Direction of movement for particles in a wave

[DWill]

Homework Statement

The question includes a graph of a wave (on a string) with several points labeled on the graph. We are asked to determine the direction of the transverse velocity of each point on the wave as well as the transverse acceleration.

Homework Equations

The Attempt at a Solution

I know that the direction of transverse acceleration depends on the curvature of the wave, i.e. for negative curvature the transverse acceleration is also negative. However I'm not sure how to figure out which direction the velocity is moving in. I guessed that the points whose y-coordinates were larger than 0 had negative transverse velocity because that is the direction of equilibrium, and similarly points below the x-axis had positive transverse velocity. But this assumes the equilibrium position of the string is at x = 0, how do you tell for general case? Thanks

 

[anathema]

Thank you for your question. To determine the direction of transverse velocity of each point on the wave, we can use the following equation:

v = Aωcos(kx ± ωt + φ)

where A is the amplitude, ω is the angular frequency, k is the wave number, x is the position of the point on the wave, t is the time, and φ is the phase constant.

The sign in front of the ωt term determines the direction of the wave's motion. If the sign is positive, then the wave is moving in the positive x-direction, and if the sign is negative, then the wave is moving in the negative x-direction.

For the points above the x-axis, the y-coordinate is positive, which means that the wave is above the equilibrium position. This corresponds to a positive value for cos(kx ± ωt + φ), which means that the transverse velocity is in the same direction as the wave's motion. Similarly, for points below the x-axis, the y-coordinate is negative, which corresponds to a negative value for cos(kx ± ωt + φ), meaning that the transverse velocity is in the opposite direction of the wave's motion.

To determine the transverse acceleration, we can use the following equation:

a = -Aω^2sin(kx ± ωt + φ)

This equation tells us that the transverse acceleration is always in the opposite direction of the displacement, which means that it is always directed towards the equilibrium position.

I hope this helps answer your question. If you have any further inquiries, please feel free to ask. Happy studying!

 

[baba89]

I would first clarify the type of wave being discussed as there are different types of waves with different properties. Assuming we are discussing a transverse wave on a string, the direction of movement for particles in this type of wave can be determined by looking at the direction of displacement. When a particle on the string is displaced upwards, its transverse velocity will be in the direction of the displacement, and vice versa for a downward displacement. This can be seen in the graph where the particles above the equilibrium position have a positive displacement and therefore a positive transverse velocity, while the particles below the equilibrium position have a negative displacement and a negative transverse velocity.

As for determining the direction of transverse acceleration, it is correct that it depends on the curvature of the wave. However, it is important to note that the direction of acceleration is always towards the equilibrium position, regardless of the curvature. This means that particles with a negative curvature will have a negative transverse acceleration, while particles with a positive curvature will have a positive transverse acceleration.

In conclusion, the direction of movement for particles in a transverse wave on a string can be determined by looking at the direction of displacement, while the direction of transverse acceleration can be determined by considering the curvature of the wave.