Direction of of the velocity vector for particles in a sound wave

In summary: Yes, I found it. The correct expression should have been ##kx-\omega t## and I have ##kx_1-\omega t## in a couple of places.
  • #1
vcsharp2003
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Homework Statement
What would be the direction of velocity for particles close to ## x_1## and on either side of it as shown in the sound wave diagram below? The hollow dots show equilibrium positions of displaced particles. Solid dots are the displaced particles. The arrow shown against solid dots indicates the displacement direction from equilibrium position.

(The sound wave diagram is taken from University Physics by Sears and Zemansky)
Relevant Equations
## y(x,t) = A \cos {(kx - wt)}##
## v_p (x) = \frac {\partial y} {\partial t}##
Using the equations mentioned under this question, I came up with following analysis and directions of velocities on either side of ##x_1##. Also, I'm not sure if there is an easier qualitative way to know the velocity directions rather than do a detailed Calculus based analysis?

Sound question answer.jpg


Sound waves as Pressure wave.JPG
 
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  • #2
vcsharp2003 said:
if there is an easier qualitative way
Just consider what the graph will look like a moment later, as it shifts to the right. What does that imply for the change in displacements in the vicinity of ##x_1##?
 
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  • #3
haruspex said:
Just consider what the graph will look like a moment later, as it shifts to the right. What does that imply for the change in displacements in the vicinity of ##x_1##?

Below is what I have come up with. The dashed waveform is ##\delta t## time interval after the initial solid waveform.

##A##, ##A^{'}## and ##B##, ##B^{'}## are corresponding points on the waveform.
Particles ##P_1##, ##P_2## are the particles just before and after particle ##x_1##.

##P_1##'s displacement increases, so it must be going away from its equilibrium position i.e. towards the right.

##P_2##'s displacement also increases and therefore it's moving towards the left.

Does that look right?

IMG_20220304_142258__01.jpg
 
Last edited:
  • #4
vcsharp2003 said:
##P_2##'s displacement also increases and therefore it's moving towards the left.
Displacements are scalars or vectors, but not magnitudes.
An increasing displacement here, perhaps from very negative to less negative, means moving to the right.
 
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  • #5
haruspex said:
Displacements are scalars or vectors,
I thought displacement is always a vector.
haruspex said:
An increasing displacement here means moving to the right.
You're right. Below is my reasoning.

The particle on right was initially displaced towards left with a negative displacement, but a moment later it's displacement will be less negative i.e displacement y increases with time and therefore it moved closer towards equilibrium position. This means it's moving towards right.

Was my Calculus based analysis correct since it gave correct results?
 
  • #6
vcsharp2003 said:
Was my Calculus based analysis correct since it gave correct results?
Yes, but in a couple of places you have ##x_1## instead of x.
 
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  • #7
haruspex said:
Yes, but in a couple of places you have ##x_1## instead of x.
Ok. I couldn't find these errors. I'll keep looking.
 
  • #8
haruspex said:
Yes, but in a couple of places you have ##x_1## instead of x.
Yes, I found it. The correct expression should have been ##kx-\omega t## and I have ##kx_1-\omega t## in a couple of places.
 

Related to Direction of of the velocity vector for particles in a sound wave

1. What is the direction of the velocity vector for particles in a sound wave?

The direction of the velocity vector for particles in a sound wave is parallel to the direction of the wave's propagation. This means that the particles move back and forth in the same direction that the sound wave is traveling.

2. Does the direction of the velocity vector change in a sound wave?

No, the direction of the velocity vector for particles in a sound wave remains constant throughout the wave's propagation. This is because sound waves are longitudinal waves, meaning that the particles vibrate back and forth along the same axis as the wave's direction of travel.

3. How does the direction of the velocity vector affect the speed of a sound wave?

The direction of the velocity vector does not directly affect the speed of a sound wave. The speed of sound is determined by the medium through which it travels, such as air or water, and is not affected by the direction of the velocity vector for particles in the wave.

4. Can the direction of the velocity vector for particles in a sound wave be reversed?

Yes, the direction of the velocity vector for particles in a sound wave can be reversed. This can occur when the sound wave reflects off a surface, causing the particles to vibrate in the opposite direction. However, the overall direction of the wave's propagation remains the same.

5. How is the direction of the velocity vector related to the amplitude of a sound wave?

The direction of the velocity vector is not directly related to the amplitude of a sound wave. Amplitude refers to the magnitude or strength of the wave, while the direction of the velocity vector refers to the direction of particle movement. However, the amplitude of a sound wave can affect the displacement of particles and therefore the direction of the velocity vector for those particles.

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