Mode 2: (l ¼ 2)

λl ¼ 2π ¼ 2L ! λ2 ¼ L kl l 2 2

Mode 3: (l ¼ 3)

λl ¼ 2π ¼ 2L ! λ3 ¼ L kl l 2 3

Note that mode shapes in a pipe are standing waves which are equivalent to two traveling waves with the same amplitude and traveling in opposite directions.

7.32D Boundary Conditions Between Four Walls

7.3.12D Standing Wave Solutions of the Wave Equation

The two-dimensional wave equation in Cartesian coordinate is:

2p ¼ 1 ∂2p c2 ∂t2

!∂2p þ ∂2p ¼ 1 ∂2p

∂x2 ∂y2 c2 ∂t2

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

psðx, y, tÞ ¼ TðtÞXðxÞ YðyÞ

Substituting the above solution into the acoustic wave equation above yields the three 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 is separated into two independent variables kx and ky for two independent differential equations in two independent directions:

1 T}ðtÞ ¼ k2 c2 TðtÞ

X}ðxÞ ¼ k2

XðxÞ x

Y}ðyÞ ¼ k2

YðyÞ y

where the new variable k can be considered a combined wavenumber that related to the wavenumbers kx and ky as:

k2 ¼ k2x þ k2y

With the new variables k, kx, and ky, the solutions of the temporal and the spatial differential equations are:

TðxÞ ¼ At cos ðωt þ θtÞ

XðxÞ ¼ Ax cos ðkxx þ θxÞ

YðxÞ ¼ Ay cos kyy þ θy

Therefore, the standing wave solution of the 2D wave equation is:

psðx, y, tÞ ¼ TðtÞXðxÞYðyÞ

¼ AtAxAy cos ðωt þ θtÞ cos ðkxx þ θxÞ cos kyy þ θy

The general 2D standing wave pressure and velocity are shown below:

psðx, y, tÞ ¼ AtAxAy cos ðωt þ θtÞ cos ðkxx þ θxÞ cos kyy þ θy

!

usðx, y, tÞ ¼ uxðx, y, tÞbex þ uyðx, y, tÞbey

where:

uxðx, y, tÞ ¼ ρ1 kx AtAxAy sin ðωt þ θtÞ sin ðkxx þ θxÞ cos kyy þ θy 0c k

uyðx, y, tÞ ¼ ρ1 ky AtAxAy sin ðωt þ θtÞ cos ðkxx þ θxÞ sin kyy þ θy 0c k

The boundary condition at the left wall (CLOSED at x = 0):

uxðx, y, tÞjx¼0 ! θx ¼ 0

The boundary condition at the right wall (CLOSED at x = Lx):

And the discretized sound pressure becomes: