Question

Two transverse sinusoidal waves combining in a medium are described by the wave functions y1 = 1.00 sin π(x + 0.500t) y2 = 1.00 sin π(x − 0.500t) where x, y1, and y2 are in centimeters and t is in seconds. Determine the maximum transverse position of an element of the medium at the following positions. (a) x = 0.130 cm |ymax| = ? (b) x = 0.460 cm |ymax| = ? (d) Find the three smallest values of x corresponding to antinodes. (Enter your answers from smallest to largest.) ?cm ?cm ?cm

Answer 1

The resultant wavefunction will be

y = y₁ + y₂

y = 1.00 sin π(x + 0.500t) + 1.00 sin π(x − 0.500t)

y = 1 (sin(πx) cos(0.5π t) + cos(πx) sin(0.5π t) ) + 1 (sin(πx) cos(0.5π t) - cos(πx) sin(0.5π t) )

y = 2 sin(πx) cos(0.5π t)

Maximum y for any position x will when cos(0.5π t) is maximum and maximum value of cos(0.5π t) is 1

y_max =  2 sin(πx)

a)x = 0.13 cm

y_max =  2 sin(π X 0.13)

y_max = 0.79 cm

b) x = 0.46 cm

y_max =  2 sin(π X 0.46)

y_max = 1.98 cm

c) Antinodes are points where amplitude is maximum so sin(πx) must be maximum i.e 1

sin(πx) = +- 1

x = 0.5 cm

x = 1.5 cm

x = 2.5 cm