Question

# Two waves traveling in opposite directions on a stretched rope interfere to give the standing wave...

Two waves traveling in opposite directions on a stretched rope interfere to give the standing wave described by the following wave function:

y(x,t) = 4 sin⁡(2πx) cos⁡(120πt),

where, y is in centimetres, x is in meters, and t is in seconds. The rope is two meters long, L = 2 m, and is fixed at both ends.

In terms of the oscillation period, T, at which of the following times would all elements on the string have a zero vertical displacement, y(x,t) = 0, for the first time:

t = T/8

t = T/4

t = T/2

t = 3T/4

t = T

Other:

If the oscillation frequency is decreased by a factor of four, f_new = f/4, while keeping the tension force, the length of the rope, and the linear mass density constants, then how many loops would appear on the rope?

One loop

Two loops

Three loops

Four loops

No loops appear, because the conditions are not satisfied for the standing waves to exist.

Other:

The wave functions of two waves traveling in the same direction are given below. The two waves have the same frequency, wavelength, and amplitude, but they differ in their phase constant.

y1 (x,t) = 2 sin⁡(2πx ‒ 20πt), and

y2 (x,t) = 2 sin(2πx ‒ 20πt + φ),

where, y is in centimetres, x is in meters, and t is in seconds.

Which of the following wave functions represents the resultant wave due to the interference between the two waves:

y_result (x,t) = 4 sin⁡(2πx ‒ 20πt + φ)

y_result (x,t) = 2 sin⁡(πx ‒ 10πt + φ/2)

y_result (x,t) = 4 cos⁡(φ/2) cos⁡(πx ‒ 10πt - φ/2)

y_result (x,t) = 4 cos⁡(φ/2) sin⁡(2πx ‒ 20πt + φ/2)

y_result (x,t) = 2 sin⁡(φ/2) sin⁡(2πx ‒ 20πt + φ/2)

Other:

If the amplitude of the resultant wave is A_res = 2 cm, then which of the following values of φ is correct:

π/6

π/3

π/2

2π/3

π

Other:

The speed of the resultant wave due to the interference between y_1 and y_2 is:

80π cm/s

10 m/s

20 m/s

80π cos⁡(φ/2) cm/s

0.1 m/s

Other:

Assume that the two waves start to propagate at the same instant, t_0,1 = t_0,2 = 0 sec, but at different initial positions from an observation point. Wave-1 travels a distance x1 to get to the observation point, while wave-2 travels a distance x2. The amplitude of the resultant wave due to interference would be maximum at the observation point if the path difference between the two waves, ∆x=x2-x1, is:

π m

4 cm

1.5 m

0.5 m

1 m

Other:

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