Part (d)

The formula of the sound power of spherical waves is derived in Chap. 5 as:

w ¼ 2πA2 ¼ 2π 10:42 ¼ 1:64 ½w& ρoc 415

6.4Acoustic Waves from a Point Source

6.4.1Point Sources Formulated with Source Strength

The small spherical source in the previous section can be further reduced to a point source. When a small spherical source is reduced to a point source, the radius and velocity of the surface of the sphere are eliminated and replaced by acoustic source strength.

The difference between a small spherical source and a point source in the formulation of a radiation wave is that in small spheres, the radiation wave is formulated with radius and vibration at the surface of a small sphere. In point sources, the radiation wave is formulated with an acoustic source strength.

Unlike small spheres, point sources do not have a physical body. A point source is a hypothetical source that is an acoustic source strength that can radiate spherical waves of any frequency. Since the pressure and velocity from a small spherical sphere can be presented in terms of source strength Qs as:

where θo is a given phase and Qs is source strength and is defined as:

Z

Qs Unds ¼ 4πa2Ua

s

Qs represents the volume of fluid flowing into the acoustic medium from the source. The definition of the source strength factor removes the explicit dependence of pressure on source size as well as the source surface velocity and replaces it with the volume of fluid flow into the acoustic medium. This abstraction of the source also

removes the requirement that the source surface must be spherical. It only assumes that the radiation wave from the source is omnidirectional regardless of its shape. Hence, point sources are mathematical tools that can collectively represent very complex waves radiating from general geometrics.

6.4.2Flow Rate as Source Strength

Source strength Qs is defined as:

I

Qs Unds ¼ 4πa2Ua

s

where s is an arbitrary closed surface enclosing the source, Un is the amplitude of the velocity at the normal direction to the closed surface, and Ua is the amplitude of velocity at the surface of this pulsating sphere (r ¼ a) introduced in Sect. 6.2 and shown below for reference:

all particles passing through the closed surface per unit time. When the air density

the definition, the source strength, Qs, is also the flow rate of the source:

It can be shown that the source strength is independent of the closed surface. This means that the source strength is the same for every closed surface:

source

Two enclosed surfaces

The following example will demonstrate this statement by calculating the source strength by integrating over the surface at a distance of (i) one radius (1a) and (ii) two radii (2a):

(i) s at r ¼ 1a (at the surface of the pulsating sphere, r ¼ a):

II

(ii) s at r ¼ 2a (at a distance of two times the radius of the sphere, r ¼ 2a):

II

It thus follows that the source strength is the same at the surface of the sphere and one radius distance away from the surface of the sphere.

6.4.3Point Source in an Infinite Baffle

A point source is placed in an infinite baffle as shown in the figure below:

Point source in the infinite baffle

The source strength of the hemispherical point source is expressed as

QH ¼ 2Qs

Since the surface area is half of the complete sphere, the field pressure for a hemispherical point source QH is: