The equation of a transverse wave traveling in a string is given by y(x,t) = 10 cos (π/2)(0.0050x - 8.0t + 0.57), in which x and y are expressed in centimeters and t in seconds. Write down the equation of a wave which, when added to the given one, would produce standing waves on the string.

Question

The problem is...I don't know what to do with that phase constant in the way. Like, I know that if the equation was just:

y(x,t) = 10 cos (π/2)(0.0050x - 8.0t)

I would just have to go through some really, really obnoxious trig to figure it out. But I am at a complete loss here. How do I even figure it out with that stupid phase constant around to muck things up?

Answer 1

add the two equations..
a) 10 cos (π/2)(0.0050x - 8.0t + 0.57)
b) 10 cos (π/2)(0.0050x - 8.0t+ PC) (your equation had PC at zero.

Now notice on both equations, the wavelength is the same, and the wave speed is the same. So lets make it simpler.
a) 10 cos (π/2)(A+.057)
b) 10 cos (π/2)(A+PC)

Remember the trig formula for sum of two cosines?cos u + cos v = 2 cos(½(u+v)) cos(½(u−v))

so the sum of the two waves is
20cos(1/2 (2A+PC+.057)cos(1/2(PC-.057))

The second term is a constant, which essentially van vary the amplitude of the prime first function. Notice what happens when PC-.057 is equal to Pi/2 or an odd multiple. In any event, the phase term only reduces the amplitude, event ot zero when PC is right. The first term gives standing waves.

Answer 2

But how does that even make sense?????? The equation for a standing wave is

y'(x,t) = 2Asin(kx)cos(omega * t)

Answer 3

And how do you go about finding the other phase constant?