Omega Unknown
Omega Unknown
Difficult Wave Question!!! Please Help?
The wavefunction of a transverse sinusoidal wave on a string has the form y(x,t) = A*cos(kx +omega*t + phi), where x and y are in m, t is in s and phi is the phase constant in radians. The wave has a period T = 23.4 ms and travels in the negative x direction with a speed of 31.6 m/s. At t = 0, a particle on the string at x = 0 has a displacement of 1.85 cm and travels downward with a speed of 2.00 m/s. What is the amplitude of the wave?
Hint: You are given the transverse displacement and transverse velocity at x=0 and t=0. You can use this information to write 2 equations in which the unknowns are the amplitude and the phase constant. You may need to use a trig identity to separate the unknowns.
What is the initial phase constant, phi?
What is the maximum transverse speed of the string?
I'm going to write the equation in symbols;
Y = ACos(kx + wt + b)
You will need the velocity equation, differentiate wrt, t
V = -AwSin(kx + wt + b)
Now w is the circular frequency, related to the period as, w = 2(pi)f = 2(pi)/T , so it is known (since you know T)
You are now given ; Y = .0185 & V = -2 at (x,t) = (0,0)
Substitute these initial conditions into the Y & V equations;
.0185 = ACos(b)
-2 = -AwSin(b)
multiply both sides of (1) by w, then square both eqs and add;
w^2(.0185)^2 + 4 = (Aw)^2[(Cos(b))^2 + (Sin(b))^2]
= (Aw)^2
So;
A^2 = (.0185)^2 + 4/w^2
once you find A, get b from above, ArcCos(b) = .0185/A
Max V occurs when Sin(kx + wt +b) =1 , then Vmax = Aw
You can finish it.
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