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Homework Statement
So using the D'Alembert solution, I know the solution of the wave equation is of the form:
y(x,t) = f(x-ct) + g(x+ct)
I'm told that at t=0 the displacement of an infinitely long string is defined as y(x,t) = sin (pi x/a) in the range -a<= x <= a
and y =0 otherwise.
The string is initially at rest.
I'm told that the waves move along the string with speed c and told to sketch the displacement of the string at t=0, t=a/2c and t=a/c
Homework Equations
The Attempt at a Solution
So substituting t=0 into the d'alembert solution gives
f(x) + g(x) = sin pix/a
similarly since the string is initially at rest, we can calculate that f(x) - g(x) = const. therefore f(x) = 1/2sin pix/a + k where k is some const. and g(x) = 1/2 sin pix/a - k
So is the full solution y(x,t) = 1/2 [ sin (pi(x-ct)/a) + sin(pi(x+ct)/a) ] ? Isn't this a stationary wave..? I am not sure how to sketch for t = a/2c etc...thanks :)