# 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)


The correct answer is (2).

• Where did you learn it? – kiss my armpit May 13 '13 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. – ja72 May 13 '13 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 May 12 '13 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. – Triatticus May 17 '13 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. (From Wikipedia)

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).