124 5 Solutions of Spherical Wave Equation

5.4Flow Velocity

Similar to the plane wave, the flow velocity can be obtained from sound pressure using Euler’s force equation.

The one-dimensional Euler’s force equation in the spherical coordinates can be obtained by replacing the vector gradient operator with the spherical coordinate gradient operator:

5.4.1Flow Velocity in Real Format

Sound pressure p(r, t) in the spherical coordinate system was derived in the previous section as:

rpðr, tÞ ¼ A cos ðωt kr þ θÞ

Flow velocity u(r, t) in spherical coordinates can be derived from Euler’s force equation and expressed in real explicit phase [REP] format as:

The following will derive the above equation in real explicit phase [REP] format:

Flow velocity expressed in real explicit phase [REP] format can be derived independently using Euler’s force equation:

∂∂r pðr, tÞber ¼ ρo ∂∂t uðr, tÞ

For a given pressure p(x, t), flow velocity u(x, t) can be calculated as:

Flow velocity in real number format will be derived in this section:

Flow velocity formulated in real explicit phase [REP] format is derived from pressure p(r, t) as:

rpðr, tÞ ¼ A cos ðωt kr þ θÞ

Substituting the above pressure p(r, t) into the formula for flow velocity yields:

Combining the cosine and sine functions into one cosine function yields:

where:

ϕ ¼ tan 1 1 kr

Eq. (5.4) is the flow velocity of spherically symmetric acoustic wave, for a given pressure function, as described in Eq.(5.3).

5.4.2Flow Velocity in Complex Format

Sound pressure p(r, t) in the spherical coordinate system was derived in the previous ion as:

rpðr, tÞ ¼ A cos ðωt kr þ θÞ

It can be formulated in a complex explicit phase [CEP] as shown below using Euler’s formula:

The corresponding flow velocity in a complex exponential format as shown below can be derived using Euler’s force equation:

where:

ϕ ¼ tan 1 1 kr

The derivation of the flow velocity formulated in complex explicit phase [CEP] format is not derived here. The derivation is part of the homework in this chapter.

Example 5.2

What is the phase difference between the pressure and the flow velocity of a

spherical wave at 3 [m] from the origin at 1000 [Hz]?

Use 340 ms for the speed of sound (c) in air.

Example 5.2 Solution The general form of pressure of a spherical wave is:

rpðr, tÞ ¼ A cos ðωt kr þ θÞ

The formula of flow velocity is derived from pressure using Euler’s force equation:

A

ruðr, tÞ ¼ ρoc cos ϕ cos ðωt kr þ θ ϕÞ

where:

ϕ ¼ tan 1 1 kr

Hence, by inspecting the exponents of pressure and flow velocity, the phase difference between pressure and flow velocity of a spherical wave is equal to ϕ, with flow velocity lagging behind the pressure.

For the given conditions:

f ¼ 1000 ½Hz&