Example 4.4

Use Euler’s force equation to get the flow velocity u+(x, t) from the given pressure as shown below:

Based on the two examples above, we can conclude that:

1

u ðx, tÞ ¼ ρoc p ðx, tÞ

Remark:

Note that for general complex waves, there are no linear relationships between pressure and velocity:

4.2RMS Pressure

The root-mean-square (RMS) pressure is defined as:

Z T

p2RMS T1 0 p2dt

where T is the period of the pressure wave.

Formulas of RMS pressure for traveling waves p (x, t) and standing waves ps(x, t) will be developed based on the definition of RMS pressure shown above. The following is a summary of formulas of RMS pressure that will be derived in this section:

90 4 Acoustic Intensity and Specific Acoustic Impedance

4.2.1RMS Pressure of BTW

The RMS pressure of traveling waves is defined as follows:

where A is the amplitude of the traveling waves of the four basic traveling waves:

p ðx, tÞ ¼ A ∙ cos ðωt þ kxÞ pþðx, tÞ ¼ A ∙ cos ðωt kxÞ p ðx, tÞ ¼ A ∙ sin ðωt þ kxÞ pþðx, tÞ ¼ A ∙ sin ðωt kxÞ

The following is an example of validating these relationships:

For forward traveling wave p+(x, t) ¼ cos (ωt kx), the RMS pressure is by definition:

For general traveling waves, RMS pressure is independent of the choice of location. Therefore, we can calculate the square of RMS pressure at x ¼ 0. Therefore:

Because ω ¼ 2Tπ, replace ω with 2Tπ to get:

This means that the square of RMS pressure is an integral over a period 2π. Note that the geometry relationship is used to yield:

4.2.2 RMS Pressure of BSW

The square of the RMS pressure of standing waves can be illustrated as:

The following is an example to validate one of the above relationships between pressure and RMS pressure:

The square of the RMS pressure is defined as:

Bringing the time-independent function cos(kx) outside of the integral yields:

Example 4.5 (Traveling Wave with Known Velocity)

Use the following harmonic motion as velocity u(t) to answer every question of this problem:

A large, rigid wall is vibrating with a velocity u(t) perpendicular to its planar surface. The motion of the wall creates plane waves that propagate into the air. Considering that the flow velocity of the plane wave on the surface of the wall (x ¼ 0) would be equal to the velocity of the wall:

= 0

a)Obtain the expressions for the flow velocity at any point in space (not just on the wall surface) of the plane wave radiating from the wall in terms of the wall vibration velocity.

b)Obtain the expressions for the sound pressure at any point in space (not just on the wall surface) of the plane wave radiating from the wall in terms of the wall vibration velocity.

c)What is the root-mean-square (RMS) pressure at any point in space if the wall velocity Vo ¼ 0.01 m/s.

Example 4.5 Solution

a) The general solution of the forward plane wave with an amplitude of A:

uðx, tÞ ¼ A cos ðωt kx þ θÞ

Substituting the boundary condition into general solution yields:

A ¼ Vo,

Therefore, the general expression of the particle velocity at any point to the right of the wall is:

b)The acoustic pressure can be calculated from the flow velocity using Euler’s force equation.

For a forward traveling wave, the acoustic pressure is related to the flow velocity as follows:

For traveling waves, forward or backward, the RMS pressure is independent of space.

Hence: