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Wolfgang Ahnert on computer-aided sound system design and room-acoustical fundamentals for auditoriums and concert halls, the Handbook for Sound Engineers is a must for serious audio and acoustic engineers. The fifth edition has been updated to reflect changes in the industry, including added emphasis on increasingly prevalent technologies such as software-based recording systems, digital recording using MP3, WAV files, and mobile devices.

This edition has been honed to bring you the most up-to-date information in the many aspects of audio engineering. Glen is owner of Innovative Communications, a company that specializes in room acoustics and sound system design. For this purpose one just has to separate the real parts from the imaginary parts in eqns 1. This holds, for instance, for any kind of random noise. In this case the integrals would not converge. Nowadays it is most conveniently achieved by using digital computers. To give at least an idea of it we suppose that N is an even number.

We note that the sum in eqn 1. But while the calculation according to eqn 1. Thus, the saving of computing time provided by FFT is considerable, particularly if N is large. For a more detailed description of the fast Fourier transform, the reader is referred to the extensive literature on this subject e. The power spectrum, which is an even function of the frequency, does not contain all the information on the original signal s t , because it is based on the absolute value of the spectral function only, whereas all phase information has been eliminated.

Inserted into eqn 1. This is the case when these functions 20 Room Acoustics are random or nearly random. In a certain sense a sine or cosine signal can be considered as an elementary signal; it is unlimited in time and steady in all its derivatives, and its spectrum consists of a single line. Accordingly, any signal can be considered as a close succession of very short pulses, as indicated in Fig.

Likewise, any two points in an enclosure may be considered as the input and output terminal of an acoustic transmission system. Linearity means that multiplying the input signal with a factor results in an output signal which is augmented by the same factor. Since the response cannot precede the excitation, the impulse response of any causal system must vanish for t Figure 1. The transfer function G f has also a direct meaning: if a harmonic signal with frequency f is applied to a transmission system, its amplitude will be changed by the factor G f and its phase will be shifted by the phase angle of G f.

For this reason it would be impractical to characterise the strength of a sound signal by its sound pressure or its intensity. More information may be found in Ref. One of the most obvious facts of human hearing is that the ear is not equally sensitive to sounds of different frequencies. Generally, the loudness at which a sound is perceived depends, of course, on its objective strength, i.

Furthermore, it depends in a complicated manner on the spectral composition of the sound signal, on its duration and 24 Room Acoustics Figure 1. The dashed curve corresponds to the average hearing threshold. Figure 1. The numbers next to the curves indicate the loudness level. The lowest, dashed curve which corresponds to a loudness level of 3 phons, marks the threshold of hearing. However, at Hz its loudness level would be only 24 phons whereas at 50 Hz it would be almost inaudible.

Using these curves, the loudness level of any pure tone can be determined from its frequency and its sound pressure level. In order to simplify this somewhat tedious procedure, electrical instruments have been constructed which measure the sound pressure level. Several weighting functions are in use and have been standardised internationally; the most common of them is the A-weighting curve.

Sound waves, sources and hearing 25 When such an instrument is applied to a sound signal with more complex spectral structure, the result may deviate considerably from the true loudness level. The reason for such errors is the fact that, in our hearing, weak spectral components are partially or completely masked by stronger ones and that this effect is not modelled in the above-mentioned sound level meters. Apart from masking in the frequency domain, temporal masking may occur in nonstationary signals.

In particular, a strong time-variable signal may mask a subsequent weaker signal. This effect is very important in listening in closed spaces as will be described in more detail in Chapter 7. A more fundamental shortcoming of the above-mentioned measurement of loudness level is its unsatisfactory relation to our subjective perception.

In fact, doubling the subjective sensation of loudness does not correspond to twice the loudness level as should be expected. Instead it corresponds only to an increase of about 10 phons. Nowadays instruments as well as computer programs are available which are able to measure or to calculate the loudness of almost any type of sound signal, taking into account the above-mentioned masking effect. Another important property of our hearing is its ability to detect the direction from which a sound wave is arriving, and thus to localise the direction of sound sources.

For sound incidence from a lateral direction it is easy to understand how this effect is brought about: an originally plane or spherical wave is distorted by the human head, by the pinnae and—to a minor extent—by the shoulders and the trunk. This distortion depends on sound frequency and the direction of incidence. As a consequence, the sound signals at both ears show characteristic differences in their amplitude and phase spectrum or, to put it more simply, at lateral sound incidence one ear is within the shadow of the head but the other is not.

The interaural amplitude and phase differences caused by these effects enable our hearing to reconstruct the direction of sound incidence. Such transfer functions have been measured by many researchers. Sound waves, sources and hearing 27 However, if the sound source is situated within the vertical symmetry plane, this explanation fails since then the source produces equal sound signals at both ear canals. But even then the ear transfer functions show characteristic differences for various elevation angles of the source, and it is commonly believed that the way in which they modify a sound signal enables us to distinguish whether a sound source is behind, above or in front of our head.

We do not consider loudspeakers here because they reproduce sound signals originating from other sources. It is a common feature of all these sources that the sounds they produce have a more or less complicated spectral structure—apart from some rare exceptions. In fact, it is the spectral content of speech signals phonems which gives them their characteristics. Similarly, the timbre of musical sounds is determined by their spectra.

The signals emitted by most musical instruments, in particular by string and wind instruments, including the organ, are nearly periodic. It is the fundamental which determines what we perceive as the pitch of a tone. This means that our ear receives many harmonic components of quite different frequencies even if we listen to a single tone. Likewise, the spectra of many speech sounds, in particular vowels and voiced consonants, have a line structure. As an example, Fig. For normal speech the fundamental frequency lies between 50 and Hz and is identical to the frequency at which the vocal chords vibrate.

The total frequency range of conversational speech may be seen from Fig. Sound waves, sources and hearing 29 Figure 1. The transmission of the fundamental vibration, on the other hand, is less important since our hearing is able to reconstruct it if the sound signal is rich in higher harmonics. Among musical instruments, large pipe organs have the widest frequency range, reaching from 16 Hz to about 9 kHz.

There are some instruments, especially percussion instruments, which produce sounds with even higher frequencies. The piano follows, having a frequency range which is smaller by about three octaves, i. The frequencies of the remaining instruments lie somewhere within this range. This is true, however, only for the fundamental frequencies. Since almost all instruments produce higher harmonics, the actual range of frequencies occurring in music extends still further, up to about 15 kHz.

In music, unlike speech, all frequencies are of 30 Room Acoustics almost equal importance, so it is not permissible deliberately to suppress or to neglect certain frequency ranges. On the other hand, the entire frequency range is not the responsibility of the acoustical engineer. At frequencies lower than 50 Hz geometrical considerations are almost useless because of the large wavelengths of the sounds; furthermore, at these frequencies it is almost impossible to assess correctly the sound absorption by vibrating panels or walls and hence to control the reverberation.

This means that, in this frequency range too, room acoustical design possibilities are very limited. On the whole, it can be stated that the frequency range relevant to room acoustics reaches from 50 to 10 Hz, the most important part being between and Hz. The acoustical power output of the sound sources as considered here is relatively low by everyday standards. Table 1. The human voice generates a sound power ranging from 0. A full symphony orchestra can easily generate a sound power of 10 W in fortissimo passages.

It may be added that the dynamic range of most musical instruments is about 30 dB woodwinds to 50 dB string instruments. A large orchestra can cover a dynamic range of dB. An important property of the human voice and musical instruments is their directionality, i. The lower the sound frequency, the less pronounced is the reduction of sound intensity by the head, because with decreasing frequencies the sound waves are increasingly diffracted around the head.

Unfortunately general statements are almost impossible, since the directional distribution of the radiated sound changes very rapidly, not only from one frequency to the other; it can be quite different for instruments of the same sort but different manufacture. This is true especially for string instruments, the bodies of which exhibit complicated vibration patterns, particularly at higher frequencies. The radiation from a violin takes place in a fairly uniform way only at frequencies lower than about Hz; at higher frequencies, however, matters become quite involved.

For the room acoustician, however, it is important to know that strong components, particularly from the strings but likewise from the piano, the woodwinds and, of course, from the tuba, are radiated upwards. For further details we refer to the exhaustive account of J. Autocorrelation measurements on speech and music have been performed by several authors. Two of his results are depicted in Fig. These values are indicated in Table 1. They range from about 10 to more than ms. The variety of possible noise sources is too large to discuss in any detail. A typical noise source in halls is the air conditioning system; some of the noise produced by the machinery propagates in the air ducts and is radiated into the hall through the air outlets.

The arrow points in the viewing direction: a in the horizontal plane; b in the vertical plane. Sound waves, sources and hearing 33 Figure 1. The Fourier Transform and its Applications. Singapore: McGrawHill, Psychoacoustics—Facts and Models. Berlin: Springer-Verlag, Spatial Hearing. Speech communication.

In: Crocker MJ ed. New York: John Wiley, Speech Analysis Synthesis and Perception. Acoustics and the Performance of Music. Acoust Soc Am ; Chapter 2 Reflection and scattering Up to now we have dealt with sound propagation in a medium which was unbounded in every direction. The sound absorption by a wall will be dealt with mainly from a formal point of view, whereas the discussion of the physical causes of sound absorption and of the functional principles of various absorbent arrangements will be postponed to Chapter 6. Throughout this chapter we shall assume that the incident, undisturbed wave is a plane wave.

In reality, however, all waves originate from a sound source and are therefore spherical waves or superpositions of spherical waves. More on this matter 36 Room Acoustics may be found in the literature see, for instance, Ref. Its absolute value as well as its phase angle depend on the frequency and on the direction of the incident wave. Soft walls, however, are very rarely encountered in room acoustics and only in limited frequency ranges.

Another quantity which is even more closely related to the physical behaviour of the wall and to its construction is based on the particle velocity normal to the wall which is generated by a given sound pressure at the surface. As explained in Section 1. This holds also for the wall impedance. A simple example of such a curve is shown in Fig. Furthermore, we must reverse the sign of k because of the reversed direction of travel. For a completely absorbent wall the impedance equals the characteristic impedance of the medium. Inserting eqn 2. The pressure amplitude in this wave is found by adding eqns 2.

According to eqns 2. So, by measuring the pressure amplitude as a function of x, we can evaluate the wavelength. Figure 2. A sound wave arriving from the left will excite two plane waves in the layer travelling in the opposite x-direction. It should be noted that eqn 2. The new situation is depicted in Fig.

Suppose we replace in eqn 2. According to the well-known formulae for coordinate Figure 2. For the calculation of the wall impedance we require the velocity component normal to the wall, i. It is obtained from eqn 1. This applies if the normal component of the particle velocity at any wall element depends only on the sound pressure at that element and not on the pressure at neighbouring elements. In practice, surfaces with local reaction are rather the exception than the rule. They are encountered whenever the wall itself or the space behind it is unable to propagate waves or vibrations in a direction parallel to its surface.

Obviously this is not true for a panel whose neighbouring elements are coupled together by bending stiffness. Moreover, this does not apply to a porous layer with an air space between it and a rigid rear wall. In the latter case, however, local reaction of the various surface elements of the arrangement can be brought about by rigid partitions which obstruct the air space in any lateral direction and prevent sound propagation parallel to the surface.

Using eqn 2. Any pressure difference between the two sides of the layer forces an air stream through the pores with an air velocity vs. Then it follows from eqn 2. If the thickness d of the air space is much smaller than the wavelength, i. In Fig. Minimum absorption occurs for all such frequencies at which the distance d between the porous layer and the rigid rear wall is an integral multiple of half the wavelength. This can be easily understood since, at these distances or frequencies, the standing wave behind the porous layer has a zero of particle velocity in the plane of the layer, but energy losses can take place only if the air is moving in the pores of the layer.

Accordingly, rs in eqn 2. This can be achieved by hanging or stretching the fabric in pleats as is usually done with draperies. In the next example we consider an arrangement similar to that shown in Fig. For the sake of simplicity we assume that the porous layer cannot vibrate as a whole, i. Again we suppose that the direction of the incident sound wave is perpendicular to both layers. The motion of the non-porous layer is controlled by its mass; according to eqn 2. According to eqn 2.

Using eqns 2. Practical resonance absorbers as applied in room acoustical design will be discussed in Section 6.

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By inserting this into eqns 2. If the phases of the waves incident on a wall are randomly distributed one can neglect all phase relations and the interference effects caused by them. Then the components are called incoherent.

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In the following it is convenient to use a spherical polar coordinate system as depicted in Fig. Its origin is the centre of a wall element dS; the wall normal is its polar axis. Accordingly, the sound pressure level close to the wall would surpass that measured far away by 3 dB. For the same reason, the sound absorption of an absorbent surface adjacent and perpendicular to a rigid wall is higher near the edge than at a distance of several wavelengths from the wall.

When the sinusoidal excitation signal is replaced with random noise of limited bandwidth the pressure distribution is obtained by averaging eqn 2. As an example, the dashed curve of Fig.


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Now the standing wave has virtually levelled out for all 54 Room Acoustics Figure 2. Now we again consider a wall element with area dS. It should be mentioned that the validity of the Paris formula has been called into question by Makita and Hidaka. This, of course, would also modify eqn 2. It spreads more or less in all directions. If this wall is exposed 56 Room Acoustics Figure 2. This would indeed be true if the acoustical wavelength were vanishingly small. Behind the wall, i.

And in region C the plane wave is disturbed by interferences with the diffraction wave. On the whole, there is a steady transition from the undisturbed, i. This is shown in Fig. Of course, the extension of this transition depends on the angular wave number k and the distance d.

A similar effect occurs at the upper boundary of region A. Consider the sound pressure at point P. Both P and S are situated on the middle axis of the disc at distances R1 and R2 from its centre, respectively. For lower frequencies its effect is much smaller. Generally, any body or surface of limited extension distorts a primary sound wave by diffraction unless its dimensions are very small compared to the wavelength.

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One part of the diffracted sound is scattered more or less in all directions. The role of sound scattering by the human head in hearing has already been mentioned in Section 1. In the opposite case of short wavelengths, the scattering cross-section approaches twice its visual crosssection, i. In the opposite case, i. If the wall has an irregular surface structure, a noticeable fraction of the incident sound energy will be scattered in all directions.

In this case we speak of a 60 Room Acoustics d a b c Figure 2. In Section 8. As an example of a sound scattering boundary, we consider the ceiling of a particular concert hall. This measurement has been carried out on a model ceiling. To understand this we return to Fig. Therefore we can conclude that any change of wall impedance creates a diffraction wave. A practical example of this kind are walls lined with relatively thin panels which are mounted on a rigid framework.

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Between these points, however, the lining is more compliant because it can perform bending vibrations, particularly if the frequency of the exciting sound is close to the resonance frequency of the lining see eqn 2. Scattering will be even stronger if adjacent partitions are tuned to different resonance frequencies due to variations of the panel masses or the depths of the air space behind them. We assume that d is noticeably smaller than the wavelength. If a plane wave arrives normally at the wall, it will excite all strips with equal amplitude and phase, and each of them will react to it by emitting a secondary wave or wavelet.

Complete randomness, however, would require a very large number of elements. A similar effect can be reached with so-called pseudorandom sequences of phase angles. If these sequences are periodic, arrangements of this kind act as phase gratings, with the grating constant Nd if N denotes the number of elements within one period. Probably the most popular kind of them is based on quadratic residues. Suppose that the period N is a prime number. Other useful sequences exploit the properties of primitive roots of prime numbers or the index function.

A sound wave hitting such an arrangement will excite secondary waves in the wells. Each of these waves travels towards the rigid bottom of the well. On the other hand, there are critical frequencies at which no scattering takes place at all. This occurs when all depths are integral multiples of the acoustical wavelength. It goes without saying that such a diffuser works not only for normal incident sound but also for waves arriving from oblique directions.

A second design parameter is the width d of the wells. If it is too small, the number of allowed diffraction orders according to eqn 2. Furthermore, narrow wells show increased losses due to the viscous and thermal boundary layer on their walls. If, on the other hand, the wells are too wide, not only the fundamental wave mode 64 Room Acoustics Figure 2.

Practical diffusers have well widths of a few centimetres. The concept of Schroeder diffusers is easily applied to two-dimensional structures which scatter the incident sound into all diffraction orders within the solid angle. They consist of periodic arrays of parallel and rigidwalled channels. The explanation of pseudorandom diffusers as presented here is only qualitative since it neglects all losses and assumes all wells behave independently. Here x is the coordinate running parallel to the scattering surface while the y-axis points into the surface. This leads us to N linear inhomogeneous 66 Room Acoustics equations for the amplitudes Cm.

For a more detailed description of this method the reader is referred to Ref. The phase grating diffusers invented by Schroeder are certainly based on an ingenious concept. Nevertheless, they suffer from the fact that the scattered energy is concentrated in a number of grating lobes which are separated by large minima. This is caused by the periodic repetition of a base element. One way to overcome this disadvantage is to use aperiodic number sequences. These and other possibilities are discussed in Ref. Another peculiarity of this type of diffusers is their relatively high sound absorption.

More on the cause of this absorption will be said in Section 6. References 1 Mechel F, Schallabsorber S. Acustica ; Theoretical Acoustics. New York: McGraw-Hill, Zur Akustik der neuerbauten Beethovenhalle in Bonn. Binaural dissimilarity and optimum ceilings for concert halls: more lateral sound diffusion. J Acoust Soc Am, ; Acustica ; 7 Schroeder M. Number Theory in Science and Communication, 2nd edn. The wide-angle diffuser—a wide-angle absorber? Acoustic Absorbers and Diffusers. London: Spon Press, Now we shall try to obtain some insight into the complicated distribution of sound pressure or sound energy in a room which is enclosed on all sides by walls.

We shall return to it in the next chapter. In this chapter we shall choose a different way of tackling our problem which will lead to a solution in closed form—at least a formal one. This advantage is paid for by a higher degree of abstraction, however. Characteristic of this approach are certain boundary conditions which have to be set up along the room boundaries and which describe mathematically the acoustical properties of the walls, the ceiling and the other surfaces.

Then solutions of the wave equations are sought which satisfy these boundary conditions. It will turn out that this method in its exact form too can only be applied to highly idealised cases with reasonable effort. The rooms with which we are concerned in our daily life, however, are more or less irregular in shape, partly because of the furniture, which forms part of the room boundary. Rooms such as concert halls, theatres or churches deviate from their basic shape because of the presence of balconies, galleries, pillars, columns and other wall irregularities. Then even the formulation of boundary conditions may turn out to be quite involved, and the solution of a given problem 68 Room Acoustics requires extensive numerical calculations.

For this reason the immediate application of wave theory to practical problems in room acoustics is very limited. That is to say, we assume, as earlier, a harmonic time law for the pressure, the particle velocity, etc. Here r is used as an abbreviation for the three spatial coordinates. Whenever the boundary or a part of it has non-zero absorption both the eigenfunctions and the eigenvalues are complex.

An important example will be given in the next section. At this point we need to comment on the wave number k in the boundary condition 3. If the room contains a sound source operating at a certain frequency, k is given by this frequency.

In this case the eigenfunctions as well as the eigenvalues are frequency-dependent. Otherwise, one can identify k with kn , the eigenvalue to be evaluated by solving the boundary problem. If all the eigenvalues and eigenfunctions were known, we could—at least in principle—evaluate any desired acoustical property of the room, for instance, its steady-state response to arbitrary sound sources. Suppose the sound sources are distributed continuously over the room according to a density function q r , where q r dV is the volume velocity of a volume element dV at r.

For this purpose we insert both series into eqn 3.

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It is interesting to note that it is symmetric in the coordinates of the sound source and of the point of observation. If we put the sound source at r, we observe at point r0 the same sound pressure as we did before at r, when the sound source was at r0. Thus eqn 3. Sound waves in a room 71 As mentioned before, the eigenvalues are in general complex quantities. In practice, rooms with exactly this shape do not exist.

On the 72 Room Acoustics other hand, many concert halls or other halls, churches, lecture rooms and so on are much closer in shape to the rectangular room than to any other of simple geometry, and so the results obtained for strictly rectangular rooms can be applied at least qualitatively to many rooms encountered in practice. Therefore our example is not only intended for the elucidation of the theory discussed above but also has some practical bearing. As far as the properties of the wall are concerned, we start with the simplest case, namely that of all the walls being rigid.

That is to say, that at the surface of the walls the normal components of the particle velocity must vanish. In cartesian coordinates the Helmholtz equation 3. If this product is inserted into the Helmholtz equation, the latter splits up into three ordinary Figure 3. Sound waves in a room 73 differential equations. The same is true for the boundary conditions.

The pressure amplitude is zero at all points at which at least one of the cosines becomes zero. The numbers nx , ny and nz indicate the numbers of nodal planes perpendicular to the x-axis, the y-axis and the z-axis, respectively. For non-rectangular rooms the surfaces of vanishing sound pressure are generally no planes. On either side of a nodal line the sound pressures have opposite signs. Figure 3. Sound waves in a room 75 Table 3. If there is only one nonzero integer n, the propagation is parallel to one of the coordinate axes, i.

We can get an illustrative survey on the arrangement, the types and the number of the eigenvalues by the following geometrical representation.

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We interpret kx , ky and kz as cartesian coordinates in a k-space. Each of the allowed values of kx , given by eqn 3. The same statement holds for the values of ky and kz , given by eqns 3. These three equations therefore represent three sets of equidistant, mutually orthogonal planes in the k-space. Sound waves in a room 77 Figure 3. The arrow pointing from the origin to an eigenvalue point indicates the direction of one of the eight wave planes which the corresponding mode consists of see Fig. Negative values obviously do not yield additional eigenvalues, since eqn 3. The lattice points corresponding to tangential and to axial modes are situated on the coordinate planes and on the axes, respectively.

The straight line connecting the origin of the coordinate system to a certain lattice point has—according to eqn 3. This representation allows a simple estimate of the number of eigenfrequencies which are located between the frequency 0 and some other given frequency f. Considered geometrically, eqn 3. Hence only half of them have been accounted for so far. Similarly, all lattice points on the coordinate axes related to axial modes are common to four octants; therefore only one fourth of them are contained in eqn 3.

The number of all lattice point corresponding to tangential modes can be calculated in about the same way which led us to eqn 3. Thus, one correction term to eqn 3. However, we should keep in mind that half of these points are already contained in the expression above while another quarter of them have been, as mentioned, counted in eqn 3. For each of them eqn 3.

Since this equation is linear in V, the total number of eigenfrequencies is just the sum of all Ni. We bring this section to a close by applying eqns 3. The rectangular room for which the eigenfrequencies listed in Table 3. For an upper frequency limit of Hz, eqn 3. Using the more accurate formula 3. That means that we must not neglect the corrections due to tangential and axial modes when dealing with such small rooms at low frequencies. This might be a large concert hall, for instance. In the frequency range from 0 to 10 Hz there are, according to eqn 3.

At Hz the number of eigenfrequencies per hertz is about ; thus, the average distance of two eigenfrequencies on the frequency axis is less than 0. But now we consider a room the walls of which are not completely rigid. This means, the normal components of particle velocity may have non-vanishing values along the boundary. Accordingly, we have to replace the boundary condition 3. As in the preceding section the solution of the wave equation consists of three factors px , py and pz , each of which depends on one spatial coordinate only.

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By inserting px into the boundary conditions 3. Once the allowed values of kx have been determined, the ratio of the two constants C1 and D1 can be evaluated from eqns 3. 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. 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. Here nx is an arbitrary integer.

Suppose that the wall is reactive, i. Conversely, a compliance wall, i. If the allowed values of kx are denoted by kxnx , 82 Room Acoustics the eigenvalues of the original differential equation are given as earlier by see eqn 3. 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. Sound waves in a room 83 walls; thus, a pure standing wave is not possible. By comparing this with our earlier eqn 3. Therefore the stationary sound pressure in a room and at one single exciting frequency proves to be the combined effect of numerous resonances.

The half-widths of the resonance curves according to eqn 2. Furthermore, we see from this that the halfwidths according to eqn 3. If, on the contrary, the average half-width of the resonances is much larger than the average spacing of the eigenfrequencies, there will be strong overlap of resonances and the latter cannot be separated. Instead, at any frequency several or many terms of the sum in eqn 3. The room volume V has to be expressed in cubic metres. In large halls the Schroeder frequency is typically below 50 Hz; hence there is strong modal overlap in the whole frequency range of interest, and there is no point in evaluating any eigenfrequencies.

It is only in small rooms that a part of the important frequency range lies below fs , and in this range the acoustic properties are determined largely by the values of individual eigenfrequencies. To calculate the expected number Nfs of eigenfrequencies in the range from zero to fs , eqn 3. This example illustrates Sound waves in a room 85 the somewhat surprising fact that the acoustics of small rooms are in a way more complicated than those of large ones.

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