Question

These two waves travel along the same string:

y₁ = (4.14 mm) sin(2.31πx - 430πt)

y₂ = (5.79 mm) sin(2.31πx - 430πt + 0.771πrad).

What are (a) the amplitude and (b) the phase angle (relative to wave 1) of the resultant wave? (c) If a third wave of amplitude 5.24 mm is also to be sent along the string in the same direction as the first two waves, what should be its phase angle in order to maximize the amplitude of the new resultant wave?

Answer 1

y = y1 + y2

y = 4.14 sin(2.31πx - 430πt) + (5.79 mm) sin(2.31πx - 430πt + 0.771πrad)

y = 4.14*sin(2.31πx - 430πt) + (5.79 mm) sin(2.31πx - 430πt )*cos0.771π + (5.79 mm) cos(2.31πx - 430πt )*sin0.771π

y = sin(2.31πx - 430πt)*[4.14 + (5.79 *cos0.771π) ] + cos(2.31πx - 430πt )* (5.79*sin0.771π )

R*costheta = 4.14 + (5.79 *cos0.771π)

R*sintheta = (5.79*sin0.771π )

y12 = R*sin(2.31πx - 430πt+theta)

R = sqrt(4.14^2+5.79^2+(2*4.14*5.79*cos(0.771pi))

R = 3.821 mm

+++++++++++++++++++++++++

phase angle = tan^-1((5.79*sin0.771π )/(4.14 + (5.79 *cos0.771π)))

phase angle = -1.514 rad