The Physics Philes, lesson 103: Of Nodes and Anitnodes
Last week we looked at what happens when a wave pulse is reflected off a barrier. But what happens when a sinusoidal wave on a string is reflected? That’s our topic today as we discuss standing waves on a string.
In order to do this, we need to use a concept we learned last week called the principle of superposition. To jog your memory, the principle of superposition basically says that we can add two wave functions together to describe the resulting motion of two waves that overlap. Now, we consider two waves running through a string. One of the waves is the initial (or incident) wave and the other is that wave reflected from a string’s fixed end. As a wave in simple harmonic motion is propagated through the string and reflected back, the string will look subdivided into segments. It’s easy to see in this video:
You might notice some interesting things as you watch the video. There are certain spots on the string that don’t move at all. Those spots are called nodes. The points where the amplitude of the wave is at its greatest are called – you guessed it – antinodes. If we were just looking at a wave on a string, the amplitude would be constant and the wave pattern would move at a speed equal to the wave speed. But in this new situation, the wave pattern stays in the same position and the amplitude fluctuates. This is why these types of waves are called “standing waves.”
So we have these waves traveling through a string. Surely at some point they will get in each others way, right? What happens then? That is when constructive and destructive interference come in. This video illustrates constructive and destructive interference nicely:
The green and blue waves are the waves traveling in opposite directions, and yellow wave is the wave as we would see it on the string. There are certain times in this cycle in which the two opposing waves are out of phase. The total wave at that time is zero. The displacement is always zero at the nodes. At the nodes, the displacement of the two waves are equal and opposite, thus cancel each other out. That kind of interference is called destructive interference.
The opposite is true at the antinodes, or points of greatest amplitude. At those points, the two waves are identical, so the resulting amplitude is large. This is called constructive interference.
To figure out the wave function of the standing wave, all we have to do is add the wave functions of the initial and reflected wave (if those waves have equal amplitude, period, and wavelength). Let’s have a wave function y1(x,t) that represents the initial wave traveling to the left from x=0 and a wave function y2(x,t) that represents the reflected wave traveling to the right from x=0. Since a wave reflected from a fixed end is inverted, one of the waves includes a negative sign:
Since the wave function for the resulting wave is just the sum of the initial and reflected wave, we can write it like this:
Using some fancy trig identities, this can be re-written in a prettier way:
A_SW is the standing wave amplitude, and it’s equal to twice the amplitude of the two original waves.
Notice that this new wave function has two factors, one as a function of x and one as a function of t. The part of the function in the parentheses tells use that the shape of the string is a sine curve. The wave shape stays the same, but it oscillates up and down according to singωt. If this wave was just traveling along a string, we would expect that there be phase differences between the oscillations of points that are right next to each other. That’s not true for a standing wave. For standing sinusoidal waves, all of the point between pairs of nodes are oscillating in phase.
A standing wave is different from a traveling wave in another way. A standing wave doesn’t transfer energy from one end to the other. The two waves carry equal amounts of power in opposite direction. The energy will flow from node to antinode and back, but the average energy transfer rate is zero everywhere. Pretty weird, right?
Welp, we’re almost done with our discussion of mechanical waves. Only two more posts in this section left. Next week we’ll start talking about our last subject in this section: normal modes on a string.
Featured image credit: YouTube