A wave propagates on a string in the positive x-direction at a velocity v. The shape of the string at t = t0 is given by g(x, t0) = A sin (xa) Write the wave equation for a general time t.
A sin[x−v(t−t0)a]
Here given is g(x,t0)=A sin(xa) at t = t0
Comparing with the general equation g(x,t)=A sin(kx−ωt+ϕ)
We get at t = t0
k=1a
also−ωt0+ϕ=0
⇒ ϕ=ωt0
We also known that k=ωv
⇒ω=va
so we get
g(x,t)=A sin(xa−vta+vt0a)
g(x,t)=A sin[xa−vta+vt0a]
=A sin(x−v(t−t0)a)