a certain axis in space or converging to it. A special case of a cylindrical wave on a plane is a circular wave.

The front of a cylindrical wave is a cylindrical surface , on the axis of which a source is located, for example, having the shape of a thread, that is, infinitely thin and straight. The propagation of the front of such a wave in space can be compared with a cylindrical surface, continuously increasing its radius. An example of a cylindrical wave is the wave process on the surface of the water from an oscillating float, as well as an electromagnetic wave generated in the near zone by a linear in-phase antenna .

Definition

The simplest monochromatic symmetric cylindrical wave with a source in the center satisfies the two-dimensional wave equation and is described using the Hankel function of zero order:

u(r,t)=H_{0}(kr)\cdot e^{i\omega t},

{\ displaystyle u (r, t) = H_ {0} (kr) \ cdot e ^ {i \ omega t},}

u(r,t)=H_{0}(kr)\cdot e^{{i\omega t}},

(1.1)

Where $H_{0}(kr)$ ${\ displaystyle H_ {0} (kr)}$ $H_{0}(kr)$ - Hankel function of zero order;

i

{\ displaystyle i}

i

- imaginary unit ;

\omega

{\ displaystyle \ omega}

\omega

- circular frequency;

k

{\ displaystyle k}

k

- wave number ;

r

{\ displaystyle r}

r

- distance from the axis.

At large distances from the axis - that is, at $kr\rightarrow inf$ ${\ displaystyle kr \ rightarrow inf}$ $kr\rightarrow inf$ the wave field (1.1) takes the form

u({\overrightarrow {r}},t)={\cfrac {u_{0}}{\sqrt {r}}}\cdot e^{i(\omega t-kr)}.

{\ displaystyle u ({\ overrightarrow {r}}, t) = {\ cfrac {u_ {0}} {\ sqrt {r}}} \ cdot e ^ {i (\ omega t-kr)}.}

u({\overrightarrow {r}},t)={\cfrac {u_{0}}{\sqrt {r}}}\cdot e^{i(\omega t-kr)}.

(1.2)

^[one]

Properties

As you move away from the oscillator, the amplitude decreases hyperbolic ;
Since the area of the lateral surface of the cylinder $\thicksim r$ ${\ displaystyle \ thicksim r}$ then the function stream $u(r,t)$ ${\ displaystyle u (r, t)}$ remains constant;
In the recording form (1.2) , the wave amplitude can be distinguished ${\cfrac {A}{\sqrt {r}}},$ ${\ displaystyle {\ cfrac {A} {\ sqrt {r}}},}$ phase $\omega t-kr=\omega (t-{\cfrac {r}{u_{\Phi }}}),$ ${\ displaystyle \ omega t-kr = \ omega (t - {\ cfrac {r} {u _ {\ Phi}}},}$ Where $u_{\Phi }$ ${\ displaystyle u _ {\ Phi}}$ - phase velocity of a plane wave .

Notes

↑ Abramowitz M., Stegan I. Handbook of special functions. - Moscow: Science, 1979.

Links

Cylindrical wave on dic.academic.ru

Cylindrical wave

Definition

Properties

Notes

Links

See also

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