Читать книгу Vibroacoustic Simulation - Alexander Peiffer - Страница 66

The most simple geometry we might think of is a point in space. For simple derivation of the sound field of a point source the spherical coordinate system is introduced as shown in Figure 2.4 and defined by the following coordinate transformation

(2.62a)

(2.62b)

(2.62c)

Figure 2.4 Definition of a spherical coordinate system. Source: Alexander Peiffer.

Using this coordinate system and neglecting the angular components the Laplace operator Δ reads as

(2.63)

The wave equation for the velocity potential (2.30) becomes

(2.64)

The two right terms can be written in a different form using rΦ as argument

(2.65)

Equation (2.30) is the one-dimensional wave equation for the argument rΦ, so we can use the D’Alambert solution

(2.66)

Figure 2.5 Breathing sphere as source model for a monopole. Source: Alexander Peiffer.

The first term represents an outgoing wave travelling away from the source, the second an incoming wave travelling to the source. As we are interested in sound being emitted from the source we consider the outgoing harmonic solution with complex amplitude A

(2.67)

Consider a pulsating sphere of radius R in the centre with normal surface velocity vR. With the velocity potential the radial velocity can be easily derived from the solution (2.67):

(2.68)

Substituting Equation (2.67) into (2.68) and solving for A gives

(2.69)

Hence,

(2.70)

The strength Q(t) of the source is defined by the volume flow rate. This is the surface of the sphere times normal velocity vR

(2.71)

With the harmonic source strength

(2.72)

the spherical wave solution is

(2.73)

Using equations (2.25) and (2.35) pressure and velocity are given by

(2.74)

and

(2.75)