## Abstract

The noise radiated by a turbulent boundary layer over a rough wall is shown to be characterized by a dipole surface source that, if the surface pressure is spatially homogeneous, can be specified by a convolution integral combining the surface pressure wavenumber spectrum and the wavenumber spectrum of the surface roughness slope. For random roughness elements with almost vertical sides, the surface slope has a wavenumber white spectrum and the radiated sound is directly proportional to the surface pressure spectrum multiplied by an acoustic efficiency factor (*k*_{o}*h*)^{2}, where *k*_{o} is the acoustic wavenumber and *h* is the geometrical r.m.s. roughness height. The theoretical result agrees with the roughness noise measurements of Smith *et al*. (Smith *et al.* 2008 AIAA paper no. 2008-2904) for all flow speeds and roughness element heights (in wall units) of *k*_{g}^{+}<34. The theory can also be applied to wavy walls and this provides a new method for measuring the wavenumber spectrum of surface pressure fluctuations.

## 1. Introduction

Experimental evidence shows that the sound radiated by a turbulent boundary layer over a rough surface can be significantly louder than the sound of a boundary layer over a smooth wall (Cole 1980; Hersh 1983; Farabee & Geib 1991; Smol'yakov 2001; Liu & Dowling 2007; Grissom *et al*. 2007). Theoretical descriptions of the additional sound radiation from a rough wall have been proposed (Howe 1984, 1998; Smol'yakov 2001), but the applicability of these theories to different types of rough-wall surface has yet to be established. In this paper, we will present a theory for rough-wall boundary-layer noise that encompasses all types of rough-wall surfaces, and we will show how the radiated sound field from roughness elements that have almost vertical sides can be characterized in terms of the local wall pressure spectrum (table 1).

As described by Howe (1984), the sound radiation from turbulence in a free stream is of quadrupole order (Lighthill 1952) and the intensity of the far-field sound scales as *ρ*_{o}*U*^{3}*M*^{5}, where *ρ*_{o} is the fluid density; *U* is the flow speed; and *M* is the flow Mach number. For turbulence in a boundary layer over a smooth wall, the presence of the surface acts as a passive reflector, providing that the acoustic wavelength is large compared with the boundary-layer thickness, and the fluctuating shear stress can be ignored (Powell 1960; Howe 1979). The smooth surface does not alter the quadrupole nature of the source and the radiated sound scales in the same way as free turbulence. Roughness elements on the surface have three effects: the first is to enhance the turbulence in the boundary layer; the second is to increase the skin friction; and the third is to scatter propagating or convected pressure waves in many different directions. The enhanced turbulence in the boundary layer will continue to scale as a quadrupole source, but the unsteady component of the skin friction and the scattered field is typical of surface sources which behave as acoustic dipoles (Curle 1955). The far-field intensity from a compact acoustic dipole in a low Mach number flow (*M*≪1) scales as *ρ*_{o}*U*^{3}*M*^{3}, which is of order 1/*M*^{2} more efficient than the noise from turbulence, and explains the increased sound levels from boundary layers over rough surfaces.

The experimental evidence supporting the hypothesis that rough-wall boundary-layer noise is of dipole order goes back to Skudrzyk & Haddle (1960), who measured the sound from a spinning cylinder coated with either 60 grit or 180 grit sandpaper. They showed that the radiated sound field scaled with the sixth, tenth or twelfth power of the velocity, but only the first of these scalings has been confirmed by subsequent experiments. Chanaud (1969) measured the sound from a spinning disc with roughened rings on the surface and showed the sound level increased with the sixth power of the disc edge velocity as expected for a dipole source. Cole (1980) appears to have been the first to study the sound from rough surfaces in a wind tunnel. He considered smooth and rough surfaces and showed that roughness enhanced the measured spectral levels by approximately 3 dB. A more comprehensive study was carried out by Hersh (1983), who studied the sound radiated from the exit of open-ended pipes with the internal surfaces roughened by sandpaper of different grit sizes. Sound measurements were made in the acoustic far field, some distance from the pipe exit. Measurements for a smooth pipe showed that the sound field scaled with the eighth power of the flow speed as expected for a quadrupole source, and jet noise from the pipe exit. Measurements with 40 grit sandpaper on the pipe inner wall showed that the sound level scaled with the sixth power of the flow speed, as expected for a dipole source. In this experiment, Hersh (1983) was also able to measure the wall shear stress from the pressure drop along the pipe and developed scaling laws based on boundary-layer parameters including the friction velocity *u*_{τ} and the equivalent sand grain roughness height *k*_{s}. He showed that transitionally rough surfaces (*k*_{s}*u*_{τ}/*v*<50, where *v* is the kinematic viscosity) produced low-amplitude sound, but fully rough surfaces *k*_{s}*u*_{τ}/*v*>400 produced more sound, approximately 15 dB above the smooth-wall level. He found that the noise spectra scaled as *Φ*_{pp}(*ω*)/(*ρ*_{o}/*c*_{∞})^{2}*u*_{τ}^{5}*k*_{s} when plotted as a function of the non-dimensional frequency *ωk*_{s}/*u*_{τ}, where *Φ*_{pp}(*ω*) is the measured sound spectral level (in Pa^{2} s rad^{−1}); *c*_{∞} is the speed of sound; and *ω* is the frequency. More recently, Grissom *et al*. (2007) have carried out wind tunnel measurements in quiet facilities where low background noise levels permitted much more accurate measurements. Grissom *et al*. (2007) were able to identify the existence of roughness noise for many different surface types, including hydrodynamically smooth surfaces *k*_{s}*u*_{τ}/*v*<10. These results also confirm the dipole scaling of the source, but scaling on different boundary-layer parameters was found to be inconsistent. Smith *et al*. (2008) measured both the far-field sound and the wall pressure spectrum and determined that the ratio of the two was proportional to (*k*_{o}*h*)^{2}, where *k*_{o} is the acoustic wavenumber and *h* is the geometrical r.m.s. roughness height. This result demonstrates that the scaling of the wall pressure spectrum with boundary-layer parameters is identical to the scaling of the far-field sound, and the radiation efficiency is proportional to (*k*_{o}*h*)^{2} which is expected for a dipole source.

There have been relatively few theoretical studies of roughness noise. Howe (1984) argued that the most important mechanism for far-field sound radiation was caused by a scattering mechanism. He developed a model for the radiated sound caused by scattering from a single hemispherical roughness element in terms of the fluctuating Reynolds stress source terms in the boundary layer above the roughness. He assumed that fluctuating shear stresses and interstitial flows between the roughness elements could be ignored, and that ‘the dominant turbulent Reynolds stresses are situated above the roughness elements’. The sound field from multiple roughness elements was obtained by assuming a correlation length scale for the scattering from multiple elements. However, the prediction method could not be completed without information on the wavenumber spectrum of the boundary-layer turbulence, and Howe (1988) used the model given by Chase (1987) and the experiments of Hersh (1983) to complete his model. Howe's (1988) final result gives a far-field noise spectrum that scales in the same way as the data of Hersh (1983) with *k*_{s} replaced by the geometric roughness height *k*_{g}.

In Howe (1998), the Rayleigh scattering from an independently distributed array of hemispherical roughness elements, of radius *R*, below a turbulent boundary layer was considered. It was assumed that *Ru*_{τ}/*ν*>10, so that the roughness elements protrude out of the viscous sublayer. For roughness elements whose size is small compared with the boundary-layer turbulence length scale, Howe (1998) provides a result that shows the ratio of the far-field sound spectral level to the surface pressure spectrum scales as (*k*_{o}*R*)^{2}(*ωR*/*U*_{c})^{2}.

Smol'yakov (2001) developed a roughness noise model that assumed that the noise was generated by the same mechanisms as a bluff body in a free stream. This model does not include the scattering mechanism suggested by Howe (1984), but includes the sources that are caused by the direct interaction of the roughness with the mean flow. He used a series of measurements in a quiet wind tunnel to develop an empirical scaling law, which, on rearrangement (Grissom *et al*. 2006), can be shown to be identical to Howe's (1988) scaling, but his empirical spectrum has a peak frequency that is almost twice the peak frequency of Howe's model.

While there have been a number of experimental studies and some theoretical mechanisms postulated for roughness noise, there is no theory that includes all of the possible source mechanisms and surface shapes. In §2, we will address this issue by deriving a unified theory based on Lighthill's acoustic analogy, making the assumptions that quadrupole sources can be ignored, the surface pressure is spatially homogeneous and that the boundary-layer thickness is small compared with the acoustic wavelength. The theory can be applied to any type of rough surface and we will consider separately the case of random sandpaper-type roughness and a wavy wall. In §4, the resulting physics will be discussed and compared with measurements.

## 2. A unified theory for roughness noise

Problems in hydroacoustics can be described using Lighthill's (1952) acoustic analogy and the solution to the acoustic wave equation in the presence of impermeable surfaces. We will work from the solution to Lighthill's equation given by Goldstein (1976), which is(2.1)The left-hand side of this equation represents the acoustic pressure perturbation, , which radiates away from a region of turbulent flow in a volume *V* bounded by surfaces *S*. The normal component of the surface velocity is given by *V*_{n} and *ρ*_{o} is the mean fluid density. If the surface can be considered rigid, then the velocity normal to the surface is zero and the first term may be eliminated. The function *G*(** x**,

*t|*

**,**

*y**τ*) represents the Green's function that is a solution to the wave equation and may be chosen to satisfy different boundary conditions. The last term on the right-hand side (the volume integral) depends on the Lighthill stress tensor

*T*

_{ij}, and the double derivative of the Green's function indicates that it is a quadrupole source term. The second term on the right-hand side has a single derivative of the Green's function and represents the dipole source term, which depends on the force per unit area applied to the fluid by the surface

*f*

_{i}. The force includes the effect of viscosity and is defined as

*f*

_{i}=

*p*

_{s}

*n*

_{i}−

*σ*

_{ij}

*n*

_{j}, where

*p*

_{s}=

*p*−

*p*

_{o}is the unsteady component of the pressure;

*n*

_{i}is the surface normal (directed into the volume

*V*); and

*σ*

_{ij}is the viscous stress tensor. The surface integral is applied over the surface bounding the flow (figure 1) and so, for a rough wall, the integral must be carried out over the actual surface, which adds a level of complexity. It will be shown below that this complexity can be reduced without significant approximation, so the integral is carried out over the flat surface from which the roughness protrudes.

Equation (2.1) is the basic equation that applies to all flow noise problems over rigid surfaces and will be used here to describe the noise from rough-wall boundary layers. However, extracting the magnitude of the source terms and simplifying the evaluation of the integrals is not trivial and needs careful attention.

Aeroacoustic theory shows that if the time scale of the flow is inversely proportional to the flow speed, then the quadrupole term in equation (2.1) scales as the eighth power of the velocity, whereas the dipole term scales as the sixth power of the velocity. The ratio of the acoustic power generated by the quadrupole and dipole sources is proportional to the square of the flow Mach number. In underwater applications, the quadrupole sources can be ignored if dipole sources are present because they generate acoustic power that is approximately 10^{−3} less than the dipole sources. In low-speed wind tunnel applications, this ratio is 4×10^{−2}, which is still small enough to render the quadrupole source of second order. The prevailing theoretical viewpoint is that quadrupole sources can be neglected in comparison with surface source terms that are of dipole order.

If we apply the above scaling approximations to equation (2.1), then we can drop the volume quadrupole term, and the surface dipole term can be assumed to account for the radiated sound field. Note that there is no place in the theory for a surface quadrupole and, in general, quadrupole sources only exist in the volume of the fluid where turbulent flow is dominant.

For a rough wall that lies in the *y*_{1},*y*_{3} plane and is defined by *y*_{2}=*ξ*(*y*_{1},*y*_{3}), we can use a Green's function that satisfies the non-penetration boundary condition on the surface *y*_{2}=0, so ∂*G*/∂*y*_{2}=0 when *y*_{2}=0. If the surface displacement is small compared with the acoustic wavelength, then the Green's function on the actual surface can be obtained from a Taylor series expansion as(2.2)The Green's function that satisfies the non-penetration boundary condition on *y*_{2}=0 is defined aswhere *y*^{#}=(*y*_{1}, −*y*_{2}, *y*_{3}) is the image source location. If we limit consideration to the leading terms in the expansions given in (2.2), and to the acoustic field where |** x**|≫|

**|, we obtain, making the usual far-field approximations,(2.3)Using this expansion, the evaluation of the dipole term in equation (2.1) gives(2.4)Note that the force parallel to the**

*y**y*

_{1}

*y*

_{3}plane has a dipole scaling and the force normal to the

*y*

_{1}

*y*

_{3}plane has quadrupole-type scaling, although it is strictly a dipole.

To gain some insight into the source mechanisms of roughness noise, we will consider the dipole noise source term in the absence of viscous shear stress (Howe 1979, 1986), so the only contribution to the compressive stress tensor comes from the unsteady pressure. The dipole sound for an observer in the acoustic far field is then given by(2.5)The integral is carried out over the actual rough surface and is more easily evaluated if the surface is projected onto the *y*_{1}*y*_{3} plane, which is parallel to the external flow. The surface is then defined as *y*_{2}*=ξ*(*y*_{1},*y*_{3}), and equation (2.5) becomes(2.6)where the scale factor *J* and the normal to the surface areConsequently, equation (2.6) can be simplified to an integral over the planform, which depends on *ξ* and ∂*ξ*/∂*y*_{i}.

We will examine the contribution from the dipole terms by considering the Fourier transform of the acoustic field with respect to time given by(2.7)where *k*_{o}=*ω*/*c*_{∞} is the acoustic wavenumber and the convention for Fourier transforms is the same as used by Howe (1998). This is the roughness noise equation that gives the far-field sound radiation from a rough surface in the frequency domain. Note that each term depends on the roughness height and so (2.7) is zero for a smooth wall. The implication is that sound radiation from a smooth wall is of quadrupole order and will be determined by the terms that are not included in (2.7). The source term is dependent on the unsteady surface pressure *p*_{s}, which can be generated by several different mechanisms. Figure 1 shows three different sources of unsteady surface pressure: (i) the near-field pressure fluctuations caused by turbulence in the outer boundary-layer flow, (ii) near-field pressure fluctuations caused by gusts distorted by the roughness, and (iii) pressure fluctuations caused by vortex shedding from the roughness elements themselves. However, equation (2.7) applies to all the source mechanisms and we can proceed without assuming that any one of the source mechanisms is dominant.

To reduce the results, we introduce the wavenumber spectrum of the surface pressure spectrum *p*_{s}(*κ*_{1}, *κ*_{3}, *ω*) which can be defined, if the surface pressure is a single-valued function of (*y*_{1},*y*_{3}), so thatNote that the surface pressure will not be a single-valued function of (*y*_{1},*y*_{3}) on surfaces where the roughness height is multivalued or a piecewise continuous function of (*y*_{1},*y*_{3}), unless it can be assumed that the surface pressure is independent of the surface height.

When the wavenumber spectrum of the surface pressure is introduced into equation (2.7), we obtain(2.8)Then, if we define the wavenumber transform of the surface roughness and its gradient as(2.9)where *h* is the r.m.s. roughness height, we obtain the convolution integral(2.10)Equation (2.10) applies to any roughness distribution, and we can derive a particularly simple form for the power spectral density in the acoustic far field if the surface pressure perturbations are statistically homogeneous. It should be noted that this assumption may not apply when the surface pressure spectra are biased by inhomogeneous turbulent flow caused by roughness elements. For example, hydrodynamically rough surfaces, which are characterized by elements protruding into the outer part of the boundary layer, may have surface pressure spectra that are dominated by the local flow around the roughness elements and could not be characterized using a homogeneous surface pressure spectrum.

If the surface pressure perturbations are statistically homogeneous over the planform, then we can express the power spectral density of the acoustic far field as(2.11)in terms of the wavenumber spectrum of the surface pressure, defined as(2.12)so that(2.13)where Φ_{PP}(*ω*) is the surface pressure spectrum at a point and the integrand is non-dimensional. The scaling in (2.13) shows that the far-field spectrum for any of the mechanisms shown in figure 1 can be characterized in terms of a non-dimensional integral, which may be frequency dependent, and will depend on the roughness distribution. However, the novel part of this result is that we have separated the acoustic coupling (or scattering) effects, which are defined by *Γ* from the source effects that determine the surface pressure through the function *Ψ*_{PP}. For a given geometry, *Γ* will be unaltered by flow speed or Reynolds number, and can be readily calculated if the details of the roughness are known. By contrast, *Ψ*_{PP} will be strongly influenced by the characteristics of the boundary layer and the height of the roughness elements in wall units or relative to the boundary-layer thickness. This function may be difficult to measure or model, especially at high Reynolds number.

An important feature of (2.13) is that the integral over the wavenumber combines a normalized wavenumber spectrum of the unsteady surface pressure with a non-dimensional wavenumber filter function *Γ*, which is determined by the characteristics of the roughness. Different types of roughness will apply different wavenumber filters to the surface pressure spectrum, and hence alter the far-field noise spectrum. Typically, we expect the surface pressure wavenumber spectrum to peak in the vicinity of the convective ridge in which *κ*_{1}∼*ω*/*U*_{c}, where *U*_{c} is an eddy convection velocity. Hence, if the filter function is constant or slowly varying for wavenumbers in this range, we can use the fact that(2.14)to show that the wavenumber integral in (2.13) integrates to a constant.

A particularly simple application of these results is to random roughness that can be modelled by roughness elements with vertical sides, as shown in figure 2*a*. On a vertical side, the roughness gradient is infinite and must be considered as a limiting case. For example, if a patch of roughness includes *N* ‘almost vertical sides’, which are randomly located at , then we can define the surface slope at each point as ∂*ξ*/∂*y*_{i}=Δξ^{(n)}/Δ*y*_{i}^{(n)}, where tends to zero. The planform area beneath each almost vertical side is , and so adding the contribution from all the points, which fall into this category, gives(2.15)Using the relationship for random surface locations , (2.16)and defining the mean square value of as *α*^{2}*h*^{2}*Σ*/*N* (where *α* is a scaling factor of *O*(1), and it is assumed that ) gives, for acoustically compact roughness (*k*_{o}*h*≪1) and an observer in the *x*_{3}*=*0 plane,(2.17)which is independent of wavenumber. However, if the roughness has vertical sides, then the surface pressure will not be a single-valued function of (*y*_{1},*y*_{3}) unless it is independent of roughness height. The variation of pressure over a vertical surface will be small if the turbulence length scale (∼*U*_{c}/*ω*) is large compared with the roughness height. If we limit consideration to surfaces where *ωh*/*U*_{c}<1, then we can assume that the surface pressure will be a single-valued function of (*y*_{1},*y*_{3}) and equation (2.13) reduces to(2.18)The constant *C* should be the same for all geometrically similar random surfaces.

Equation (2.18) is particularly important since it provides a straightforward formula for the prediction of roughness noise for discontinuous or near discontinuous surfaces. Many naturally occurring and man-made rough surfaces fall into this category. The predictions require only elementary information about the rough surface, the observer location and an estimate of the single-point wall pressure spectrum. It is likely that, in many circumstances, existing correlations for smooth walls (such as Goody 2004) can be used to provide an adequate first estimate of the pressure spectrum.

Another model that can be used to represent a rough surface is a distribution of hemispherical bosses as shown in figure 2*b*. This was the model considered by Howe (1998), and we will show that the equations given above may be used to reproduce his result for this surface. If each roughness element in figure 2*b* has a radius *R*^{(n)} and is randomly located at points *y*^{(n)} on a flat surface, then we can use (2.9) to givewhere the surface integral is only carried out over the area beneath each boss. At the edge of each boss, the surface displacement is zero and so we can integrate by parts to obtainFollowing Howe (1998), we will assume that, at the wavenumbers of interest, the size of the roughness element is compact, so |*k**R*^{(n)}|≪1 and the exponential term under the integrand can be dropped. The remaining integral gives the volume of the roughness element and soFor an independent distribution of roughness elements, we can use (2.16) to obtain *Γ* aswhere the mean boss radius is and we have normalized using . Using this result in (2.13) then exactly yields the result for rough-wall boundary-layer noise given by Howe (1998), providing that proper account is taken for difference between the incident pressure field (used by Howe 1998) and the surface pressure on the roughness element.

The two examples given above show different scalings for the wavenumber filter function *Γ*. For the piecewise continuous wall illustrated in figure 2*a*, the wavenumber filter is a constant and the far-field sound scales with frequency as (*k*_{o}*h*)^{2}. By contrast, the wall that consists of individual roughness elements, as shown in figure 2*b*, has a wavenumber filter function *Γ* that is proportional to and, if the wall pressure spectrum is dominated by wavenumber components that are on the convective ridge, where *k*_{1}∼*ω*/*U*_{c}, then we expect the far-field sound to scale with frequency as , which is to be expected because each of the roughness elements is acoustically compact. By contrast, the discontinuities in the piecewise continuous wall scatter sound much more efficiently because there is no cancellation of pressure fluctuations on the front and back of the scattering object.

## 3. Sound from a boundary layer over a wavy wall

Another simple case to consider is flow over a wavy wall for which the roughness is only a function of *y*_{1} and periodic with wavenumber *k*_{w}, so(3.1)where the coefficients *a*_{n} are normalized, so that . It follows that(3.2)and so from (2.10),(3.3)

The radiated sound spectrum is then(3.4)The acoustic field from a wavy wall therefore depends on the sum of discrete wavenumber components of the wavenumber surface pressure spectrum. If the wall height is chosen to be sinusoidal, or almost sinusoidal, then the first term in the series will dominate the far-field spectrum, and it will be directly proportional to the frequency dependence of the wavenumber spectrum at the wavenumber *k*_{w}. We can therefore measure the wavenumber surface pressure spectrum using(3.5)This provides an indirect measurement method for the wall pressure wavenumber spectrum which can be used to verify the empirical models, such as those by Corcos (1964) and Chase (1987), at far smaller wavenumbers than has previously been possible.

## 4. Discussion

In §§2 and 3, we derived a series of equations that relate the far-field sound generated by a turbulent boundary layer over a rough surface to the surface pressure wavenumber spectrum. It was shown that for randomly rough surfaces, with elements that include almost vertical sides, the far-field sound is directly proportional to the surface pressure spectrum, with a scaling factor that depends on (*k*_{o}*h*)^{2}. This result applies regardless of the source mechanism that generates the surface pressure and regardless of the scaling of the surface pressure, but it was assumed that the scattering from each roughness element could be summed incoherently and that the surface pressure perturbations are statistically homogeneous over the planform. The analysis highlights the important distinction between the acoustic radiation efficiency, which depends only on the roughness geometry, and the hydrodynamic scaling of the surface pressure, which depends on boundary-layer scaling parameters and Reynolds number. By normalizing the measured far field by the surface pressure spectrum, it follows that the impact of the hydrodynamic scaling is eliminated, which greatly simplifies the interpretation of the results. To verify this, figure 3 shows roughness noise measurements for sandpaper roughness taken by Smith *et al*. (2008) in a quiet wall jet wind tunnel for a range of roughness sizes at a fixed tunnel speed, plotted in the non-dimensional form(4.1)Equation (2.18) shows that *K*_{pp}(*ω*)=20 log(*k*_{o}*h*), and figure 3 demonstrates that the proposed scaling works very effectively with a scatter of less than 2 dB. The data given in figure 3 are for non-dimensional roughness heights of 6<<18 (where and *k*_{g} is the nominal roughness grain size, approx. 4*h* in this case), so it covers the range of roughness sizes from hydrodynamically smooth to transitionally rough. The data were collapsed by empirically determining the value of the constant *C*, by matching the normalized spectra over the measured frequency range of 10–11 kHz where all the spectra showed a good signal-to-noise ratio. The value of the constant was found to be in the range −15.9<10 log(*C*)<−12.1.

For a wavy wall, the analysis shows that the radiated sound is directly proportional to the wavenumber spectrum of the surface pressure at a fixed wavenumber. Models for the wavenumber spectrum have been developed by Corcos (1964) and Chase (1987) and are given in appendix A. In the Corcos (1964) model, the spectra scale on a Strouhal number *ω*/*k*_{w}*U*_{c}, where *U*_{c} is the convection speed of disturbances in the boundary layer. By contrast, the model for the wavenumber spectrum developed by Chase (1987) includes additional scaling parameters and is sensitive to the value *k*_{w}*δ*, which determines the width of the spectral peak. Figure 4 shows the normalized far-field spectra for flow over a wavy wall and compares it with the models of Chase and Corcos. These data, made as part of the same experiment as the sandpaper measurements reported by Smith *et al*. (2008), are for a wall with a rectified sine-wave shape, with *k*_{w}*δ*≈70 and *h*/*δ*≈0.01. The comparison is shown for a range of boundary-layer thickness Reynolds number 9100<*Re*_{δ}<18 300, for which 4.7<*h*^{+}<10.4. The Chase model works quite well, and shows that the measurement approach given here is consistent with earlier studies of surface pressure wavenumber spectra. However, we note that, as the flow speed is decreased, the peaks in the spectra shown in figure 4 move to higher non-dimensional frequencies, indicating that there is a Reynolds number effect.

Previous studies have failed to find a universal scaling law for roughness noise based on boundary-layer parameters (Grissom *et al*. 2007), and the analysis given here helps to explain why this was the case. The frequency scaling of the far-field noise depends on *k*_{o}*h* and the hydrodynamic scaling of the boundary-layer pressure fluctuations. For sandpaper-type surfaces with sharp roughness elements, the separation of these two effects is quite simple, but for more complex geometries, including periodic arrays of identical roughness elements, the coupling between the roughness geometry and the hydrodynamic scaling will be more involved.

## 5. Conclusions

We have derived relationships that show how the far-field noise generated by a boundary layer over a rough surface depends on the surface pressure wavenumber spectrum and the roughness geometry. It has been shown that if the surface pressure perturbations are statistically homogeneous over the planform, then the acoustic radiation efficiency depends only on the geometry of the roughness and can be separated from the scaling of the surface pressure wavenumber spectrum. A convenient way to represent this effect is to consider the acoustic radiation efficiency as a wavenumber filter applied to the surface pressure wavenumber spectrum. For surfaces with a random distribution of roughness elements with almost vertical sides, such as sandpaper, the wavenumber filter is white and the far-field sound is directly proportional to the surface pressure spectrum multiplied by a factor of (*k*_{o}*h*)^{2}. The constant of proportionality is the same for geometrically similar shapes. For deterministic roughness elements, the wavenumber filter is more complex but readily calculated from the details of the roughness element geometry. This approach has separated the scaling of the far-field noise from the scaling surface pressure spectrum, which is mechanism dependent, and is known to scale on different flow parameters at high and low frequencies (Blake 1970).

## Acknowledgments

This work was supported by the Office of Naval Research and the authors would wish to thank Dr Ki-Han Kim for his support under grants N00014-05-1-0516, N00014-07-1-0458 and N00014-08-1-0493. The authors would also wish to thank Mr Nathan Alexander, Dr Ben Smith and Dr Dustin Grissom for their contributions to the experimental measurements presented in this paper.

## Appendix A

Using the Corcos (1964) model, as specified by Howe (1998) in equation (3.4), gives(A1)It follows that the normalized spectrum is(A2)which scales on *ω*/*k*_{w}*U*_{c} and has a peak in the vicinity of *ω*/*k*_{w}*U*_{c}*=*1.

Alternatively, we can use the Chase (1987) model for the wavenumber spectrum, which gives(A3)This can be used directly in equation (3.5).

## Footnotes

- Received December 12, 2008.
- Accepted February 2, 2009.

- © 2009 The Royal Society