For zero forcing, the given equation for $v_y$ is
$$
\left(\frac{D^2}{Dt^2}+\kappa_r^2-v_s^2\nabla^2\right)v_y=0.
$$
Use the <shearing-wave ansatz>
$$
v_y=\Re\left\{\widetilde v(t)
e^{i[k_x(t)x+k_yy]}\right\}.
$$
The explicit $x$ dependence cancels from the <material derivative> when
$$
\boxed{\dot k_x=Sk_y,
\qquad k_x(t)=k_x(0)+Sk_yt}.
$$
The amplitude then obeys the time-dependent <harmonic oscillator> equation
$$
\boxed{\ddot{\widetilde v}+g(t)\widetilde v=0,
\qquad
g(t)=\kappa_r^2+v_s^2[k_x(t)^2+k_y^2]}.
$$
This evolving <wavevector> is the characteristic signature of a <shearing wave>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 2 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-321/2/c/solution