Two transverse sinusoidal waves combining in a medium are described by the wave

Question

Two transverse sinusoidal waves combining in a medium are described by the wave functions

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 = 0.260 cm
|ymax| = ___cm

(b) x = 0.380 cm
|ymax| = ___cm

(c) x = 1.70 cm
|ymax| = ___cm

(d) Find the three smallest values of x corresponding to antinodes. (Enter your answers from smallest to largest.)

Explanation / Answer

The combined wave function is: y = y1 + y2

Using: sin(a ± b) = sin(a) cos(b) ± cos(a) sin(b)

we get for the combined wave:

y = 2 sin(pi x) cos ( 0.7 pi t)

a) To find the maximum transverse position of an element of the medium; we start by setting cos(0.700 pi t) = +1, we then get at x = 0.26 cm:

y_max = 2 sin ( 0.26 pi) = 1.458 cm

(b) At x = 0.38 cm:

y_max = 1.86 cm

c)

At x = 1.70 cm

we take cos(0.700t) = 1:

y_max = 1.62 cms

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