Final answer:
The student's question from Physics concerns the study of waves on a string with given initial conditions. The standard technique to find the wave's displacement function involves mathematical methods such as separation of variables or Fourier series. Without the full context or formula, we cannot provide a detailed solution or graph.
Step-by-step explanation:
The question at hand appears to be from Physics, likely at the College level and pertains to the study of waves and vibrations in a string. The initial conditions for the problem are a string of length L = 1 and c² = 1. The initial velocity of the string is zero, and its initial deflection is given by a small constant k multiplied by a sinusoidal function of the position. This type of problem is typically solved using techniques from the study of partial differential equations and boundary value problems, specifically, the wave equation in one dimension which is derived from Newton's second law applied to the oscillations of the string.
To find u(x, t), the solution of the wave equation with the given initial conditions, one would use the method of separation of variables or the Fourier series expansion. The approach involves expressing the initial shape of the string as a sum of sinusoidal modes, each of which evolves in time according to the wave equation. For a zero initial velocity, each mode's time dependence would be a cosine function of time. Once the time evolution of each mode is determined, these can be added back together to find the complete solution u(x, t).
Unfortunately, without a complete formula or more context from the text (such as Fig. 291 mentioned in the student's request), it is not possible to provide a step-by-step solution to this problem or to sketch the graph of u(x, t). It is recommended that the student refers back to their coursework and textbook for the specific methods used in these calculations, as well as graphical examples that might be similar to Fig. 291.