  In our manuscript, we develop a new approach for stability analysis of
one-dimensional wave equation with time delay. The major contribution of our
work is to develop a new method for spectral analysis. We derive sufficient and
necessary conditions for the feedback gain and time delay which guarantee the
exponential stability of the closed-loop system. Comparing with similar
conditions developed in the past literatures, we discuss all the situation when
the time delay is positive, including when it is irrational. We prove that the
exponential stability can be achieved if and only if the time delay is an even
number. We also get the general formula term of the stability region of the
coupling gain for different even multiples of time delay, and from this we
easily obtain the shrink of the stability region as time delay increases. In
addition, we explore the impact of slight perturbations in time delay on high
frequency robustness.

Analysis of PDEs