2.1 The mathematical model

Consider an aerofoil with trailing-edge serrations, modelled as a flat plate as shown in figure 1, with an infinitesimal thickness and an averaged chord length $c$ and spanwise length $d$ . Let $x^{\prime }$ , $y^{\prime }$ and $z^{\prime }$ denote the streamwise, spanwise and normal to the plate coordinates respectively. The observer point is located at $(x_{1},x_{2},x_{3})$ . The profile function $H(y^{\prime })$ is used to describe the serrated edges. The origin of the coordinates is chosen in such a way that $H(y^{\prime })$ is an oscillatory function of zero mean and that $H(y^{\prime })=0$ in the absence of serrations. Figure 1 shows a sawtooth serration with a root-to-tip amplitude of $2h$ and a wavelength of ${\it\lambda}$ .

When the sound wavelength is equal to or shorter than the chord length $c$ , the flat plate can be treated as a semi-infinite plate without a leading edge (Amiet Reference Amiet1976b ). Furthermore, the plate can be considered to be infinite in the spanwise direction provided that it has a relatively large aspect ratio (typically $d/c>3$ ) (Amiet Reference Amiet1978; Roger & Carazo Reference Roger and Carazo2010). The turbulence inside the boundary layer is assumed to be frozen, i.e. it remains statistically the same before and after passing over the trailing edge.

After implementing a spatial and time Fourier transformation, the hypothetical surface pressure beneath the turbulent boundary layer that would exist when the flat plate was infinite can be expressed as an integral of different wall pressure gust components. The incoming wall pressure gust of frequency ${\it\omega}$ , as illustrated in figure 1, takes the form of

(2.3) $$\begin{eqnarray}p_{i}=P_{i}\text{e}^{-\text{i}({\it\omega}t-k_{1}x^{\prime }-k_{2}y^{\prime })},\end{eqnarray}$$

where $P_{i}$ is the magnitude of the incident wall pressure gust and $k_{1}$ and $k_{2}$ denote the wavenumbers in the chordwise and spanwise directions respectively.

The sound sources due to the presence of solid boundaries (Curle Reference Curle1955) can be modelled as dipoles, in addition to the quadrupoles in free field (Lighthill Reference Lighthill1952). As explained in Amiet’s paper (Amiet Reference Amiet1976b ), the incident pressure produces a scattered field originating from the trailing edge, due to the change in boundary condition at the wall. The scattered field induces a pressure jump that cancels the incident pressure jump at the trailing edge and in the wake after the plate (Kutta condition). Thus, the total pressure can be decomposed into two parts, namely $p_{t}=p_{i}+p$ . The incident wall pressure is given by (2.3), and the scattered pressure field, $p$ , must satisfy the following conditions at $z^{\prime }=0$ :

(2.4) $$\begin{eqnarray}\left.\begin{array}{@{}ll@{}}\displaystyle \frac{\partial p}{\partial z^{\prime }}=0, & x^{\prime }<H(y^{\prime }),\\ \displaystyle p=-P_{i}\text{e}^{-\text{i}({\it\omega}t-k_{1}x^{\prime }-k_{2}y^{\prime })}, & x^{\prime }\geqslant H(y^{\prime }).\end{array}\right\}\end{eqnarray}$$

It is worth pointing out that the incident pressure $p_{i}$ defined in (2.3) is twice the ‘conventional incident pressure’ that would exist when the semi-infinite flat plate was absent. Therefore, the scattered pressure $p$ to be obtained on the upper surface of the plate would be twice the ‘conventional scattered pressure’ due to the linearity of the wave equation and (2.4). Since the scattering problem is anti-symmetrical, it follows that the scattered pressure $p$ is in fact, as in Amiet’s paper (Amiet Reference Amiet1976b ), the scattered pressure jump across the plate. This can be made evident through a more rigorous analysis.

In the plate-fixed frame $\{x^{\prime },y^{\prime },z^{\prime }\}$ , the flow has a uniform speed $U$ in the streamwise direction outside the boundary layer, and the wave equation governing the scattered pressure field $p$ is

(2.5) $$\begin{eqnarray}{\rm\nabla}^{2}p-\frac{1}{c_{0}^{2}}\left(\frac{\partial }{\partial t}+U\frac{\partial }{\partial x^{\prime }}\right)^{2}p=0,\end{eqnarray}$$

where $c_{0}$ denotes the speed of sound. With the assumption of harmonic perturbation, $p=P(x^{\prime },y^{\prime },z^{\prime })\text{e}^{\text{-i}{\it\omega}t}$ , the above equation reduces to

(2.6) $$\begin{eqnarray}{\it\beta}^{2}\frac{\partial ^{2}P}{\partial x^{\prime 2}}+\frac{\partial ^{2}P}{\partial y^{\prime 2}}+\frac{\partial ^{2}P}{\partial z^{\prime 2}}+2\text{i}kM_{0}\frac{\partial P}{\partial x^{\prime }}+k^{2}P=0,\end{eqnarray}$$

where $k={\it\omega}/c_{0}$ , ${\it\beta}^{2}=1-M_{0}^{2}$ and $M_{0}=U/c_{0}$ .

In order to make the boundary conditions in (2.4) independent of $y^{\prime }$ , the coordinate transformation (Roger, Schram & Santana Reference Roger, Schram and Santana2013) $x=x^{\prime }-H(y^{\prime }),\,y=y^{\prime },\,z=z^{\prime }$ is used and leads to the following differential equation:

(2.7) $$\begin{eqnarray}({\it\beta}^{2}+{H^{\prime }}^{2}(y))\frac{\partial ^{2}P}{\partial x^{2}}+\frac{\partial ^{2}P}{\partial y^{2}}+\frac{\partial ^{2}P}{\partial z^{2}}-2H^{\prime }(y)\frac{\partial ^{2}P}{\partial x\partial y}+\left(2\text{i}M_{0}k-H^{\prime \prime }(y)\right)\frac{\partial P}{\partial x}+k^{2}P=0,\end{eqnarray}$$

where $H^{\prime }(y)$ and $H^{\prime \prime }(y)$ denote the first and second derivatives of $H(y)$ . The boundary conditions now read

(2.8) $$\begin{eqnarray}\left.\begin{array}{@{}ll@{}}\displaystyle P(x,y,0)=-P_{i}\text{e}^{\text{i}(k_{1}x+k_{2}y)}\text{e}^{\text{i}k_{1}H(y)}, & x\geqslant 0,\\ \displaystyle \partial P(x,y,0)/\partial z=0, & x<0.\end{array}\right\}\end{eqnarray}$$

Since the coefficients in (2.7) are $y$ -dependent, the standard ‘separation of variables’ technique cannot be applied to solve this equation. We therefore turn to using a Fourier expansion technique in the following derivation.

2.2 Fourier expansion

As the scattering problem is periodic in the spanwise direction, one can expand the scattered pressure field using Fourier series as

(2.9) $$\begin{eqnarray}P(x,y,z)=\mathop{\sum }_{n=-\infty }^{\infty }P_{n}(x,z)\text{e}^{\text{i}k_{2n}y},\end{eqnarray}$$

where $k_{2n}=k_{2}+2n{\rm\pi}/{\it\lambda}$ .

Substitution of the above expression into the transformed wave equation, shown in (2.7), yields

(2.10) $$\begin{eqnarray}\displaystyle & & \displaystyle \left\{({\it\beta}^{2}+{H^{\prime }}^{2}(y))\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial y^{2}}+\frac{\partial ^{2}}{\partial z^{2}}-2H^{\prime }(y)\frac{\partial ^{2}}{\partial x\partial y}+\left(2\text{i}M_{0}k-H^{\prime \prime }(y)\right)\frac{\partial }{\partial x}+k^{2}\right\}\nonumber\\ \displaystyle & & \displaystyle \qquad \times ~\mathop{\sum }_{n=-\infty }^{\infty }P_{n}(x,z)\text{e}^{\text{i}k_{2n}y}=0.\end{eqnarray}$$

By multiplying (2.10) by $\text{e}^{\text{-i}k_{2n^{\prime }}y}$ , then integrating it over $y$ from $-{\it\lambda}/2$ to ${\it\lambda}/2$ , it can be readily shown that

(2.11) $$\begin{eqnarray}\displaystyle & & \displaystyle \left\{{\it\beta}^{2}\frac{\partial ^{2}}{\partial x^{2}}\right.+\left.\frac{\partial ^{2}}{\partial z^{2}}+2\text{i}kM_{0}\frac{\partial }{\partial x}+(k^{2}-k_{2n^{\prime }}^{2})\right\}P_{n^{\prime }}(x,z)\qquad \nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{1}{{\it\lambda}}\int _{-{\it\lambda}/2}^{{\it\lambda}/2}\mathop{\sum }_{n=-\infty }^{\infty }\left\{H^{\prime 2}(y)\frac{\partial ^{2}}{\partial x^{2}}-(H^{\prime \prime }(y)+2\text{i}k_{2n}H^{\prime }(y))\frac{\partial }{\partial x}\right\}P_{n}(x,z)\text{e}^{\text{i}[2(n-n^{\prime }){\rm\pi}/{\it\lambda}]y}\,\text{d}y=0.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

It should be noted that when both $H^{\prime }(y)$ and $H^{\prime \prime }(y)$ are constant within an entire sawtooth wavelength, the summation over different modes in (2.11) can be dropped and we obtain a fully decoupled differential equation for mode $n^{\prime }$ . However, this means that the flat plate has a straight or swept trailing edge. For serrations of an arbitrary profile, both $H^{\prime }(y)$ and $H^{\prime \prime }(y)$ generally depend on $y$ . Thus, (2.11) becomes a coupled differential equation, i.e. more than one mode appears in each differential equation. The physical interpretation of this mode coupling will be discussed later.

Figure 2.

The schematic of sawtooth serrations.

In this paper, we only focus on the sawtooth serration, which has been shown to be effective in reducing the trailing-edge noise (Howe Reference Howe1991b ). The method can, however, also be used for other serrations. Consider a sawtooth centred around the coordinate origin, and let $({\it\chi}_{0},{\it\epsilon}_{0})$ , $({\it\chi}_{1},{\it\epsilon}_{1})$ and $({\it\chi}_{2},{\it\epsilon}_{2})$ be the Cartesian coordinates of the tip and roots of the sawtooth, as shown in figure 2. The serration profile function $H(y)$ can therefore be defined as

(2.12) $$\begin{eqnarray}H(y)=\left\{\begin{array}{@{}ll@{}}\displaystyle {\it\sigma}_{0}(y-{\it\chi}_{0}-m{\it\lambda})+{\it\epsilon}_{0},\quad & {\it\chi}_{0}+m{\it\lambda}<y\leqslant {\it\chi}_{1}+m{\it\lambda},\\ \displaystyle {\it\sigma}_{1}(y-{\it\chi}_{1}-m{\it\lambda})+{\it\epsilon}_{1},\quad & {\it\chi}_{1}+m{\it\lambda}<y\leqslant {\it\chi}_{2}+m{\it\lambda},\end{array}\right.\end{eqnarray}$$

where ${\it\sigma}_{j}=({\it\epsilon}_{j+1}-{\it\epsilon}_{j})/({\it\chi}_{j+1}-{\it\chi}_{j})$ , $j=0,1$ and $m=0,\pm 1,\pm 2,\pm 3,\ldots \,$ . Let ${\it\sigma}$ , defined as ${\it\sigma}=|{\it\sigma}_{j}|=4h/{\it\lambda}$ , denote the sharpness of the sawtooth serrations. For a sawtooth profile, as mentioned above, $H^{\prime }(y)$ is not continuous and $H^{\prime \prime }(y)$ is thus singular at the joint points. We use the conventional generalized function ${\it\delta}(x)$ to describe the singularities at these points, i.e.

(2.13a ) $$\begin{eqnarray}\displaystyle & \displaystyle H^{\prime }(y)=\left\{\begin{array}{@{}ll@{}}\displaystyle {\it\sigma}_{0},\quad & {\it\chi}_{0}+m{\it\lambda}<y\leqslant {\it\chi}_{1}+m{\it\lambda},\\ \displaystyle {\it\sigma}_{1},\quad & {\it\chi}_{1}+m{\it\lambda}<y\leqslant {\it\chi}_{2}+m{\it\lambda},\end{array}\right. & \displaystyle\end{eqnarray}$$

(2.13b ) $$\begin{eqnarray}\displaystyle & \displaystyle H^{\prime \prime }(y)=\mathop{\sum }_{m=-\infty }^{\infty }(-1)^{m+1}2{\it\sigma}{\it\delta}(y-m{\it\lambda}/2). & \displaystyle\end{eqnarray}$$

By substituting the serration profile function and its derivatives, (2.12) and (2.13), into the wave equation, (2.11), and making use of the fact that $\int _{-\infty }^{\infty }f(x){\it\delta}(x-{\it\tau})\,\text{d}x=f({\it\tau})$ , we obtain

(2.14) $$\begin{eqnarray}\displaystyle & & \displaystyle \left\{\left({\it\beta}^{2}+{\it\sigma}^{2}\right)\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial z^{2}}+2\text{i}kM_{0}\frac{\partial }{\partial x}+(k^{2}-k_{2n^{\prime }}^{2})\right\}P_{n^{\prime }}(x,z)\nonumber\\ \displaystyle & & \displaystyle \qquad =-\frac{4{\it\sigma}}{{\it\lambda}}\mathop{\sum }_{n-n^{\prime }=odd}\left(1-\frac{k_{2}{\it\lambda}+2n{\rm\pi}}{(n-n^{\prime }){\rm\pi}}\right)\frac{\partial P_{n}(x,z)}{\partial x}.\end{eqnarray}$$

To make the above equation more compact, we write the set of differential equations obtained above in a matrix form. Let a linear operator

(2.15) $$\begin{eqnarray}\text{D}=\left\{\left({\it\beta}^{2}+{\it\sigma}^{2}\right)\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial z^{2}}+2\text{i}kM_{0}\frac{\partial }{\partial x}\right\}\end{eqnarray}$$

and a vector of functions

(2.16) $$\begin{eqnarray}\boldsymbol{P}=\left(\cdots P_{-n^{\prime }}(x,z),P_{-n^{\prime }+1}(x,z),\ldots ,P_{n^{\prime }-1}(x,z),P_{n^{\prime }}(x,z),\ldots \right)^{\text{T}},\end{eqnarray}$$

then the coupled equations can be written as

(2.17) $$\begin{eqnarray}\text{D}\boldsymbol{P}-\unicode[STIX]{x1D63C}\boldsymbol{P}=\unicode[STIX]{x1D63D}\frac{\partial \boldsymbol{P}}{\partial x},\end{eqnarray}$$

where the symbol $\text{T}$ in (2.16) represents the transpose of a matrix. The matrices $\unicode[STIX]{x1D63C}$ and $\unicode[STIX]{x1D63D}$ denote the coefficient matrices of $\boldsymbol{P}$ and $\partial \boldsymbol{P}/\partial x$ respectively. Here, $\unicode[STIX]{x1D608}_{ml}$ and $\unicode[STIX]{x1D609}_{ml}$  representing the entries corresponding to mode $m$ in row and $l$ in column of matrices $\unicode[STIX]{x1D63C}$ and $\unicode[STIX]{x1D63D}$ are

(2.18a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D608}_{ml}=(k_{2m}^{2}-k^{2}){\it\delta}_{ml},\quad \unicode[STIX]{x1D609}_{ml}=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{4{\it\sigma}}{{\it\lambda}}\frac{m+l+k_{2}{\it\lambda}/{\rm\pi}}{l-m},\quad & m-l\text{ is odd},\\ 0,\quad & m-l\text{ is even},\end{array}\right.\end{eqnarray}$$

where ${\it\delta}_{ml}$ represents the Kronecker delta. It should be noted that the indices of matrices $\unicode[STIX]{x1D63C}$ and $\unicode[STIX]{x1D63D}$ are from $-m$ to $m$ and $-l$ to $l$ rather than from $1$ to $2m+1$ and $1$ to $2l+1$ given the fact that the mode numbers are symmetric with respect to $0$ .

By substituting the profile geometry, (2.12), into the boundary conditions, (2.8), and performing the same Fourier expansions, one obtains

(2.19) $$\begin{eqnarray}\left.\begin{array}{@{}ll@{}}\displaystyle P_{n}(x,0)=-P_{i}a_{n}\text{e}^{\text{i}k_{1}x}, & x>0,\\ \displaystyle \frac{\partial P_{n}}{\partial z}(x,0)=0, & x\leqslant 0,\end{array}\right\}\end{eqnarray}$$

where the $a_{n}$ are defined as

(2.20) $$\begin{eqnarray}a_{n}=\frac{1}{{\it\lambda}}\int _{-{\it\lambda}/2}^{{\it\lambda}/2}\text{e}^{\text{i}k_{1}H(y)}\text{e}^{\text{-i}2n{\rm\pi}y/{\it\lambda}}\,\text{d}y.\end{eqnarray}$$

In (2.17), as $\unicode[STIX]{x1D63D}$ is not a diagonal matrix, the term $\unicode[STIX]{x1D63D}(\partial \boldsymbol{P}/\partial x)$ contains coupling terms, in the sense that different modes, $P_{n}$ for example, appear in the governing equation of $P_{m}$ . This means that every mode is interacting with the other modes and cannot be solved independently. From the expression of $\unicode[STIX]{x1D63D}$ in (2.18), it can be seen that the strength of the coupling is proportional to ${\it\sigma}/{\it\lambda}$ . This indicates that sharper serrations have stronger coupling between different modes. It should also be noted that $\unicode[STIX]{x1D63C}$ is a diagonal matrix, so if $\unicode[STIX]{x1D63D}\approx 0$ , i.e. the serrations are very wide, then there is no coupling effect and one can solve each mode independently.

At very low frequencies, the contribution of higher modes is expected to become gradually negligible compared with mode $0$ . Thus, it can be reasonably assumed that in the case of the governing equation of $P_{0}$ (mode 0), the coupling with higher modes is weak and one can solve $P_{0}$ individually. The coupling effect becomes more pronounced at high frequencies and for sharp serrations. To solve these coupled equations at relatively high frequencies, one can use an iterative procedure, to be introduced in the next section.

2.3 The iterative solution procedure

2.3.1 Scattered surface pressure

To obtain the scattered surface pressure, (2.17), together with the boundary conditions in (2.19), needs to be solved. By analogy with the solution of a system of linear algebraic equations (Süli & Mayers Reference Süli and Mayers2003), which can be obtained iteratively, we shall solve our system of partial differential equations in an iterative manner.

By substituting a known initial value $\boldsymbol{P}^{(0)}$ into the coupling term in (2.17), one can write

(2.21) $$\begin{eqnarray}\text{D}\boldsymbol{P}-\unicode[STIX]{x1D63C}\boldsymbol{P}=\unicode[STIX]{x1D63D}\frac{\partial \boldsymbol{P}^{(0)}}{\partial x}.\end{eqnarray}$$

Solving (2.21) yields a new set of solutions $\boldsymbol{P}^{(1)}$ . By replacing $\boldsymbol{P}^{(0)}$ in (2.21) with $\boldsymbol{P}^{(1)}$ , we obtain a new wave equation,

(2.22) $$\begin{eqnarray}\text{D}\boldsymbol{P}-\unicode[STIX]{x1D63C}\boldsymbol{P}=\unicode[STIX]{x1D63D}\frac{\partial \boldsymbol{P}^{(1)}}{\partial x}.\end{eqnarray}$$

Solving (2.22) gives a new set of solutions $\boldsymbol{P}^{(2)}$ . This process is repeated to obtain a solution sequence, $\boldsymbol{P}^{(0)}$ , $\boldsymbol{P}^{(1)}$ , $\boldsymbol{P}^{(2)}$ , $\boldsymbol{P}^{(3)}\ldots \,$ . If the sequence is convergent, then its limit satisfies (2.17).

The initial value $\boldsymbol{P}^{(0)}$ used to start the first iteration can be obtained, as mentioned in the previous section, by ignoring all of the coupling terms, i.e. with $\unicode[STIX]{x1D63D}=0$ , and by solving each equation individually via the standard Schwarzschild technique, as shown in (2.2). The solution to each equation in the decoupled system of equations

(2.23) $$\begin{eqnarray}\text{D}\boldsymbol{P}-\unicode[STIX]{x1D63C}\boldsymbol{P}=0\end{eqnarray}$$

can be found as follows.

After use is made of the transformation $P_{n^{\prime }}=\bar{P}_{n^{\prime }}\text{e}^{-\text{i}kM_{0}x/({\it\beta}^{2}+{\it\sigma}^{2})}$ , the individual equations in (2.23) reduce to

(2.24) $$\begin{eqnarray}\left\{({\it\beta}^{2}+{\it\sigma}^{2})\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial z^{2}}+K_{n^{\prime }}^{2}({\it\beta}^{2}+{\it\sigma}^{2})\right\}\bar{P}_{n^{\prime }}=0,\end{eqnarray}$$

where

(2.25) $$\begin{eqnarray}K_{n^{\prime }}=\sqrt{k^{2}(1+{\it\sigma}^{2})-k_{2n^{\prime }}^{2}({\it\beta}^{2}+{\it\sigma}^{2})}/({\it\beta}^{2}+{\it\sigma}^{2}).\end{eqnarray}$$

Making use of $X=x,Z=\sqrt{{\it\beta}^{2}+{\it\sigma}^{2}}z$ , one can show that (2.24) reduces to a standard Schwarzschild problem, and the solution can be found using the Schwarzschild integral described in (2.2), as

(2.26) $$\begin{eqnarray}P_{n^{\prime }}^{(0)}=P_{i}\text{e}^{\text{i}k_{1}x}a_{n^{\prime }}((1-\text{i})\text{E}(-{\it\mu}_{n^{\prime }}x)-1),\end{eqnarray}$$

where $P_{n^{\prime }}^{(0)}$ is the element of the vector $\boldsymbol{P}^{(0)}$ corresponding to the $n^{\prime }$ th mode, and

(2.27a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\mu}_{n^{\prime }}=K_{n^{\prime }}+k_{1}+\frac{kM_{0}}{{\it\beta}^{2}+{\it\sigma}^{2}}, & \displaystyle\end{eqnarray}$$

(2.27b ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{E}(x)=\int _{0}^{x}\frac{\text{e}^{\text{i}t}}{\sqrt{2{\rm\pi}t}}\,\text{d}t. & \displaystyle\end{eqnarray}$$

The initial solutions obtained by ignoring all of the coupling terms denote the decoupled part of the exact solution of each mode, which implies that the $n$ th mode excitation ( $x>0$ ) produces only an $n$ th mode response ( $x<0$ ). The iteration procedure will add a coupled part to the solution of each mode. The coupled part implies that an $n$ th mode input ( $x>0$ ) will also produce some $m$ th mode responses ( $x<0$ ), where $m\neq n$ . It can be expected that the coupling contributions from closer modes will be stronger than those from remote modes. For sawtooth serrations, as will be shown in (2.30), the coupling strength decays quadratically with respect to the difference between their ‘mode numbers’, i.e. $|m-l|$ .

By substituting $\boldsymbol{P}^{(0)}$ into the coupling terms on the right-hand side of (2.17), one obtains some inhomogeneous equations that can no longer be solved using the standard Schwarzschild technique. However, if one can transform these equations into homogeneous ones, then Schwarzschild’s method can again be applied. It should be noted that $\boldsymbol{P}^{(0)}$ satisfies (2.23); hence, for $x\neq 0$ , where $\boldsymbol{P}^{(0)}$ is first-order continuously differentiable, the following equation holds:

(2.28) $$\begin{eqnarray}\text{D}\frac{\partial \boldsymbol{P}^{(0)}}{\partial x}-\unicode[STIX]{x1D63C}\frac{\partial \boldsymbol{P}^{(0)}}{\partial x}=0.\end{eqnarray}$$

Making use of (2.28), (2.21) can be equivalently written as

(2.29) $$\begin{eqnarray}\text{D}\left(\boldsymbol{P}+\unicode[STIX]{x1D66B}\frac{\partial \boldsymbol{P}^{(0)}}{\partial x}\right)-\unicode[STIX]{x1D63C}\left(\boldsymbol{P}+\unicode[STIX]{x1D66B}\frac{\partial \boldsymbol{P}^{(0)}}{\partial x}\right)=0,\end{eqnarray}$$

where $\unicode[STIX]{x1D66B}$ is a coefficient matrix whose entries are

(2.30) $$\begin{eqnarray}\unicode[STIX]{x1D637}_{ml}=\frac{\unicode[STIX]{x1D609}_{ml}}{k_{2m}^{2}-k_{2l}^{2}}=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{-4h}{{\rm\pi}^{2}(m-l)^{2}},\quad & m-l=\text{odd},\\ 0,\quad & m-l=\text{even}.\end{array}\right.\end{eqnarray}$$

It is worth pointing out that (2.29) only holds when $x\in \boldsymbol{R}$ and $x\neq 0$ , and in order to apply the Schwarzschild technique, it must be valid over the whole domain. However, since the singularity of $\partial \boldsymbol{P}^{(0)}/\partial x$ only exists at $x=0$ , similar to the differentiation of $H(y)$ , we may again make use of the generalized function to account for this singularity. Let $\partial \hat{\boldsymbol{P}}^{(0)}/\partial x$ denote the generalized differentiation, which allows the presence of generalized functions at the singular point $x=0$ but equals $\partial \boldsymbol{P}^{(0)}/\partial x$ elsewhere, then the equation

(2.31) $$\begin{eqnarray}\text{D}\frac{\partial \hat{\boldsymbol{P}}^{(0)}}{\partial x}-\unicode[STIX]{x1D63C}\frac{\partial \hat{\boldsymbol{P}}^{(0)}}{\partial x}=0\end{eqnarray}$$

needs to hold over $x\in \boldsymbol{R}$ . The Schwarzschild technique suggests that if (2.31) does hold, then the routine application of the steps described from (2.24) to (2.26) will recover the value of $\partial \boldsymbol{P}^{(0)}/\partial x$ for $x<0$ . Thus, one can verify that the intended $\partial \hat{\boldsymbol{P}}^{(0)}/\partial x$ can indeed be found as

(2.32) $$\begin{eqnarray}\frac{\partial \hat{P}_{n^{\prime }}^{(0)}}{\partial x}(x,0)=\frac{\partial P_{n^{\prime }}^{(0)}}{\partial x}(x,0)+P_{i}a_{n^{\prime }}(1-\text{i})(-\sqrt{{\it\mu}_{n^{\prime }}})\sqrt{2{\rm\pi}x}{\it\delta}(x),\end{eqnarray}$$

where $\partial \hat{P}_{n^{\prime }}^{(0)}/\partial x$ denotes the element of $\partial \hat{\boldsymbol{P}}^{(0)}/\partial x$ corresponding to the $n^{\prime }$ th mode and

(2.33) $$\begin{eqnarray}\int _{0}^{\infty }{\it\delta}(x)\,\text{d}x=\frac{1}{2}.\end{eqnarray}$$

Now, the first iterated solution can be obtained by solving the equation

(2.34) $$\begin{eqnarray}\text{D}\left(\boldsymbol{P}+\unicode[STIX]{x1D66B}\frac{\partial \hat{\boldsymbol{P}}^{(0)}}{\partial x}\right)-\unicode[STIX]{x1D63C}\left(\boldsymbol{P}+\unicode[STIX]{x1D66B}\frac{\partial \hat{\boldsymbol{P}}^{(0)}}{\partial x}\right)=0\end{eqnarray}$$

via the steps described from (2.24) to (2.26).

Solving (2.34) gives the values of $\boldsymbol{P}^{(1)}$ . Continuing this iteration process gives $\boldsymbol{P}^{(2)}$ , $\boldsymbol{P}^{(3)}\ldots \,$ . The exact solutions $\boldsymbol{P}$ can also be expressed as

(2.35) $$\begin{eqnarray}\boldsymbol{P}(x,0)=\boldsymbol{N}(x)+\boldsymbol{C}^{(1)}(x)+\boldsymbol{C}^{(2)}(x)+\boldsymbol{C}^{(3)}(x)+\cdots \,,\end{eqnarray}$$

where $\boldsymbol{N}$ is the non-coupled part, and the coupled parts are denoted by $\boldsymbol{C}^{(i)}=\boldsymbol{P}^{(i)}-\boldsymbol{P}^{(i-1)}$ ( $i=1,2,3,\ldots \,$ ). The entries of $\boldsymbol{N}$ and $\boldsymbol{C}^{(1)}$ corresponding to mode $n^{\prime }$ are given by

(2.36) $$\begin{eqnarray}\displaystyle & \displaystyle N_{n^{\prime }}(x)=P_{i}\text{e}^{\text{i}k_{1}x}a_{n^{\prime }}\left((1-\text{i})\text{E}(-{\it\mu}_{n^{\prime }}x)-1\right), & \displaystyle\end{eqnarray}$$

(2.37) $$\begin{eqnarray}\displaystyle & \displaystyle C_{n^{\prime }}^{(1)}(x)=P_{i}\text{e}^{\text{i}k_{1}x}(1-\text{i})\mathop{\sum }_{m=-\infty }^{\infty }v_{n^{\prime }m}a_{m}\left(\vphantom{\sqrt{\frac{{\it\mu}_{m}}{-2{\rm\pi}x}}}\text{i}k_{1}(\text{E}(-{\it\mu}_{n^{\prime }}x)-\text{E}(-{\it\mu}_{m}x))\right. & \displaystyle \nonumber\\ \displaystyle & \displaystyle \left.-\,\sqrt{\frac{{\it\mu}_{m}}{-2{\rm\pi}x}}(\text{e}^{\text{-i}{\it\mu}_{n^{\prime }}x}-\text{e}^{\text{-i}{\it\mu}_{m}x})\right).\hspace{40.00006pt} & \displaystyle\end{eqnarray}$$

The elements of the second-order function $\boldsymbol{C}^{(2)}$ are provided in appendix A. Since $v_{ml}\propto h$ it can be readily shown that $C_{n^{\prime }}^{(i)}\propto h^{i}$ for $i=1,2,3,\ldots$ . This means that the solution presented in (2.35) is a perturbation (Taylor expansion) series with respect to half of the root-to-tip amplitude $h$ . Therefore, a smaller value of $h$ compared with the sound wavelength yields faster convergence. It should be noted that the function $\boldsymbol{C}^{(i)}$ becomes more and more complex as $i$ increases. However, if $\boldsymbol{C}^{(i)}$ vanishes sufficiently quickly, higher orders can be neglected without causing significant errors. This appears to be the case for the frequencies relevant to trailing-edge noise, see § 3.2. By substituting (2.36), (2.37) and (A 2) into (2.35), a second-order approximation of the exact solutions is obtained.

The scattered surface pressure is obtained by summing $P_{n^{\prime }}(x,0)$ over all different modes and transforming back to the physical coordinate system, namely

(2.38) $$\begin{eqnarray}P(x^{\prime },y^{\prime },0)=\mathop{\sum }_{n^{\prime }=-\infty }^{\infty }P_{n^{\prime }}(x^{\prime }-H(y^{\prime }),0)\text{e}^{\text{i}k_{2n^{\prime }}y^{\prime }}.\end{eqnarray}$$

Here, $P_{n^{\prime }}$ is the solution obtained from the iteration procedure mentioned above,

(2.39) $$\begin{eqnarray}P(x^{\prime },y^{\prime },0)=\mathop{\sum }_{n^{\prime }=-\infty }^{\infty }(N_{n^{\prime }}+C_{n^{\prime }}^{(1)}+C_{n^{\prime }}^{(2)}+\cdots \,)(x^{\prime }-H(y^{\prime }))\text{e}^{\text{i}k_{2n^{\prime }}y^{\prime }},\end{eqnarray}$$

where $N_{n^{\prime }}$ and $C_{n^{\prime }}^{(1)}$ are defined in (2.36) and (2.37), and $C_{n^{\prime }}^{(2)}$ can be found in appendix A. It should be noted that the terms in the second set of parentheses are the arguments for the $N_{n^{\prime }}$ and $C_{n^{\prime }}^{(i)}\,(i=1,2,3,\ldots )$ functions. It is worth pointing out that in the limiting case when $H(y^{\prime })=0$ , $\boldsymbol{C}^{(i)}$ vanishes, and (2.39) reduces to the result obtained by Amiet (Reference Amiet1976b ) for a straight edge.

As shown in (2.39), the scattered pressure field can now be expressed in terms of an infinite series. By inspection of (2.39), one can show that at sufficiently low frequencies, i.e. $k_{1}h<{\rm\pi}^{2}/4$ , the infinite series is absolutely convergent. At higher frequencies, the series still appears to be convergent, but to obtain a satisfactory approximation a higher truncation number and higher-order iterations may be required. The convergence of the series will be discussed in the following sections by comparing the far-field sound predicted using different-order approximations.

2.3.2 Far-field sound pressure

As illustrated in figure 1, the observer point is located at $(x_{1},x_{2},x_{3})$ and the flat plate has an averaged chord length $c$ and span length $d$ . The far-field sound can be found using the surface pressure integral, as mentioned in Amiet’s model (Lamb Reference Lamb1932; Curle Reference Curle1955; Amiet Reference Amiet1975),

(2.40) $$\begin{eqnarray}p_{f}(\boldsymbol{x},{\it\omega})=\frac{-\text{i}{\it\omega}x_{3}}{4{\rm\pi}c_{0}S_{0}^{2}}\iint _{s}{\rm\Delta}P(x^{\prime },y^{\prime })\text{e}^{\text{-i}kR}\,\text{d}x^{\prime }\,\text{d}y^{\prime },\end{eqnarray}$$

where ${\rm\Delta}P=P$ denotes the pressure jump, $S_{0}^{2}=x_{1}^{2}+{\it\beta}^{2}(x_{2}^{2}+x_{3}^{2})$ and

(2.41) $$\begin{eqnarray}R=\frac{M_{0}(x_{1}-x^{\prime })-S_{0}}{{\it\beta}^{2}}+\frac{x_{1}x^{\prime }+x_{2}y^{\prime }{\it\beta}^{2}}{{\it\beta}^{2}S_{0}}.\end{eqnarray}$$

By substituting the solution obtained in (2.39) into (2.40), the far-field sound pressure can be expressed as

(2.42) $$\begin{eqnarray}p_{f}(\boldsymbol{x},{\it\omega},k_{2})=P_{i}\left(\frac{-\text{i}{\it\omega}x_{3}c}{4{\rm\pi}c_{0}S_{0}^{2}}\right){\it\lambda}\frac{\sin ((N+1/2){\it\lambda}(k_{2}-kx_{2}/S_{0}))}{\sin ((k_{2}-kx_{2}/S_{0}){\it\lambda}/2)}{\mathcal{L}}({\it\omega},k_{1},k_{2}).\end{eqnarray}$$

Here, $2N+1$ represents the number of sawteeth on the edge, and the far-field sound gust-response function ${\mathcal{L}}$ is defined as

(2.43) $$\begin{eqnarray}\displaystyle {\mathcal{L}}({\it\omega},k_{1},k_{2}) & = & \displaystyle (1-\text{i})\frac{1}{{\it\lambda}c}\text{e}^{\text{-i}k(M_{0}x_{1}-S_{0})/{\it\beta}^{2}}\text{e}^{\text{i}k(M_{0}-x_{1}/S_{0})h/{\it\beta}^{2}}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{n^{\prime }=-\infty }^{\infty }({\it\Theta}_{n^{\prime }}+{\it\Theta}_{n^{\prime }}^{(1)}+{\it\Theta}_{n^{\prime }}^{(2)}+\cdots \,),\end{eqnarray}$$

with (only the first two terms are given, see more results in appendix A)

(2.44) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle {\it\Theta}_{n^{\prime }}=a_{n^{\prime }}Q_{n^{\prime }n^{\prime }},\\ \displaystyle {\it\Theta}_{n^{\prime }}^{(1)}=\mathop{\sum }_{m=-\infty }^{\infty }v_{n^{\prime }m}\text{i}k_{1}a_{m}(Q_{n^{\prime }n^{\prime }}-Q_{n^{\prime }m})-v_{n^{\prime }m}\sqrt{{\it\mu}_{m}}a_{m}(S_{n^{\prime }n^{\prime }}-S_{n^{\prime }m}).\end{array}\right\}\end{eqnarray}$$

The functions $Q_{nm}$ and $S_{nm}$ in the above equations are given by

(2.45a ) $$\begin{eqnarray}\displaystyle & \displaystyle Q_{nm}=\mathop{\sum }_{j=0}^{1}\frac{1}{{\it\kappa}_{nj}}\left(\frac{1}{{\it\mu}_{A}}[\text{e}^{\text{i}{\it\kappa}_{nj}{\it\chi}_{j+1}}{\it\Gamma}(c+{\it\epsilon}_{j+1};{\it\mu}_{m},{\it\mu}_{A})-\text{e}^{\text{i}{\it\kappa}_{nj}{\it\chi}_{j}}{\it\Gamma}(c+{\it\epsilon}_{j};{\it\mu}_{m},{\it\mu}_{A})]\right.\qquad & \displaystyle \nonumber\\ \displaystyle & \displaystyle \qquad \qquad \left.-\,\frac{1}{{\it\mu}_{Bnj}}\text{e}^{\text{i}{\it\kappa}_{nj}({\it\chi}_{j}-(c+{\it\epsilon}_{j})/{\it\sigma}_{j})}[{\it\Gamma}(c+{\it\epsilon}_{j+1};{\it\mu}_{m},{\it\mu}_{Bnj})\!-\!{\it\Gamma}(c+{\it\epsilon}_{j};{\it\mu}_{m},{\it\mu}_{Bnj})]\right),\qquad & \displaystyle\end{eqnarray}$$

(2.45b ) $$\begin{eqnarray}\displaystyle & \displaystyle S_{nm}=\mathop{\sum }_{j=0}^{1}\frac{1}{\text{i}{\it\kappa}_{nj}}\left(\vphantom{\frac{1}{\sqrt{{\it\eta}_{Bmj}}}}\frac{1}{\sqrt{{\it\eta}_{Am}}}[\text{e}^{\text{i}{\it\kappa}_{nj}{\it\chi}_{j+1}}\text{E}({\it\eta}_{Am}(c+{\it\epsilon}_{j+1}))-\text{e}^{\text{i}{\it\kappa}_{nj}{\it\chi}_{j}}\text{E}({\it\eta}_{Am}(c+{\it\epsilon}_{j}))]\right. & \displaystyle \nonumber\\ \displaystyle & \displaystyle \qquad -\,\left.\frac{1}{\sqrt{{\it\eta}_{Bmj}}}\text{e}^{\text{i}{\it\kappa}_{nj}({\it\chi}_{j}-(c+{\it\epsilon}_{j})/{\it\sigma}_{j})}[\text{E}({\it\eta}_{Bmj}(c+{\it\epsilon}_{j+1}))-\text{E}({\it\eta}_{Bmj}(c+{\it\epsilon}_{j}))]\right), & \displaystyle\end{eqnarray}$$

where the function ${\it\Gamma}$ is defined by

(2.46) $$\begin{eqnarray}{\it\Gamma}(x;{\it\mu},{\it\nu})=\text{e}^{\text{-i}{\it\nu}x}\text{E}({\it\mu}x)-\sqrt{\frac{{\it\mu}}{{\it\mu}-{\it\nu}}}\text{E}(({\it\mu}-{\it\nu})x)+\frac{1}{1-\text{i}}(1-\text{e}^{\text{-i}{\it\nu}x}),\end{eqnarray}$$

and

(2.47) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle {\it\mu}_{A}=k_{1}+k(M_{0}-x_{1}/S_{0})/{\it\beta}^{2},\\ \displaystyle {\it\mu}_{Bnj}=k_{1}-(k_{2n}-kx_{2}/S_{0})/{\it\sigma}_{j},\\ \displaystyle {\it\kappa}_{nj}=k_{2n}-kx_{2}/S_{0}+k(M_{0}-x_{1}/S_{0}){\it\sigma}_{j}/{\it\beta}^{2},\\ \displaystyle {\it\eta}_{Am}=K_{m}+kM_{0}/({\it\beta}^{2}+{\it\sigma}^{2})-k(M_{0}-x_{1}/S_{0})/{\it\beta}^{2},\\ \displaystyle {\it\eta}_{Bmj}=K_{m}+kM_{0}/({\it\beta}^{2}+{\it\sigma}^{2})+(k_{2n}-kx_{2}/S_{0})/{\it\sigma}_{j}.\end{array}\right\}\end{eqnarray}$$

It should be noted that (2.42) is the far-field sound induced by the scattered pressure only. When the incident pressure is also incorporated, as pointed out by Amiet (Reference Amiet1978), the number $1$ appearing in the parentheses of the function ${\it\Gamma}$ should be omitted, i.e. the third term on the right-hand side of (2.46) should be replaced by $-\text{e}^{\text{-i}{\it\nu}x}/(1-\text{i})$ .

2.3.3 Statistical formulation

The hypothetical surface pressure of frequency ${\it\omega}$ beneath a turbulent boundary layer on the plate surface that would exist when the plate was infinite can be expressed as a Fourier integral,

(2.48) $$\begin{eqnarray}P_{int}({\it\omega},x^{\prime },y^{\prime })=\iint \widetilde{P}_{int}({\it\omega},k_{1},k_{2})\text{e}^{\text{i}(k_{1}x^{\prime }+k_{2}y^{\prime })}\,\text{d}k_{1}\,\text{d}k_{2}.\end{eqnarray}$$

Generally, for a given frequency ${\it\omega}$ , $k_{1}$ can have different values (Amiet Reference Amiet1976b ). However, experiments (Willmarth Reference Willmarth1959) have shown that $P_{i}({\it\omega},k_{1},k_{2})$ peaks in the vicinity of $k_{1}={\it\omega}/U_{c}$ , where $U_{c}$ denotes the convection velocity of the wall pressure gusts and is only a weak function of ${\it\omega}$ . Hence, upon defining $P_{i}({\it\omega},k_{2})=\int _{-\infty }^{\infty }\widetilde{P}_{int}({\it\omega},k_{1},k_{2})\,\text{d}k_{1}$ , (2.48) reduces to

(2.49) $$\begin{eqnarray}P_{int}({\it\omega},x^{\prime },y^{\prime })=\int _{-\infty }^{\infty }P_{i}({\it\omega},k_{2})\text{e}^{\text{i}(\bar{k}_{1}x^{\prime }+k_{2}y^{\prime })}\,\text{d}k_{2},\end{eqnarray}$$

where $\bar{k}_{1}={\it\omega}/U_{c}$ .

As shown in the preceding section, a wall pressure gust of

(2.50) $$\begin{eqnarray}P_{i}({\it\omega},k_{2})\text{e}^{\text{i}(k_{1}x^{\prime }+k_{2}y^{\prime })}\end{eqnarray}$$

will induce a far-field sound pressure

(2.51) $$\begin{eqnarray}\left(\frac{-\text{i}{\it\omega}x_{3}c}{4{\rm\pi}c_{0}S_{0}^{2}}\right){\it\lambda}\frac{\sin ((N+1/2){\it\lambda}(k_{2}-kx_{2}/S_{0}))}{\sin ((k_{2}-kx_{2}/S_{0}){\it\lambda}/2)}{\mathcal{L}}({\it\omega},k_{1},k_{2})P_{i}({\it\omega},k_{2}).\end{eqnarray}$$

Thus, the wall pressure defined by (2.49) will induce a far-field sound pressure of

(2.52) $$\begin{eqnarray}p_{f}(\boldsymbol{x},{\it\omega})=\left(\frac{-\text{i}{\it\omega}x_{3}c}{4{\rm\pi}c_{0}S_{0}^{2}}\right)\int _{-\infty }^{\infty }{\it\lambda}\frac{\sin ((N+1/2){\it\lambda}(k_{2}-kx_{2}/S_{0}))}{\sin ((k_{2}-kx_{2}/S_{0}){\it\lambda}/2)}{\mathcal{L}}({\it\omega},\bar{k}_{1},k_{2})P_{i}({\it\omega},k_{2})\,\text{d}k_{2}.\end{eqnarray}$$

The power spectral density (PSD) of the far-field sound is given by

(2.53) $$\begin{eqnarray}S_{pp}(\boldsymbol{x},{\it\omega})=\lim _{T\rightarrow \infty }\left(\frac{{\rm\pi}}{T}\langle p_{f}(\boldsymbol{x},{\it\omega})p_{f}^{\ast }(\boldsymbol{x},{\it\omega})\rangle \right),\end{eqnarray}$$

where the asterisk denotes a complex conjugate, and $2T$ is the time length used to obtain $p_{f}(\boldsymbol{x},{\it\omega})$ by performing Fourier transformation. Substitution of (2.52) into (2.53) yields

(2.54) $$\begin{eqnarray}S_{pp}(\boldsymbol{x},{\it\omega})=\left(\frac{{\it\omega}x_{3}c}{4{\rm\pi}c_{0}S_{0}^{2}}\right)^{2}\int _{-\infty }^{\infty }{\it\lambda}^{2}\left(\frac{\sin ((N+1/2){\it\lambda}(k_{2}-kx_{2}/S_{0}))}{\sin ((k_{2}-kx_{2}/S_{0}){\it\lambda}/2)}\right)^{2}\left|{\mathcal{L}}\right|^{2}{\it\Pi}({\it\omega},k_{2})\,\text{d}k_{2},\end{eqnarray}$$

where ${\it\Pi}({\it\omega},k_{2})$ is the wavenumber spectral density (Amiet Reference Amiet1975) of the hypothetical wall pressure beneath the turbulent boundary layer on the plate surface. For very wide serrations, i.e. $h\approx 0$ , (2.54) reduces to Amiet’s model (Amiet Reference Amiet1976b ). Equation (2.54) can be simplified by assuming a very large span, i.e. the number of serrations $(2N+1)$ is sufficiently large. Using the equation

(2.55) $$\begin{eqnarray}\lim _{N\rightarrow \infty }{\it\lambda}^{2}\frac{\sin ^{2}((N+1/2){\it\lambda}(k_{2}-kx_{2}/S_{0}))}{\sin ^{2}((k_{2}-kx_{2}/S_{0}){\it\lambda}/2)}\sim 2{\rm\pi}d\mathop{\sum }_{m=-\infty }^{\infty }{\it\delta}(k_{2}-kx_{2}/S_{0}+2m{\rm\pi}/{\it\lambda}),\end{eqnarray}$$

where ${\it\delta}(x)$ is the conventional generalized function defined in § 2, one can show that the PSD of the far-field sound in the plane $y^{\prime }=0$ is given by

(2.56) $$\begin{eqnarray}S_{pp}(\boldsymbol{x},{\it\omega})=\left(\frac{{\it\omega}x_{3}c}{4{\rm\pi}c_{0}S_{0}^{2}}\right)^{2}2{\rm\pi}d\mathop{\sum }_{m=-\infty }^{\infty }\left|{\mathcal{L}}({\it\omega},\bar{k}_{1},2m{\rm\pi}/{\it\lambda})\right|^{2}{\it\Pi}({\it\omega},2m{\rm\pi}/{\it\lambda}).\end{eqnarray}$$

Equation (2.56) is the fundamental result of this paper, and it is interesting to note that the infinite series in (2.56) appears to be similar to that in Howe’s model shown in (4.2). For example, both results show that the PSD of far-field sound is related to the wavenumber spectral density of the surface pressure through ${\it\Pi}({\it\omega},2m{\rm\pi}/{\it\lambda})$ ; therefore, a skewed wall pressure gust with $k_{2}=2m{\rm\pi}/{\it\lambda}$ plays an important role in sound generation.

2.4 Discussion of the effects of serration geometry

The complicated formulation of the far-field noise (2.56) and the response function (2.43) makes it very difficult to assess the effectiveness of serrations without numerical evaluation of the equations. This section attempts to derive two simple conditions for serrations to obtain effective noise reduction. In order to achieve significant sound reduction, we wish to minimize (2.56). Since $|{\mathcal{L}}({\it\omega},\bar{k}_{1},2m{\rm\pi}/{\it\lambda})|^{2}$ is very complex, we will perform an order analysis first.

Careful examination of (2.43) shows that $|{\mathcal{L}}|^{2}$ is proportional to $1/|{\it\kappa}_{n^{\prime }j}|^{2}$ . For illustration purposes, we assume that the observer is at $90^{\circ }$ above the trailing edge in the mid-span plane, i.e. $x_{1}=0$ and $x_{2}=0$ , and that the Mach number is low, e.g. $M_{0}<0.2$ . Then, ${\it\kappa}_{mj}\approx 0$ when $m$ satisfies $k_{2}+2m{\rm\pi}/{\it\lambda}\approx 0$ , and thus the value of $|{\mathcal{L}}({\it\omega},\bar{k}_{1},-2m{\rm\pi}/{\it\lambda})|^{2}$ is dominated by mode $m$ , i.e.

(2.57) $$\begin{eqnarray}|{\mathcal{L}}({\it\omega},\bar{k}_{1},-2m{\rm\pi}/{\it\lambda})|^{2}\approx \frac{2}{{\it\lambda}^{2}c^{2}}\left|a_{m}Q_{mm}+{\it\Theta}_{m}^{(1)}+{\it\Theta}_{m}^{(2)}+\,\cdots \,\right|^{2}.\end{eqnarray}$$

Furthermore, noting that $v_{nm}=4h/({\rm\pi}^{2}(n-m)^{2})$ when $n-m$ is odd, (2.44) suggests that ${\it\Theta}_{m}^{(i)}$ may be roughly approximated by only summing over modes $m-1$ and $m+1$ , since higher orders $m\pm (2j+1)$ with $j\geqslant 1$ are at least one order of magnitude smaller due to the quadratic term in the denominator of $v_{nm}$ . Using this approximation, ${\it\Theta}_{m}^{(i)}$ varies linearly with $a_{m-1}$ and $a_{m+1}$ . From the definition of $a_{m}$ in (2.20), it can be shown that

(2.58) $$\begin{eqnarray}a_{m}=\frac{\text{e}^{\text{i}m{\rm\pi}/2}}{2}\text{sinc}(k_{1}h-m{\rm\pi}/2)+\frac{\text{e}^{-\text{i}m{\rm\pi}/2}}{2}\text{sinc}(k_{1}h+m{\rm\pi}/2).\end{eqnarray}$$

Therefore, $|a_{m-1}|$ and $|a_{m+1}|$ are of the order of $|a_{m}|$ , and $|{\it\Theta}_{m}^{(i)}|=O(|a_{m}|)$ . From (2.57), we hence have $|{\mathcal{L}}({\it\omega},\bar{k}_{1},-2m{\rm\pi}/{\it\lambda})|^{2}=O(|a_{m}|^{2})$ , and

(2.59) $$\begin{eqnarray}\mathop{\sum }_{m=-\infty }^{\infty }|{\mathcal{L}}({\it\omega},\bar{k}_{1},2m{\rm\pi}/{\it\lambda})|^{2}{\it\Pi}({\it\omega},2m{\rm\pi}/{\it\lambda})=O\left(\mathop{\sum }_{m=-\infty }^{\infty }|a_{m}|^{2}{\it\Pi}({\it\omega},2m{\rm\pi}/{\it\lambda})\right).\end{eqnarray}$$

We are now in a position to discuss the conditions for minimizing (2.59). It is clear from (2.58) that $|a_{m}|$ is maximum when $m\approx \pm {\it\nu}_{0}$ , where ${\it\nu}_{0}=2k_{1}h/{\rm\pi}$ . To minimize the right-hand side in (2.59), we therefore require that ${\it\Pi}({\it\omega},2m{\rm\pi}/{\it\lambda})\ll {\it\Pi}({\it\omega},0)$ when $m$ approaches $\pm {\it\nu}_{0}$ . Assuming frozen turbulence, ${\it\Pi}({\it\omega},2m{\rm\pi}/{\it\lambda})$ is given by

(2.60) $$\begin{eqnarray}{\it\Pi}\left({\it\omega},\frac{2{\rm\pi}m}{{\it\lambda}}\right)=\frac{1}{2{\rm\pi}}\int _{-\infty }^{\infty }S_{qq}({\it\omega},y^{\prime })\text{e}^{-\text{i}(2{\rm\pi}m/{\it\lambda})y^{\prime }}\,\text{d}y^{\prime }.\end{eqnarray}$$

When $2{\rm\pi}{\it\nu}_{0}l_{y^{\prime }}/{\it\lambda}=k_{1}l_{y^{\prime }}{\it\sigma}\gg 1$ , the integrand in (2.60) for $m$ close to (or larger than) ${\it\nu}_{0}$ oscillates rapidly within the length scale $l_{y^{\prime }}$ of $S_{qq}({\it\omega},y^{\prime })$ , which corresponds to the spanwise correlation length given by

(2.61) $$\begin{eqnarray}l_{y^{\prime }}({\it\omega})=\frac{1}{S_{qq}({\it\omega},0)}\int _{-\infty }^{\infty }S_{qq}({\it\omega},y^{\prime })\,\text{d}y^{\prime }.\end{eqnarray}$$

The integral in (2.60) therefore evaluates to a small value compared with ${\it\Pi}({\it\omega},0)$ . Thus, a condition for noise reduction is that $k_{1}h_{e}\gg 1$ , where we have defined an effective root-to-tip amplitude $2h_{e}={\it\sigma}l_{y^{\prime }}$ that describes the correlated serration amplitude.

As $2k_{1}h_{e}=k_{1}{\it\sigma}l_{y^{\prime }}({\it\omega})$ , the decay rate of the spanwise correlation length $l_{y^{\prime }}({\it\omega})$ with respect to frequency is critical. If one makes use of Corcos’s correlation length model (Corcos Reference Corcos1964), $l_{y^{\prime }}({\it\omega})\approx 2.1U_{c}/{\it\omega}$ , $k_{1}h_{e}$ will reduce to a constant, in this case $2.1{\it\sigma}$ , a sole function of the serration sharpness factor and independent of frequency. This is consistent with the findings of Howe (Reference Howe1991b ). However, if the decay rate were faster than that given by Corcos, no sound reduction or even some sound increase would occur at high frequencies. An accurate description of the characteristics of the surface pressure fluctuation beneath a boundary layer is therefore critical for the model to accurately predict the sound reduction at high frequencies.

It should be noted that the condition $k_{1}h_{e}\gg 1$ is only a necessary condition because when $k_{1}h\rightarrow 0$ there is no noise reduction. This can be seen from (2.59), since $a_{m}=0$ when $k_{1}h\rightarrow 0$ except when $m=0$ , so the right-hand side in (2.59) reduces to ${\it\Pi}({\it\omega},0)$ , which corresponds to the straight-edge case. From (2.58), for a given integer $m$ away from ${\it\nu}_{0}$ , $|a_{m}|$ tends to zero when $k_{1}h\gg 1$ . This provides another necessary condition for noise reduction. Physically, the root-to-tip amplitude of the serrations must be sufficient for them to be seen by the incoming hydrodynamic waves.

We have thus obtained two necessary conditions for noise reduction, $k_{1}h_{e}\gg 1$ and $k_{1}h\gg 1$ , which are consistent with those proposed by Howe. These conditions will be further investigated in § 5.