alt="Schematic illustration of the dependence on frequency of bending-wave speed cb on a beam or panel and wave speed in air c."/>
Figure 3.24 Dependence on frequency of bending‐wave speed cb on a beam or panel and wave speed in air c.
Above this frequency, fc, the coincidence effect is observed because the bending wavelength λb is greater than the wavelength in air λ (Figure 3.25), and trace wave matching always occurs for the sound waves in air at some angle of incidence (see Figure 3.26). This has important consequences for the sound radiation from structures and also for the sound transmitted through the structures from one air space to the other (see Chapter 12 of this book).
Figure 3.25 Variation with frequency of bending wavelength λb on a beam or panel and wavelength in air λ.
Figure 3.26 Diagram showing trace wave matching between waves in air of wavelength λ and waves in panel of trace wavelength λT.
For free‐bending waves on infinite plates above the critical frequency, the plate radiates efficiently, while below this frequency (theoretically) the plate cannot radiate any sound energy at all [27]. For finite plates, reflection of the bending waves at the edges of the plates causes standing waves that allow radiation (although inefficient) from the plate corners or edges even below the critical frequency. In the plate center, radiation from adjacent quarter‐wave areas cancels. But radiation from the plate corners and edges, which are normally separated sufficiently in acoustic wavelengths, does not cancel. At very low frequency, sound is radiated mostly by corner modes, then up to the critical frequency, mostly by edge modes. Above the critical frequency the radiation is caused by surface modes with which the whole plate radiates efficiently (see Figure 3.27). Radiation from bending waves in plates and cylinders is discussed in detail in Refs. [9, 27] and chapter 10 of the Handbook of Acoustics [1]. Figure 3.28 shows some comparisons between theory and experiment for the level of the radiation efficiencies for sound radiation for several practical cases of simply-supported and clamped panel structures with acoustical and point mechanical excitation.
Figure 3.27 Wavelength relations and effective radiating areas for corner, edge, and surface modes. The acoustic wavelength is λ; while λbx and λby are the bending wavelengths in the x‐ and y‐directions, respectively. (see also the Handbook of Acoustics [1], chapter 1.)
Figure 3.28 Comparison of theoretical and measured radiation ratios σ for a mechanically excited, simply-supported thin steel plate (300 × 300 × 1.22 mm). (—) Theory (simply-supported), (‐ ‐ ‐) theory (clamped edges), (⋅⋅⋅⋅⋅⋅⋅) theory [28], and (○) measured [29] (see [30]).
Example 3.15
Determine the critical frequency for a 3 mm thick steel plate.
Solution
The longitudinal wave speed in aluminum is cL = 5100 m/s. Now, replacing κ = h/(12)1/2 in Eq. (3.85) yields
Figure 3.29 Measured radiation ratios of unstiffened and stiffened plates for (a) point mechanical excitation and (b) diffuse sound field excitation.
(Source: Reproduced from Ref. [31] with permission. See [30].)
Sound transmission through structures is discussed in Chapter 12 of this book and chapters 66, 76, and 77 of the Handbook of Acoustics [1].
Figure 3.29a and b show the logarithmic value of the radiation efficiency 10log σ plotted against frequency for stiffened and unstiffened plates. See Ref. [27] for further discussion on the radiation efficiency σrad, which is also known as radiation ratio.
3.17 Standing Waves
Standing‐wave phenomena are observed in many situations in acoustics and the vibration of strings and elastic structures. Thus they are of interest with almost all musical instruments (both wind and stringed) (see Part XIV in the Encyclopedia of Acoustics [19]); in architectural spaces such as auditoria and reverberation rooms; in volumes such as automobile and aircraft cabins; and in numerous cases of vibrating structures, from tuning forks, xylophone bars, bells and cymbals to windows, wall panels, and innumerable other engineering systems including aircraft, vehicle, and ship structural members. With each standing wave is associated a mode shape (or shape of vibration) and an eigen (or natural) frequency. Some of these systems can be idealized as simple one‐, two‐, or three‐dimensional systems. For example, with a simple wind instrument such as a flute, Eq. (3.1) together with the appropriate spatial boundary conditions can be used to predict the predominant frequency of the sound produced. Similarly, the vibration of a string on a violin can be predicted with an equation identical to Eq. (3.1) but with the variable p replaced by the lateral string displacement. With such a string, solutions can be obtained for the fundamental and higher natural frequencies (overtones) and the associated standing wave mode shapes (which are normally sine shapes). In such a case for a string with fixed ends, the so‐called overtones are just integer multiples (2, 3, 4, 5, …) of the fundamental frequency.
The
