Chapter 1, Problem 7P

A wave traveling along a string in the +x direction is given by

y₁(x, t) = A cos(ωt − βx),

where x = 0 is the end of the string, which is tied rigidly to a wall, as shown in Fig. P1.7.

Figure P1.7 Wave on a string tied to a wall at x = 0 (Problem 1.7).

When wave y₁(x, t) arrives at the wall, a reflected wave y₂(x, t) is generated. Hence, at any location on the string, the vertical displacement y_s is the sum of the incident and reflected waves:

y_s(x, t) = y₁(x, t) + y₂(x, t).

1. (a) Write an expression for y₂(x, t), keeping in mind its direction of travel and the fact that the end of the string cannot move.
2. (b) Generate plots of y₁(x, t), y₂(x, t) and y_s(x, t) versus x over the range −2λ ≤ x ≤ 0 at ωt = π/4 and at ωt = π/2.