Let the continuous function $u(x, y)$ be defined for $x \geqslant 0$ and $0 \leqslant y \leqslant b$ by the boundary value problem
$$
\begin{array}{ll}
u_{x x}+u_{y y}+k^2 u=0 & (x>0,0<y<b), \\
u_y(x, 0)=0 & (x \geqslant 0), \\
u_y(x, b)=0 & (x>0), \\
u(0, y)=1 & (0 \leqslant y \leqslant a), \\
u_x(0, y)=0 & (a<y \leqslant b), \\
u \sim \tau \mathrm{e}^{i k x} & (x \rightarrow \infty) .
\end{array}
$$
Here $k$ is a given real number satisfying $0<k b<\pi$ and it is required to obtain $\tau \in \mathbb{C}$.
Use separation of variables to show that $u$ is of the form
$$
u(x, y)=\tau \mathrm{e}^{\mathrm{ik} x}+\sum_{n=1}^{\infty} a_n \mathrm{e}^{-\beta_n x} \cos (n \pi y / b)
$$
where $\beta_n=\left\{(n \pi / b)^2-k^2\right\}^{\frac{1}{2}}(n \in \mathbb{N})$. Deduce that the function $\phi(y)=-u_x(0, y) /$ $\left((1-\tau) b^{\frac{1}{4}}\right)(0 \leqslant y \leqslant a)$ satisfies the integral equation
$$
\phi_0(y)=\sum_{n=1}^{\infty} \beta_n^{-1} \phi_n(y) \int_0^a \phi_n(t) \phi(t) \mathrm{d} t \quad(0 \leqslant y \leqslant a)
$$
where $\phi_0(y)=b^{-\frac{1}{t}}$ and $\phi_n(y)=(2 / b)^{\ddagger} \cos (n \pi y / b) \quad(n \in \mathbb{N})$. Show also that $\alpha=-\mathrm{i} k \tau(1-\tau)^{-1}$ is given by
$$
\alpha=\int_0^a \phi_0(t) \phi(t) \mathrm{d} t
$$
By using (8.41) to show that the imaginary part of $\phi$ is zero, show that $\alpha$ is real. Suppose that upper and lower bounds for $\alpha$ are determined in the form $\alpha_1 \geqslant \alpha \geqslant \alpha_0 \geqslant 0$. Show that
$$
\alpha_0^2\left(\alpha_0^2+k^2\right)^{-1} \leqslant|\tau|^2 \leqslant \alpha_1^2\left(\alpha_1^2+k^2\right)^{-1}
$$
and that $\theta=\arg (\tau)$ is such that $\cos \theta=|\tau|$.
Consider the particular case in which $b=2 a=\pi$ and $k=\frac{1}{2}$. Use the upper and lower bounds given in the previous problem with the test functions $p_1(x)=1\left(0 \leqslant x \leqslant \frac{1}{2} \pi\right), q_1(x)=x-\frac{1}{2} \pi\left(\frac{1}{2} \pi \leqslant x \leqslant \pi\right)$ and show that $0.89 \leqslant|\tau| \leqslant 0.97$. (The sums appearing in the bounds can be evaluated to two decimal places using a calculator and an estimate of the error in truncating the series.) (The boundary value problem represents the scattering of a plane sound wave in a duct which is partially closed by a plane barrier. The quantity $\tau$ is the (complex) amplitude of the plane wave transmitted beyond the obstructing barrier; $|\tau|$ is the real amplitude of the transmitted wave and $\arg (\tau)$ its phase. The physical problem can be used to suggest more sophisticated test functions but these inevitably lead to bounds which are a good deal more difficult to evaluate.)