In one dimension, reflection of an incident wave from a fixed end of the string represents the simplest type of the wave-boundary interaction and is governed by the following boundary condition $w(0,t)=0$ where $w(x,t)$ is the transversal displacement.
Let us use the
method of the images to solve the problem. Consider a displacement pulse $f(x-ct)$ propagating in the positive $x$−direction towards the fixation
point as it is depicted in the figure below. Imagine now that the boundary at $x = 0$ is
removed and the string is extended towards positive infinity. Let us consider this
infinite string and in addition to the incident pulse $f(x-ct)$, introduce an “image”
pulse so that the superposition of these two pulses would satisfy the above boundary condition. It is easy to understand that to reach this goal, the “image” pulse
should be introduced symmetrically (with respect to $x=0$) to the incident pulse, be
opposite in sign to the incident pulse and propagate in the negative $x$−direction with
the wave speed $c$ as shown in the figure.
Source: A. Metrikine and A. Vrouwenvelder, Dynamics of Structures CT4140 Wave Dynamics, Delft University of Technology