Fundamentals of General Linear Acoustics by Jacobsen Finn Juhl Peter Moller & Peter Moller Juhl

Author:Jacobsen, Finn, Juhl, Peter Moller & Peter Moller Juhl
Language: eng
Format: epub
Publisher: Wiley
Published: 2013-05-08T04:00:00+00:00

Figure 8.4 Number of natural frequencies in a 10-Hz band compared with the smoothed modal density

8.2.3 The Green's Function in an Enclosure

An important property of the modes in a room can be deduced as follows. Since every mode satisfies the Helmholtz equation, we can write

8.17a, 8.17b

and therefore,

8.18

However, this equation can be rewritten in the form

8.19

Integrating over the volume of the room and applying Gauss's theorem2 on the first term gives

8.20

where is the surface of the room. At the rigid walls the normal component of the gradient of any eigenfunction is zero, and therefore the left-hand term is zero. It follows that the eigenfunctions are orthogonal, that is, that

8.21

unless ; hence the term ‘normal mode’. It is customary to normalise the eigenfunctions so that

8.22

which implies that

8.23

Note that these considerations have not been limited to the special case of a rectangular room.

It is interesting to study how a source placed at a certain position in the room will excite the various modes. The Green's function for the sound field in a room with rigid walls can be derived as follows. We are looking for solutions to the inhomogeneous Helmholtz equation

8.24

with the boundary condition

8.25

on the walls (cf. Equation (7.162)). Any sound field in the room can be expressed in terms of the modes of the room, that is, functions that satisfy the equation

8.26

and the boundary condition mentioned above. Therefore,

8.27

The source term (the right-hand side of Equation (8.24)) can also be expanded into a sum of modes,

8.28

Multiplying with and integrating over the volume of the room gives, if we make use of the fact that the modes are orthogonal (cf. Equation (8.22)),

8.29

which shows that

8.30

and thus

8.31

It now follows that

8.32

from which we deduce that

8.33

Finally we can write the Green's function as

8.34

Note the symmetry with respect to source and receiver position, in agreement with the reciprocity principle. A point source placed on a nodal surface of a given mode does not excite the mode. Note also that each mode, not surprisingly, contributes most to the sound field when driven near its natural frequency (). It can be seen that the response is unlimited if the frequency of the excitation coincides with one of the natural frequencies.

In practice there are, of course, losses in any enclosure, even in a room with walls of solid concrete.3 As a result the eigenvalues of the problem become complex, with small imaginary parts equal to , where is the time constant of the 'th mode (see Section 8.4.1). This leads to the expression

8.35

where the second approximation has the advantage over the first one that it corresponds to a real-valued, causal time function.4 The first expression corresponds to complex eigenvalues caused by losses at the boundaries; the other versions correspond to a medium with losses.

It can be seen from Equation (8.35) that the response of any mode is limited also when it is driven at its natural frequency. It is also easy to show that the 3-dB bandwidth of the 'th mode is in radians per second and in hertz. We will study the effect of losses in Section 8.

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