1. The following expression is proposed as a solution of the wave equation:y(x, t) =Asin(kx+φ) cos(ωt),whereA,k,ω, andφare constants.a) Show that it is a solution of the wave equation, or more precisely,find the condition (involving some of the four constants) that en-sures it is a solution.b) Suppose that this expression represents the motion of a string thatis tied down at two ends (a standing wave). One end is atx= 0and the coordinateyis zero at that point for all times. What doesthis condition determine about any of the four constants?c) The other end of the string atx=Lis also tied down so thatyiszero for all times. What does this condition determine about anyof the four constants?Solution:a) The wave equations is on the front cover:∂2y∂x2=1v2∂2y∂t2∂y∂x=kAcos(kx+φ) cos(ωt);∂2y∂x2=-k2Asin(kx+φ) cos(ωt) =-k2y(x, t)∂y∂t=-ωAsin(kx+φ) sin(ωt);∂2y
This is the end of the preview.
access the rest of the document.