Non-rigid walls

Sound waves in a room

3.3 Non-rigid walls

Sound waves in a room 79 (or even infinitely many) rectangular rooms. For each of them eqn (3.20) yields the numberN_iof eigenfrequencies. Since this equation is linear inV, the total number of eigenfrequencies is just the sum of allN_i.

We bring this section to a close by applying eqns (3.20) and (3.21) to two simple examples. The rectangular room for which the eigenfrequencies listed in Table 3.1 have been calculated has the volume 59.7 m³. For an upper frequency limit of 116 Hz, eqn (3.20) indicates—with c=343 m/s—10 eigenfrequencies as compared with the 20 we have listed in the table. Using the more accurate formula (3.20a) we obtain 20 eigenfrequencies. That means that we must not neglect the corrections due to tangential and axial modes when dealing with such small rooms at low frequencies.

Now we consider a rectangular room with dimensions 50 m ×24 m× 14 m whose volume is 16 800 m³. (This might be a large concert hall, for instance.) In the frequency range from 0 to 10 000 Hz there are, according to eqn (3.20), about 1.7×10⁹ eigenfrequencies. At 1000 Hz the number of eigenfrequencies per hertz is about 5200; thus, the average distance of two eigenfrequencies on the frequency axis is less than 0.0002 Hz. These figures underline the impossibility of evaluating the sound field in a room by calculating normal modes.

which is equivalent to eqn (3.13). By inserting p_x into the boundary conditions (3.22) we obtain two linear and homogeneous equations for the constantsC₁andD₁:

C₁(k_xζ_x+k)−D₁(k_xζ_x−k)=0

C₁(k_xζ_x−k) exp (−ik_xL_x)−D₁(k_xζ_x+k) exp (ik_xL_x)=0 (3.24) which have a non-vanishing solution only if the determinant of their coefficients is zero. This leads to the following condition:

exp (ik_xL_x)= ±k_xζ_x−k

k_xζ_x+k (3.25)

Solving this equation for k_xζ_x (separately for both signs) and using the relation (1.15) yields two transcendent equations

tanu=ikL_x

2uζ_x and tanu=i2uζ_x

kL_x (3.25a)

with u=1

2k_xL_x

In general, eqn (3.25a) must be solved numerically. Once the allowed values ofk_xhave been determined, the ratio of the two constantsC₁ andD₁ can be evaluated from eqns (3.24); for instance, from the first of them:

C₁

D₁ =k_xζ_x−k

k_xζ_x+k= ±exp (ik_xL_x) (3.26)

the latter equality resulting from eqn (3.25). Thus thex-dependent factor of the eigenfunction reads, apart from a constant factor:

p_x(x)= cos

k_x

x−L_x/2

(even solution) sin

k_x

x−L_x/2

(odd solution) (3.27)

Evidently, these functions are symmetric or antisymmetric, depending on the sign of the exponential in eqn (3.26). Of course, this is a consequence of the symmetry of wall properties. As mentioned before, the complete eigenfunction is made up of three such factors.

From now on we restrict the discussion to enclosures with a nearly rigid boundary, i.e. to the limiting case |ζ| 1. Then we expect that the

Sound waves in a room 81 eigenvalues and eigenfunctions are not very different from those of the rigid-walled room.

An approximate solution of eqn (3.25) can be found by employing the series expansion of the exponential function and to truncate it after the second term:

exp (±z)=1±z+z²

2! ±...≈1±z forz1

Sincekk_xζ_xthis relation can be applied to right-hand side of eqn (3.25), with the result:

exp (ik_xL_x)≈ ±exp

− 2k k_xζ_x

One might be tempted to solve this equation by equating the exponents.

However, before doing so we should remember that the exponential function with imaginary argument is a periodic function with the period 2π,exp (iz)= exp (iz+i2π). Thus, to obtain the complete solution of the last equation its right-hand side must be multiplied by the factor exp (iπn_x), which also includes both signs. Heren_xis an arbitrary integer. This leads us to the result:

k_xL_x≈n_xπ+i 2k k_xζ_x

According to our assumption the second term on the right is much smaller than the first one; hence we can replacek_xin the denominator withn_xπ/L_x:

k_x≈n_xπ

L_x +i 2k

n_xπζ_x (3.28)

Comparing this result with eqn (3.14a) confirms our expectation that the allowed values ofk_x are not much different from those of the hard-walled room. With increasing ‘order’n_x of the mode the difference becomes even smaller.

Suppose that the wall is reactive, i.e. that it is free of absorption. Then its specific impedance is purely imaginary and the correction term is real.

If Im ζ is positive, which indicates that the motion of the wall is mass- controlled, then the allowed value is higher than in the rigid-walled room.

Conversely, a compliance wall, i.e. a wall with the impedance of a spring, will lower the allowed valuek_x.

Whenever the specific impedance has a non-vanishing real part indicating wall lossesk_xis complex; the imaginary part ofk_xis related to the damping constant (see also eqn (3.9)). If the allowed values ofk_xare denoted byk_xnx,

the eigenvalues of the original differential equation are given as earlier by (see eqn (3.12)):

k_nxnynz=

k²_xnx+k²_yny+k²_znz1/2

(3.29) In Fig. 3.5 the absolute value of the x-dependent factor of a certain eigenfunction is represented for three cases: for rigid walls (ζ_x= ∞), for mass- loaded walls with no energy loss (ζ_x=i), and for walls with real impedance.

In the second case, the standing wave is simply shifted together, but its shape remains unaltered. On the contrary, in the third case of lossy walls, there are no longer exact nodes and the pressure amplitude is different from zero at all points. This can easily be understood by keeping in mind that the walls dissipate energy, which must be supplied by waves travelling towards the

Figure 3.5 One-dimensional normal mode, pressure distribution forn_x=4: (a)ζ= ∞; (b)ζ=i; (c)ζ=2.

Sound waves in a room 83 walls; thus, a pure standing wave is not possible. This situation is compara- ble to a standing wave in front of a single plane with a reflection factor less than unity as shown in Fig. 2.3.