Two transverse waves travel along the same taut string. Wave 1 is described by y1(x, t) = A sin(kx - ωt), while wave 2 is described by y2(x, t) = A sin(kx + ωt + φ). The phases (arguments of the sines) are in radians, as usual, and A = 2.1 cm and φ = 0.35π rad.

Choose the answer that correctly describes the waves’ directions of travel.

MultipleChoice :

1) Both waves travel in the negative x direction.

2) Due to the φ term in the phase of wave 2, it does not travel. Wave 1 travels in the negative x-direction.

3) Due to the φ term in the phase of wave 2, it does not travel. Wave 1 travels in the positive x-direction.

4) Both waves travel in the positive x-direction.

5) Wave 1 travels in the negative x-direction, while wave 2 travels in the positive x-direction.

6) Wave 1 travels in the positive x-direction, while wave 2 travels in the negative x-direction.

7) There is not enough information.

6) Wave 1 travels in the positive x-direction, while wave 2 travels in the negative x-direction.

Explanation:

What matters is the part , the other parts of the equation don't affect time and space variations. We know that when the sign is - the wave propagates to the positive direction while when the sign is + the wave propagates to the negative direction, but here is an explanation of this:

For both cases, + and -, after a certain time (), the displacement y of the wave will be determined by the term. For simplicity, if we imagine we are looking at the origin (x=0), this will be simply .

To know which side, right or left of the origin, would go through the origin after a time (and thus know the direction of propagation) we have to see how we can achieve that same displacement y not by a time variation but by a space variation (we would be looking where in space is what we would have in the future in time). The term would be then , which at the origin is . This would mean that, when the original equation has , we must have that for to be equal to , and when the original equation has , we must have that for to be equal to

Note that their values don't matter, although they are a very small variation (we have to be careful since all this is inside a sin function), what matters is if they are positive or negative and as such what is possible or not .

In conclusion, when , the part of the wave on the positive side () is the one that will go through the origin, so the wave is going in the negative direction, and viceversa.

area of focus

force tends to be low and long diustance masses are high