B.2.1 The Blocked Forces Interpretation

The following term was described in Section B.2 as the blocked force

 (B.36)

It is worth illustrating this fact in more detail. The incoming wave generates the force ${\mathit{F}}_{\text{in}}^{\prime }$ given by the boundary conditions of the edge and an edge motion ${\mathit{q}}_{e,\mathrm{i}\mathrm{n}}$. In order to achieve the bocked boundary condition, a force must act in such a way on the boundary that it excites an out-going that matches exactly the incoming wave with negative sign, thus:

 (B.37)

This force is given by

 (B.37)

So in total, the edge must act with force $\left\{\begin{array}{c}{\mathit{F}}_{\text{in}}^{\prime }\end{array}\right\}-\left[\begin{array}{c}{\mathit{D}}_{\text{dir}}^{\prime }\end{array}\right]\left\{\begin{array}{c}{\mathit{q}}_{e,\mathrm{i}\mathrm{n}}\end{array}\right\}$ in order to block the edge motion of an incoming wave. From the force balance follows that the incoming and reflected wave acts with the force as shown in Equation (B.26).

There are further consequences. First, for the calculation of waves that are reflected in one sense but transmitted into a different wave type but the same plate, the reflected blocked force components must by subtracted. The easiest way to do so is to subtract ${\mathit{q}}_{e,\mathrm{i}\mathrm{n}}$ from the global solution in Equation (B.28). Second, for the realization of the blocked force of one wave type, the in-plane wave requires both in-plane waves. Thus, there is an inert exchange of energy between both subystems that breaks the rule for the diffuse field reciprocity.

In the remainder of Section B.2, the blocked forces for each wave type are derived in detail. According to the diffuse field reciprocity (7.19), this blocked force is also given by the free field radiation stiffness of each wave type. This separation into waves is clear for bending waves, but not for the in-plane waves where the edge motion usually generates a combination of both wave types, shear and longitudinal waves.

B.2.2 Bending Waves

The wave solution for incoming bending waves relies on the positive and real wavenumber solution $k_B$ and ${\mu }_{\text{ in},B2}=+\sqrt{k_x^2-k_B^2}=-{\mu }_{B2}$. Using this and setting the evanescent wave ${\mathbf{\Psi }}_{B1}=0$, the incoming wave reads as

 (B.39)

 (B.40)

With $y=0$ we get the edge displacement:

 (B.41)

and for the traction force vector

 (B.42)

The reaction to the edge displacement follows from the radiation stiffness times the edge motion

 (B.43)

Using these equations in (B.36) gives the blocked reverberant force due to bending waves

 (B.44)

For further consideration it is better to replace the propagation constants ${\mu }_{B1/2}$ by wavenumber expressions

 (B.45)

Using the this convention

 (B.46)

leads to the cross spectral density matrix:

 (B.47)

This cross spectrum is per square length, as the force is a specific force per length.

For the consideration of the reciprocity, the modal density of plates and expression for the total energy is required. As explained in section 8.2.4.1, the plane wave formulation with the projected wavenumber as degree of freedom has to be treated as a one-dimensional system.

 (B.48)

From Figure B.4 it follows that the effective length $L$ follows from $L_y$ and the wavenumber ratio.

 (B.49)

Together with $c_{\text{gr}}=2c_B$, we get the modal density

 (B.50)

The energy in the plate subsystem is given by

 (B.51)

In this case the energy length density is required, thus

 (B.52)

For the determination of the radiation stiffness, the same convention of the propagation constants (B.45) is used

 (B.53)

With the modified length specific diffuse field reciprocity (7.19) we get finally:

This proves the diffuse field reciprocity in case of bending waves of homogeneous plates.

B.2.3 Longitudinal Waves

The wave solution for incoming longitudinal waves relies on the positive solution ${\mu }_{\text{in},L}=+\sqrt{k_x^2-k_L^2}=-{\mu }_L$. Using this the incoming wave reads as:

 (B.54)

 (B.55)

with $y=0$ we get the edge displacement

 (B.56)

and for the traction force vector

 (B.57)

The reaction to the edge displacement follows from the radiation stiffness times the edge motion

 (B.58)

Using these equations in (B.36) gives the blocked force due to longitudinal waves

 (B.59)

For further consideration the propagation constants ${\mu }_{L/S}$ are replaced by wavenumber expressions

 (B.60)

Using this convention

 (B.61)

leads to the cross spectral density matrix

 (B.62)

With the effective length

 (B.63)

and the fact that for $L$ and $S$-waves, the group velocity equals the phase velocity $c_{\text{gr}}=c_L$, we get the modal density

 (B.64)

The energy in the plate subsystem is given by

 (B.65)

Note that the amplitude for in-plane waves is given as a product including the wavenumber. So, the longitudinal wave amplitude displacement is given by ${\hat{\mathrm{\Psi }}}_L{k}_L$. The energy length density reads as

 (B.66)

The diffuse field reciprocity requires the following expression to relate the radiation stiffness to the cross spectrum

In the above equation the mass per area can be replaced using the expression for the wavenumber (3.167)

With the identity $k_s^2=2k_L^2/(1-\nu )$ we get

 (B.67)

With the modified length specific diffuse field reciprocity (7.19) we get finally

 (B.68)

This matrix can be used as imaginary of the radiation stiffness, but this is not derived from diffuse field reciprocity. The radiation stiffness is defined by a coupled motion of both waves, making it impossible to identify a pure longitudinal wave radiation.

B.2.4 Shear Waves

Following the similar procedure as in section B.2.3 with an incoming shear wave, the blocked force reads

 (B.69)

As illustrated in Figure B.2, there is an angle or wavenumber range where both propagation constants are complex, but also a range where ${\mu }_L$ is real and only ${\mu }_S$ remains imaginary. For $k_x\le k_L$ we use (B.60)

 (B.70)

giving the following blocked force matrix

 (B.71)

Applying shear wave quantities to the one-dimensional modal density and energy gives

 (B.72)

Using ${\omega }^2{\rho }_{0}h=k_S^{2}S$ we get

 (B.73)

In the wavenumber range $k_L<k_x\le k_S$ only $k_S$ remains real, and we keep the longitudinal propagation constant

 (B.74)

The cross correlation provides

 (B.75)

We note that the diffuse field reciprocity is impossible to fulfill here, because the matrix contains imaginary components. However, if we ignore this fact, and with the diffuse field reciprocity factor (B.72), we get

 (B.76)

B.2.5 In-plane Waves

Both waves can be combined by adding both radiation imaginary parts of the radiation stiffnesses (B.68) and (B.73)

 (B.77)

In contrast to this the imaginary of the in-plane stiffness matrix (8.121) reads

 (B.78)

So, except the off diagonal components, the blocked force cross correlation is equal to the imaginary radiation stiffness. This equation is valid for $k_x\le k_L$; for $k_L<k_x\le k_S$, Equation (B.76) must be used as there is no propagating longitudinal wave. For the comparison to the combined in-plane stiffness matrix (8.121), we use ${\mu }_S=-j{k}_{yS}$ and keep ${\mu }_L$ getting

 (B.79)

Extension of the denominator and extracting the imaginary part gives

 (B.80)

Besides the off diagonals the equation corresponds also to the result from the blocked forces cross correlation (B.76). Thus, the radiation stiffness including both in-plane waves can be used for the calculation of the cross correlation function from the diffuse field reciprocity.

B.3.1 Derivation of Stiffness Matrix from Transfer Matrix

The stiffness matrix of the noise control treatment can be calculated from the transfer matrix of the infinite layer (9.96)

due to the discussions in section 9.1.1. Reordering the variables links the transfer matrix to the impedance matrix by:

 (B.81)

With ${{\mathit{v}}_z}_i=j\omega {\mathit{u}}_i$ the final transformation to the stiffness matrix reads as:

 (B.82)