Question

Two sound sources emitting plane waves separated by (12) meters vibrate according to law (y₁ = 3cos(4π t)) and (y₂ = 4sin(4π t)) respectively. A point in the medium is at a position (8) meters from the first source and (4) meters from the second on the line joining two sources. The sources output waves of velocity (32 ,m/s). Find the equation of the motion of the particles.

A) (y_R = 5sin(4π t + tan⁻¹(4/3)))
B) (y_R = -5sin(4π t + tan⁻¹(3/4)))
C) (y_R = -5sin(4π t + tan⁻¹(4/3)))
D) (y_R = 5sin(4π t + tan⁻¹(3/4)))

Answer 1

The motion of particles from two interfering plane waves is described by the equation yᵣ = 5sin(4π t + tan⁻¹(3/4)) , which is answer choice D) from the provided options.

Step-by-step explanation:

The question involves a physics concept related to the interference of waves and we need to find the equation of the motion of the particles resulting from the combination of two plane waves emitted by two sound sources. The waves emitted have the equations y₁ = 3cos(4π t) and y₂ = 4sin(4π t). To solve for the equation of motion, we should use the principle of superposition and take into account the phase difference caused by the difference in the travel distance of the waves to the observing point.

To find the resultant amplitude and phase, we can represent the waves in terms of sine, as they have the same angular frequency. Representing y₁ as 3sin(4π t + 90°), we can combine the amplitudes by constructing a right triangle where the sides represent the amplitude of each wave and the hypotenuse would be the resultant amplitude. The phase angle is notably the arctangent of the ratio of the opposite side over the adjacent side.

The correct equation of the motion for the particles is the one that includes the amplitude 5 (derived from the Pythagorean theorem) and has the correct sign for the resultant phase shift. This results in yᵣ = 5sin(4π t + tan⁻¹(3/4)), which corresponds to answer choice D).

Answer 2

To find the equation of motion of the particles, calculate the displacement of each source from the point in the medium and find the phase difference between the two sources. Then, use the superposition principle to derive the equation of motion.

Step-by-step explanation:

To find the equation of motion of the particles, we first need to find the displacement of each source from the point in the medium. The displacement from the first source is given by y1 = 3cos(4πt) and from the second source is y2 = 4sin(4πt). Using the distance formula, we find that the displacement from the first source is 8 and from the second source is 4.

Next, we need to find the phase difference between the two sources. The phase difference is given by Φ = tan⁻¹((y2 - y1) / d), where d is the distance between the two sources. In this case, d = 12, so Φ = tan⁻¹((4 - 3) / 12) = tan⁻¹(1/12).

Finally, the equation of motion of the particles can be found using the superposition principle, which states that the total displacement at a point is the sum of the individual displacements. The equation of motion is given by yR = 5sin(4πt + tan⁻¹(1/12)).