c

f lmn ¼ 2π klmn

where klmn (combined wavenumber) is related to kxl, kym, kzn as:

k2lmn ¼ k2xl þ k2ym þ k2zn

The component wavenumbers kxl, kym, and kzn are the discretized wavenumbers of the eigenmode (l, m, n) and are related to circular frequencies as:

Four combinations of boundary conditions of a pipe will be discussed near the end of this chapter. The natural frequencies of the four cases of boundary conditions are summarized as:

7.11D Standing Waves Between Two Walls

Between two parallel walls, standing waves exist as mode shapes of air movement between two walls. The standing wave solutions of the acoustic wave equation can be obtained by solving the boundary conditions imposed by the rigid walls.

In a previous chapter, the standing wave solutions were constructed by the addition of two traveling waves with the same amplitude (A+ ¼ A ¼ A) that moved in opposite directions as:

psðx, tÞ ¼ pþðx, tÞ þ p ðx, tÞ

¼A cos ðωt kx þ θþÞ þ A cos ðωt þ kx þ θ Þ

¼2A cos ðωt þ θtÞ cos ðkx þ θxÞ

, = − +

=

In this chapter, standing waves ps(x, t) are obtained directly by solving the acoustic wave equation.

The one-dimensional wave equation in Cartesian coordinate is:

2p ¼ 1 ∂2p c2 ∂t2

!∂2p ¼ 1 ∂2p

∂x2 c2 ∂t2

The differential equation is separable if a solution can be cast in the following form that satisfies the acoustic wave equation:

pðx, tÞ ¼ TðtÞXðxÞ

Substituting the above solution into the acoustic wave equation above yields the two separate ordinary differential equations (ODEs):

where k is a new variable that relates the spatial and temporal parts of the differential equations. The new variable k gives us the two separate ordinary differential equations (ODEs) as:

1 T}ðtÞ ¼ k2 c2 TðtÞ

X}ðxÞ ¼ k2

XðxÞ

With the new variable k, the solutions of the temporal and the spatial differential equations are:

XðxÞ ¼ Ax cos ðkx þ θxÞ

Therefore, the standing wave solution of the 1D wave equation is:

psðx, tÞ ¼ TðtÞXðxÞ

¼ AtAx cos ðωt þ θtÞ cos ðkxx þ θxÞ

where the unknown constants At, Ax, θt, and θx are calculated using the boundary conditions from the two rigid walls. Since T(t) and X(x) are both real numbers, the standing wave functions are real.

Using Euler’s force equation, the velocity of the one-dimensional standing wave was derived in a previous chapter and is:

7.2 Natural Frequencies and Mode Shapes in a Pipe

If the boundary condition at the end of a pipe is either open or closed, there will be four combinations of boundary conditions of a single pipe. The four combinations of the boundary conditions of a single pipe are summarized in the table below:

Case 1 and Case 2 in the table above are demonstrated in the examples listed below. You can practice Case 3 and Case 4 in the homework exercises listed below:

Case 1: Example 7.1

Case 2: Example 7.2

Case 3: Homework Exercise 7.1

Case 4: Homework Exercise 7.2

Example 7.1 (CLOSED-CLOSED PIPE)

A pipe with CLOSED-CLOSED boundary conditions has a finite length L as shown below. Set the left end of the pipe as x¼0:

(a)Derive the following formulas for wavenumber and natural frequency of eigenmode l:

kl ¼ Lπ l,

c f l ¼ 2L l

(b) Plot the mode shapes of the first three natural frequencies.

Example 7.1 Solution Part (a)

Natural Frequencies of a CLOSED-CLOSED PIPE

The closed end represents a fixed wall that requires the flow velocity to be zero at all times.

The above boundary condition requires that the velocity is zero at x ¼ 0 at any time. Therefore:

sin ðkx þ θxÞjx¼0 ¼ 0 ! sin ð0 þ θxÞ ¼ 0 ! θx ¼ 0

Note that the sine function above is simplified from sin(kx + θx) to sin(θx) by letting x ¼ 0 at the left wall.

The boundary condition at the right wall (CLOSED):

The above boundary condition requires that the velocity is zero at x ¼ L at any time. Therefore:

Substituting kl into the velocity yields the final one-dimensional standing waves between two rigid walls:

1

usðx, tÞ ¼ ρ0c AtAx sin ðωt þ θtÞ sin ðklxÞ psðx, tÞ ¼ AtAx cos ðωt þ θtÞ cos ðklxÞ

where the wavenumber of eigenmode l is:

π

kl ¼ L l

Therefore, the natural frequency of eigenmode l is:

Part (b)

Mode Shapes of a CLOSED-CLOSED PIPE

The discretized velocity, pressure, and wavelength of the eigenmode l are relisted here for plotting the mode shapes:

1

usðx, tÞ ¼ ρ0c AtAx sin ðωt þ θtÞ sin ðklxÞ

psðx, tÞ ¼ AtAx cos ðωt þ θtÞ cos ðklxÞ

π

kl ¼ L l

The wavelength of each eigenmode is helpful for plotting the mode shapes, and the wavelengths of the first three eigenmodes are shown below:

2 λ1 ¼ 2L; λ2 ¼ L; λ3 ¼ 3 L;

Based on the wavelengths above and zero velocities on the edge, we can plot the mode shape of the mode (l, m) ¼ (2, 3) as follows:

The first two mode shapes of velocity and pressure of a CLOSED-CLOSED pipe are shown below:

=

Note that the zero velocity corresponds to the maximum pressure (zero slope) based on Euler’s force equation.

Also, the mode shapes of flow velocity are typically not plotted in 2D and 3D because flow velocities are vectors. It will require more than one figure to show a mode shape of flow velocities: one figure for each direction. Therefore, mode shapes of sound pressures are commonly used.

The first three mode shapes of the pressure of a CLOSED-CLOSED pipe are shown below:

Mode 1: (l ¼ 1)

= =

= =

Example 7.2 (OPEN-OPEN PIPE)

A pipe with OPEN-OPEN boundary conditions has a finite length L as shown below. Set the left end of the pipe as x¼0. Determine the first three natural frequencies and plot their corresponding mode shapes:

Example 7.2 Solutions

The boundary condition of an open end requires that the sound pressure be zero at all times. Note that the boundary condition of a closed end requires that the flow velocity be zero at all times.

The boundary condition at the left wall (OPEN):

psðx, tÞjx¼0 ¼ 0

! psðx ¼ 0, tÞ ¼ AtAx cos ðωt þ θtÞ cos ðkx þ θxÞjx¼0 ¼ 0

The above boundary condition requires that the pressure is zero at x ¼ 0 at any

The above boundary condition requires that the velocity is zero at x ¼ L at any time. Therefore:

π

cos kx þ 2 x¼L ¼ 0

ππ

!klL þ 2 ¼ 2 ð2l þ 1Þ

Substituting kl into the velocity yields the final one-dimensional standing waves between two rigid walls:

where the wavenumber of the eigenmode l is:

π

kl ¼ L l, l ¼ 1, 2, . . .

Therefore, the natural frequency of the eigenmode l is:

Mode Shapes of an OPEN-OPEN PIPE:

The first three mode shapes of the pressure of an OPEN-OPEN pipe are shown below:

Mode 1: (l ¼ 1)

λl ¼ 2π ¼ 2L ! λ1 ¼ L kl l 2 1