For every $\lambda>0$, $\tanh\lambda<\lambda$, so
$$
\lambda\coth\lambda>1.
$$
The radicand in part f is therefore negative precisely when its second factor is negative. The two wave speeds are then complex conjugates, one with positive imaginary part and exponential growth. Hence
$$
\boxed{\text{instability occurs exactly when }
\lambda\tanh\lambda<1}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 331 / 3 / g / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-331/3/g/solution