Assume that a guided wave excited in the straight part of a pipe propagates through a pipe bend, as shown in Fig. 4. For one pipe bend, there are two interfaces on the propagating path, as marked by \(z{'_1}\) and \(z{'_2}\) in Fig. 4. Complicated mode conversions occur at these interfaces, scattering different modes of the guided wave and causing significant confusion with the testing signals.

**A. Basic principles for theoretical derivation**

For each interface, the displacement and stress fields over the cross section should be consistent, i.e.,

\({\left. {{{\mathbf{V}}_i}} \right|_{z'=z{'_i}}}={\left. {{{\mathbf{V}}_{i+1}}} \right|_{z'=z{'_i}}}\) , \({\left. {{{\mathbf{T}}_i}} \right|_{z'=z{'_i}}}={\left. {{{\mathbf{T}}_{i+1}}} \right|_{z'=z{'_i}}}\), (\(i=1,2\)), (18)

where the subscript *i* denotes the *i*th section of the pipe shown in Fig. 4. According to the NME method, the mode structures at the interfaces can also be expanded with normal modes in either part, i.e.,

$${\left. {\mathbf{V}} \right|_{z'=z{'_i}}}=\sum\limits_{n} {\left( {{a_{s,n}}{{{\mathbf{\bar {\psi }}}}_{{k_{s,n}}}}} \right)=} \sum\limits_{n} {\left( {{a_{c,n}}{{{\mathbf{\bar {\psi }}}}_{{k_{c,n}}}}} \right)}$$

19,

$${\left. {\mathbf{T}} \right|_{z'=z{'_i}}}=\sum\limits_{n} {\left( {{b_{s,n}}{{{\mathbf{\bar {T}}}}_{{k_{s,n}}}}} \right)=} \sum\limits_{n} {\left( {{b_{c,n}}{{{\mathbf{\bar {T}}}}_{{k_{c,n}}}}} \right)}$$

20,

where *s* and *c* denote the straight pipe and curved pipe, respectively, *a* and *b* are the expansion coefficients of the normal modes, and \({\mathbf{\bar {T}}}\) is the normalized stress mode structure, which in the SAFE modeling is defined as

$${\mathbf{\bar {T}}}={\mathbf{D}}\left[ {L+{L_r}\frac{\partial }{{\partial r}}+{L_\theta }\frac{\partial }{{\partial \theta }}+{L_{z'}}\frac{\partial }{{\partial z'}}} \right]{\mathbf{\bar {\psi }}}$$

21.

Because the displacement–stress relationship [Eq. (2) or (21)] is nonlinear, the expansion coefficients of the displacement mode structures (*a**n*) are different from those of the stress mode structures (*b**n*). However, *b**n* can be calculated according to the displacement–stress relationship.

Based on the bi-orthogonality relationships of both straight pipes[25] and pipe bends, the expansion coefficients are calculated as

$${a_{s,n}}=\iint_{s} {\left( {{{\left. {{{\mathbf{V}}^*}} \right|}_{z'=z{'_i}}} \cdot {{{\mathbf{\bar {T}}}}_{{k_{s,n}}}}+j\omega \cdot {{{\mathbf{\bar {\psi }}}}_{{k_{s,n}}}} \cdot {{\left. {{{\mathbf{T}}^*}} \right|}_{z'=z{'_i}}}} \right)} \cdot {\hat {e}_{z'}}ds$$

22,

$${a_{c,n}}=\iint_{s} {\left( {{{\left. {{{\mathbf{V}}^*}} \right|}_{z'=z{'_i}}} \cdot {{{\mathbf{\bar {T}}}}_{{k_{c,n}}}}+j\omega \cdot {{{\mathbf{\bar {\psi }}}}_{{k_{c,n}}}} \cdot {{\left. {{{\mathbf{T}}^*}} \right|}_{z'=z{'_i}}}} \right)} \cdot {\hat {e}_{z'}}ds$$

23.

Note that because the normal modes are essentially the solutions to the governing equation of wave motions in waveguides, the normal modes cannot satisfy two different governing equations for different waveguides simultaneously, indicating that the displacement field of the interface cannot be expressed by modes in straight pipes and pipe bends simultaneously. This is to say, the displacement and stress field continuity principle does not hold in the NME framework. However, the NME method still reveals the inherent connections between the modes in straight pipes and pipe bends, and it gives valuable information about the mode conversions at pipe bends. Therefore, the displacement and stress field continuity principle is assumed to hold in the following derivation.

**B. Scattering study for first interface**

At interface \(z{'_1}\), each mode in straight section 1 can also be expanded with normal modes in curved section 2, i.e.,

$${{\mathbf{\bar {\psi }}}_{{k_{s,l}}}}=\sum\limits_{m} {\left( {{a_{lm}}{{{\mathbf{\bar {\psi }}}}_{{k_{c,m}}}}} \right)}$$

24,

$${a_{lm}}=\iint_{s} {\left( {{{{\mathbf{\bar {\psi }}}}^*}_{{{k_{s,l}}}} \cdot {{{\mathbf{\bar {T}}}}_{{k_{c,m}}}}+j\omega \cdot {{{\mathbf{\bar {\psi }}}}_{{k_{c,m}}}} \cdot {{{\mathbf{\bar {T}}}}^*}_{{{k_{s,l}}}}} \right)} \cdot {\hat {e}_{z'}}ds$$

25.

Then, by combining Eqs. (19) and (24) and considering the continuity principle of Eq. (18), we obtain

$$\begin{gathered} {\left. {\mathbf{V}} \right|_{z'=z{'_1}}}=\sum\limits_{l} {\left( {{a_{s,l}}{{{\mathbf{\bar {\psi }}}}_{{k_{s,l}}}}} \right)=} \sum\limits_{l} {\left( {{a_{s,l}}\sum\limits_{m} {\left( {{a_{lm}}{{{\mathbf{\bar {\psi }}}}_{{k_{c,m}}}}} \right)} } \right)} \\ =\sum\limits_{m} {\left( {\sum\limits_{l} {\left( {{a_{s,l}}{a_{lm}}} \right){{{\mathbf{\bar {\psi }}}}_{{k_{c,m}}}}} } \right)} =\sum\limits_{m} {\left( {{a_{c,m}}{{{\mathbf{\bar {\psi }}}}_{{k_{c,m}}}}} \right)} \\ \end{gathered}$$

26.

Equation (26) gives the relationship between the expansion coefficients as

$$\sum\limits_{l} {\left( {{a_{s,l}}{{{\mathbf{\bar {\psi }}}}_{{k_{s,l}}}}} \right)=} \sum\limits_{m} {\left( {\sum\limits_{l} {\left( {{a_{s,l}}{a_{lm}}} \right){{{\mathbf{\bar {\psi }}}}_{{k_{c,m}}}}} } \right)} =\sum\limits_{m} {\left( {{a_{c,m}}{{{\mathbf{\bar {\psi }}}}_{{k_{c,m}}}}} \right)}$$

27.,

which can be expressed in matrix form as

$${{\mathbf{A}}_m}={{\mathbf{A}}_l}{{\mathbf{G}}_{lm}}$$

28,

where \({{\mathbf{A}}_m}=\left( {a_{{c,1}}^{t},a_{{c,2}}^{t}, \cdots } \right)\), \({{\mathbf{A}}_l}=\left( {a_{{s,1}}^{i},a_{{s,2}}^{i}, \cdots ,a_{{s,1}}^{r},a_{{s,2}}^{r}, \cdots } \right)\), and \({{\mathbf{G}}_{lm}}\) is the transfer matrix defined as

$${{\mathbf{G}}_{lm}}=\left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1m}}}& \cdots \\ {{a_{21}}}&{{a_{22}}}& \cdots &{{a_{2m}}}& \cdots \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ {{a_{l1}}}&{{a_{l2}}}& \cdots &{{a_{lm}}}& \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots \end{array}} \right]$$

29.

All modes including the incident positive propagating modes, transmitting positive propagating modes, reflecting negative propagating modes, and non-propagating modes should be considered in the calculation. Therefore, the superscripts *i*, *r*, and *t* are introduced to denote the incident, reflecting, and transmitting modes, respectively.

Conversely, by expanding each mode in pipe section 2 with normal modes in pipe section 1 and following the same derivation procedure, we have

$${{\mathbf{A}}_l}={{\mathbf{A}}_m}{{\mathbf{G}}_{lm}}={{\mathbf{A}}_m}{{\mathbf{G}}_{lm}}'$$

30.

Combining Eqs. (29) and (30) gives

$${{\mathbf{A}}_l}={{\mathbf{A}}_m}{{\mathbf{G}}_{lm}}'={{\mathbf{A}}_l}{{\mathbf{G}}_{lm}}{{\mathbf{G}}_{lm}}'$$

31,

which implies that \({{\mathbf{A}}_l}\) is the eigenvector of \({{\mathbf{G}}_{lm}}{{\mathbf{G}}_{lm}}'\) with respect to the eigenvalue of one. Thus, by solving the eigenproblem of \({{\mathbf{G}}_{lm}}{{\mathbf{G}}_{lm}}'\), the expansion coefficients \({{\mathbf{A}}_l}\) of guided waves in pipe section 1 can be derived, and \({{\mathbf{A}}_m}\) can be calculated according to Eq. (28).

In practical inspections, usually a single mode is excited in pipe section 1. Then, by setting \(a_{{s,1}}^{i}\) in \({{\mathbf{A}}_l}\) to be one and calculating \({{\mathbf{A}}_l}\) and \({{\mathbf{A}}_m}\), the reflecting and transmitting coefficients of guided waves propagating across the first interface are derived.

Considered the acoustic field in either the straight or curved section as being linear, multimode incidence can be treated as multiple single-mode incidences, which can be done by calculating the scattering of each single-mode incidence separately and then linearly superposing these scattering acoustic fields.

**C. Scattering study for second interface**

Because multiple modes are scattered at the first interface, multimode incidence should be considered for the second interface. As mentioned before, multimode incidence is considered as multiple single-mode incidences. For each incident mode *j*, we have

$${{\mathbf{A}}_{n,j}}={{\mathbf{A}}_{m,j}}{{\mathbf{G}}_{mn,j}}$$

32,

$${{\mathbf{A}}_{m,j}}={{\mathbf{A}}_{m,j}}{{\mathbf{G}}_{mn,j}}{{\mathbf{G}}_{mn,j}}'$$

33,

where \({{\mathbf{A}}_{n,j}}=\left( {a_{{s,1}}^{{t,j}},a_{{s,2}}^{{t,j}}, \cdots } \right)\) are the expansion coefficients of normal modes in straight section 3, and \({{\mathbf{A}}_{m,j}}=\left( {a_{c}^{{i,j}},a_{{c,1}}^{{r,j}},a_{{c,2}}^{{r,j}}, \cdots } \right)\) are those in curved section 2. Also, by solving the eigenproblem of \({{\mathbf{G}}_{mn,j}}{{\mathbf{G}}_{mn,j}}'\), the transmission coefficients \({{\mathbf{A}}_{n,j}}\) and reflection coefficients \(\left( {a_{{c,1}}^{{r,j}},a_{{c,2}}^{{r,j}}, \cdots } \right)\) of the *j*th incident mode scattering at the \(z{'_2}\) interface are deduced.

By superposing all the scattering acoustic fields, the scattering at the \(z{'_2}\) interface is obtained. The reflection and transmission coefficients are

$${{\mathbf{A}}_n}=\sum\limits_{j} {a_{c}^{{i,j}}{{\mathbf{A}}_{n,j}}=} \sum\limits_{j} {a_{c}^{{i,j}}\left( {a_{{s,1}}^{{t,j}},a_{{s,2}}^{{t,j}}, \cdots } \right)=} \left( {a_{{s,1}}^{t},a_{{s,2}}^{t}, \cdots } \right)$$

34,

$${{\mathbf{A}}_m}=\sum\limits_{j} {a_{c}^{{i,j}}{{\mathbf{A}}_{m,j}}=} \sum\limits_{j} {a_{c}^{{i,j}}\left( {1,a_{{s,1}}^{{t,j}},a_{{s,2}}^{{t,j}}, \cdots } \right)=} \left( {a_{c}^{{i,1}},a_{c}^{{i,2}}, \cdots a_{{s,1}}^{t},a_{{s,2}}^{t}, \cdots } \right)$$

35.

The modes reflected at the \(z{'_2}\) interface then incident negatively on the \(z{'_1}\) interface, thereby enforcing the reflections of the latter. Because the reflections between the \(z{'_1}\) and \(z{'_2}\) interfaces are rather small in most cases, they are neglected for simplification.

Combining the scattering fields of the \(z{'_1}\) and \(z{'_2}\) interfaces gives the reflection and transmission coefficients (\({{\mathbf{A}}_l}\) and \({{\mathbf{A}}_n}\)) of guided waves traveling through the pipe bend.