Sound Radiation From Idealized Structures

The sound radiation from plates and cylinders in bending (flexural) vibration is discussed in Refs. [9, 27] and chapter 10 in the Handbook of Acoustics [1]. There are interesting phenomena observed with free‐bending waves. Unlike sound waves, these are dispersive and travel faster at higher frequency. The bending‐wave speed is c_b = (ωκc_l)^1/2, where κ is the radius of gyration h/(12)^1/2 for a rectangular cross‐section, h is the thickness, and c_L is the longitudinal wave speed {E/[ρ(1 − σ²)]}^1/2, where E is Young’s modulus of elasticity, ρ is the material density, and σ is Poisson’s ratio. When the bending‐wave speed equals the speed of sound in air, the frequency is called the critical frequency (see Figure 3.24). The critical frequency is

(3.85) equation

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 c_b on a beam or panel and wave speed in air c.

Above this frequency, f_c, the coincidence effect is observed because the bending wavelength λ_b 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).

Graph depicts the variation with frequency of bending wavelength lambda-b on a beam or panel and wavelength in air lambda. — **Figure 3.25** Variation with frequency of bending wavelength λ_b on a beam or panel and wavelength in air λ.

Schematic illustration describing trace wave matching between waves in air of wavelength lambda and waves in panel of trace wavelength lambda-T. — **Figure 3.26** Diagram showing trace wave matching between waves in air of wavelength λ and waves in panel of trace wavelength λ_T.

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.

Schematic illustration of the wavelength relations and effective radiating areas for corner, edge, and surface modes. — **Figure 3.27** Wavelength relations and effective radiating areas for corner, edge, and surface modes. The acoustic wavelength is λ; while λ_bx and λ_by are the bending wavelengths in the x‐ and y‐directions, respectively. (see also the *Handbook of Acoustics* [1], chapter 1.)

Graph depicts the comparison of theoretical and measured radiation ratios sigma for a mechanically excited, simply supported thin steel plate (300 × 300 × 1.22 mm). — **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 c_L = 5100 m/s. Now, replacing κ = h/(12)^1/2 in Eq. (3.85) yields

Graphs depict the measured radiation ratios of unstiffened and stiffened plates for (a) point mechanical excitation and (b) diffuse sound field excitation. — **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 σ_rad, which is also known as radiation ratio.

Sound Radiation From Idealized Structures

EXAMPLE 3.15

SOLUTION

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