Question 02 (a)
Consider the wave equation
with initial conditions
(a) Suppose the domain is infinite, Write down d'Alembert's solution, and sketch
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Start by writing down the d'Alembert's formula.
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
For the one dimensional wave equation
where and , the solution is given by d'Alembert's formula
and . Therefore, the solution is given by
To find , one imagines that at , the original wave form splits into two equal pieces but with half the original amplitude. Hence, in this case, each part will be like . One part will move towards the left at speed 1 and the other moves to the right at speed 1. At , one part has traveled units to the left and the other has traveled units to the right. The solution is the sum of those two parts.
Notice that in the above, the left traveling wave cancels with the right travelling wave in the interval from to .