# How to determine the direction of medium's displacement vectors of a standing wave?

Consider the following problem taken from a problem booklet. My questions are:

1. What is displacement vector?
2. And how to determine the direction of displacement vector at a certain point?
3. Where is the position with zero displacement vector?

Any material between two nodes is displaced by the same direction. So the direction of B and C has to be the same as well as the direction of A and D due to symmetry. In addition, the direction of A must be the opposite of B since they are across from a node. Similarly the direction of C and D must be opposite.

So the two possible configurations are

   A-->  <--B     <--C        D-->   (figure d)
<--A        B-->     C-->  <--D      (figure c)


• Where did you learn it? Commented May 13, 2013 at 18:58
• Intuition I guess. A node is where there is no deflection, and on opposite sides of the node the deflection has to be of opposing direction for things to balance out. Commented May 13, 2013 at 19:03

Just because a wave is a standing wave doesn't necessarily mean that the particles themselves do not move, in face if the particles themselves didn't move there wouldn't be any wave motion at all. For a longitudinal wave (made of particles that oscillate in the direction of wave propagation like here as sound waves) particles oscillate left and right but have no net displacement.

Take a look at this site on standing waves in compression/longitudinal motion, perhaps it will help you understand what the answer is and why it is the correct one.

http://www.acs.psu.edu/drussell/Demos/StandingWaves/StandingWaves.html

• Are there standing waves that do have a net displacement?
– user24082
Commented May 12, 2013 at 7:56
• As Far as I know the definition of standing wave excludes a non zero net displacement, so no there shouldn't be standing wave with net displacement. Commented May 17, 2013 at 22:03

A standing wave is a wave that has nodes. The points of the wave go up and down in some places, and remain at zero at others (the nodes). The general form of a standing wave is a sine curve that remains at a fixed position, but its amplitude changes in time between $+A_0$ and $-A_0$. Specifially, there is a time where the wave form is completely flat.

The formula is something like

$$f(x) = A_0\cos(\omega t)\,sin(kx)$$

(not the most general form). Compare to a moving wave which has a fixed amplitude, but a changing offset, so it seems to move along the axis.

$$f(x) = A_0\sin(\omega t + kx)$$

Now in your case you have a tube with air. Your waves don't go up and down (transversal), but back and forth (longitudinal). The nodes are points where the air doesn't move, anti-nodes are where the air moves maximally. Still, it can be described by the same equation. You can try to draw a sine-curve through your first figure. The $y$ value should be the air displacement at point $x$, at a fixed time ($t=0$ or $t=\pi/\omega$). The sine curve must cross the $x$ axis at the nodes, and have maxima and minima at the antinodes. There are two ways to draw the curve, which are mirrored along the $x$ axis. A positive displacement means that the air molecules are moved to the right (compared to where they should be at $t=(\pi/2)/\omega$), a negative displacement means they are moved to the left. You should be able to read off the correct displacement vectors from your drawing.

A little caveat: Don't confuse displacement and pressure, or speed. The nodes always have zero displacement, but the pressure there changes all the time. The points A, B, C, D (on the slopes of the curve) sometimes have zero displacement, when the waveform crosses the $x$-axis, but at that moment the air has the highest speed (change of displacement).