Power transmission coefficient as a function of the frequency ratio

Remarks

• The expanded middle section behaves like a low-pass filter. It filters out the wave

for frequencies above ωc ¼ 2Sc.

S2L

• The impedance in a low-pass filter changes two times in series.

12.4High-Pass Filters

A high-pass filter can be constructed by a pipe with a small hole. The cross-sections of the pipes are Si, S1, and S2 as shown below. The length of outlet Pipe 1 (open side

branch) is L1 ¼ 0. The radius of Pipe 1 is a and is related to the cross-section S1 of Pipe 1 as S1 ¼ πa2:

1 = 1 + 1

=

= =

Assumptions

(1) Side pipe can be modeled as a point source:

pðr, tÞ ¼ ρ4oπckr Qse jðωt krþθ0þπ2Þ

(2) The radius of Pipe 1, a, is much less than the wavelength λ divided by 2π as:

!ka 1

(3)The acoustic impedance Z1 is calculated at:

8a r ¼ 3π

Objective

Show that based on the three assumptions above, the power transmission Tw2 coefficients to Pipe 2 of this high-pass filter can be formulated as:

Tw = 1 þ 1ωc 2 2

ω

where ωc ¼ 2cL0 and L0 ¼ 38πSSi1 a.

Procedures

Step 1: Estimate acoustic impedance Z1 at the open side branch. Step 2: Calculate pressure ratio using equivalent acoustic impedance. Step 3: Calculate power transmission coefficient.

Step 1: Estimate Acoustic Impedance Z1 at the Open Side Branch

The acoustic impedance Z1 at the open side branch can be calculated from the pressure and velocity at the side open.

Assumption #1: Side pipe can be modeled as a point source.

Using the formulas of point sources, the pressure and the flow velocity are given by:

pðr, tÞ ¼ ρ4oπckr Qse jðωt krþθ0þπ2Þ

1 = 1 + 1

A pipe with a small hole as an open side branch

The acoustic impedance can be calculated by the above pressure and velocity:

Canceling out all common terms yields:

Z1 = ρoc cos ðϕÞe jðϕÞ

S1

where:

Assumption #2: The radius a of Pipe 1 is much less than the wavelength λ

Based on kr 1, the acoustic impedance becomes:

Assumption #3: The acoustic impedance Z1 is calculated at:

8a r ¼ 3π

Assume that the acoustic impedance is calculated at a distance r ¼ 38πa from the point source, where a is the radius of opening. Then, the above acoustic impedance

Note that the acoustic impedance is a complex number. More specifically speaking, this acoustic impedance is an imaginary number. This imaginary number has a phase separation of π2 between the pressure and velocity:

The π2 phase angle indicates that there is a 90-degree phase difference between the pressure and velocity. The phase difference is similar to the phase delay of the damping of a structure. Also, the magnitude of the complex impedance is the ratio of magnitudes of pressure to velocity.

For the purpose of simplicity, the acoustic impedance in Eq. (6) can be condensed into:

In addition, at the intersection of the main pipe, the impedance becomes:

Z2 ¼ Zi ¼ ρoc

Si

Therefore:

Step 2: Calculate Pressure Ratio Using Equivalent Acoustic Impedance

A side branch on the pipe with a small hole to the outside medium behaves like a high-pass filter when the length of the side branch is very short.

The high-pass filter can be considered as a special case of the one-to-two pipe with the length of the side branch being zero, as analyzed in the previous section:

!Zi = jkL0 þ0 1 ¼ 1 j 1 0 Zo jkL kL

Therefore, all formulas derived for a one-to-two pipe are valid for the high-pass filter. The pressure ratios for the one-to-two pipe are shown below:

Since we know all impedances of the high-pass filters, all pressures can be calculated using the above equations as the following:

For the pipe with a side branch pipe as a small hole to the outside medium, Zi and Z2 are real numbers, but Z1 and Zo are complex numbers. In addition, Pi is real but P2 is complex.

Hence, the power transmission through the main pipe is:

The open side behaves like a high-pass filter; it filters out the wave for frequency ω ¼ kc below ωc ¼ 2cL0:

Power Transmission Coefficient of High-Pass Filter

Power transmission coefficient as a function of the frequency ratio

Remarks

•Transmission pressures P1 and P2 are outward (no return) forward waves.

•Because pressures Pi,Pr, and P2 are traveling plane waves, their acoustic impedances Zi(=ρ0c/Si), Zr(= 2 ρ0c/Si), and Z2(=ρ0c/S2) are real numbers.

•The transmission pressure P1 is a spherical wave, and the acoustic impedance Z1 is a complex number.

•Si: cross-section area of the inlet pipe.

•S1: cross-section area of outlet Pipe 1 (side branch).

•S2: cross-section area of outlet Pipe 2 (main pipe).

•L1: length of outlet Pipe 1 (outlet side branch).

•pi: pressure of the incident wave in the inlet pipe.

•pr: pressure of the reflected wave in the inlet pipe.

•p1: pressure of the transmitted wave in outlet Pipe 1 (outlet side branch).

•p2: pressure of the transmitted wave in outlet Pipe 2 (outlet main branch).

•Pi ¼ PiR: complex pressure of the incident wave at the RHS of the inlet pipe.

•P1: complex pressure of the transmitted wave in the side branch pipe at the intersection.

•P2 ¼ P2L: complex pressure of the transmitted wave at the LHS of Pipe 2.

•Zi: acoustic impedance of the incident wave at the RHS of the inlet pipe.

•Z1: acoustic impedance of the transmitted wave inside the branch pipe at the intersection.

•Z2: acoustic impedance of the transmitted wave at the LHS of outlet Pipe 2.

Assume the Following Properties:

•Cross-section areas of the pipes are Si, S1, and S2 where S2 ¼ Si.

•Acoustic impedances, Z¼z/S, of the pipes are Zi, Zr, and Z2

where Zr ¼ Zi; Z2 ¼ Zi.

• Incident pressure at the intersection of the pipe is Pi.

• The length of the side branch pipe is L1 ¼ 0.

The Validation of Power Reflection and Transmission Coefficients

The formulas of power for the one-to-two pipe in the previous section can be used for the high-pass filter, as shown below:

For the pipe with a side branch pipe as a small hole to the outside medium, Zi and Z2 are real numbers, but Z1 is a complex number. In addition, Pi is real but P1 and P2 are complex.

Therefore: