6.3 Acoustic Waves from a Small Pulsating Sphere

The error of this simplification for the small spherical source can be calculated by using the amplitude and phase. The errors of amplitude and phase are:

Define a total error of the approximation as:

q

E ¼ E2amplitude þ E2phase

Depending on the application, the threshold of the total error can be determined. For example, if ka ¼ 0.1, E ¼ 0.005; if ka ¼ 0.01, E ¼ 0.00005.

6.3.1Near-Field Solutions of a Small Spherical Source (kr 1)

It is also of interest to study the behavior of sound pressure and velocity at short distances (r 1) from the source or at very low frequencies (ω 1) of a wave.

When the solution (measurement point) is very close to the source, kr and ϕ are approximately given by:

kr ¼ ωc r 0

ϕ ¼ tan 1 kr1 π2

kr

cos ðϕÞ ¼ q kr

1 þ ðkrÞ2

Substituting the above approximated values back to the pressure and velocity equations induced by the pulsating small sphere yields:

Hence, the pressure is nearly 90 out of phase with the velocity near the source.

6.3.2Far-Field Solutions of a Small Spherical Source (kr 1)

It is also of interest to study the behavior of sound pressure and velocity at large distances (r 1) from the source or at high frequencies (ω 1) of a wave. When the solution (measurement point) is far away from the source, kr and ϕ are approximated as:

kr ¼ ωc r 1

ϕ ¼ tan 1 1 0 kr

kr

cos ðϕÞ ¼ q 1

1 þ ðkrÞ2

Substituting the above approximated values back to the pressure and velocity relations induced by the pulsating small sphere yields:

A comparison of the above pressure and velocity shows that the pressure is in phase with the velocity at large distances from the source or at high frequencies of waves.

Example 6.1 (A Small Spherical Source)

An acoustic pressure p(r, t) is created by a surface vibration of a spherical source with a radius a ¼ 1001 ½m&. Given the surface velocity of the sphere as:

where the surface velocity is Ua ¼ 20 [m/s] and frequency of radiation is f ¼ 680 [Hz].

Assuming that this spherical source can be treated as a small spherical source, calculate (a) the flow velocity, (b) acoustic pressure, (c) intensity, and (d) power radiating from the small spherical source.

Use 415 [rayls] for characteristic impedance (ρ0c) and 340 [m/s] for the speed of sound. Show units in the Meter-Kilogram-Second (MKS) system.

Example 6.1 Solution

The angular frequency can be calculated as:

ω ¼ 2πf ¼ 2π 680 h1si ¼ 1360π h1si

The wavenumber can be calculated as:

Part (a)

Compare the given the surface velocity of the sphere:

to the boundary condition (BC) of a small pulsating spherical source in the summary table below:

Comparing the given BC to the formulas in the summary table above gives:

a ¼ 1001 ½m&

Ua ¼ 20½m=s&

π

θo ¼ 2

The formula of velocity from a small source as shown is shown in the summary table as:

h i

! uðr, tÞ r cos ðϕÞ cos ð1360πt 4πr ϕÞ s

where:

ϕ ¼ tan 1 1 kr

Part (b)

The formula for acoustic pressure from a small source is shown in the summary table as:

Part (c)

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