Vector Sum of a Standing Wave Confusion

[jdkeeley]

Hi,
I was taught that a standing wave is formed when a progressive wave meets a boundary and is reflected. I was also taught that waves that meet a fixed end, reflect on the opposite side of the axis to the side that they met it at. (I hope that makes sense)
If this is true, when the wave is reflected, I don't understand why any wave is formed at all. Surely, if the two waves are opposites in the way they are, their amplitudes would interfere destructively and not form a wave at all.
I have been looking around for answer to this question elsewhere online but cannot seem to come up with one and wasn't sure where else to post this.
I have a physics exam on Monday and I don't understand this particular bit!
Thanks

 

[Impulse]

Standing wave diagrams are a still frame of where the wave will have traveled over time as it travels to one end of a medium and back.

This is just like the graph of a wave on a coordinate plane. Some parts of the wave are crests, some are troughs, but they don't interfere because the crests and troughs are never in the same place at the same time.

A standing wave doesn't interfere with itself because, although a crest and a trough do exist in the same place, they do not exist in that place at the same time.

EDIT:

I wrote my response thinking of a single wave pulse, in which case no interference would occur at all.

In the case of a standing wave, we create multiple pulses at regular time intervals, and constructive and destructive interference occur at different points along the medium.

The reason the wave doesn't completely cancel itself out is the dimension of time. The relative location of each crest and trough along the wire in time explains the resulting interference pattern.

 

[dauto]

as the incident wave and reflected wave travel in opposite direction there will be times when one wave's crests coincide with the other wave's valleys. When that happens the two waves cancel out as you described. But at some later time the waves move in opposite directions and eventually one wave's crest will coincide with the other wave's crest. At that instant the two waves add up to a bigger wave (constructive interference). You will see the wave go back and forth between those two opposite possibilities which gives a wave that expands and contracts without moving known as a standing wave.

 

[ehild]

A very simplified derivation will be shown:
A wave traveling along the x-axis to the right can be represented by the function y1=Asin(wt-kx). A wave traveling in the opposite direction, having the same amplitude and opposite phase is y2=-Asin(wt+kx).

The resultant function is Y=y1+y2=Asin(wt-kx)-Asin(wt+kx). Apply the addition formula to expand the sines:

Y=A[ sin(wt)cos(kx)-cos(wt)sin(kx)-sin(wt)cos(kx)-cos(wt)sin(kx)]= -2Acos(wt)sin(kx)

The result is a standing wave. It describes the vibration at each x with angular frequency w, and amplitude 2Asin(kx). At certain points the amplitude is zero (where kx=integer multiple of pi) these are the nodes. At other points, where kx is odd multiple of pi/2, the amplitude is maximum, 2A. These are the antinodes of the standing wave.

ehild

 

[HalfLostAndSearching]

for reaching out with your question. It's great that you are seeking clarification on this concept before your exam. Let me try to explain the vector sum of a standing wave in more detail.

First, it's important to understand that a standing wave is formed when two waves with the same frequency and amplitude travel in opposite directions and interfere with each other. This interference creates points of constructive and destructive interference, resulting in the characteristic pattern of nodes and antinodes in a standing wave.

Now, when a progressive wave meets a boundary and is reflected, it undergoes a phase change of 180 degrees. This means that the reflected wave is now inverted compared to the original wave. However, when this reflected wave interferes with the original wave, it does not result in destructive interference. This is because the reflected wave is not simply a mirror image of the original wave, but it has also undergone a phase change.

To understand this better, let's consider the two waves as vectors. When they meet at the boundary, their amplitudes add together to form a resultant vector. When the reflected wave is added to this resultant vector, they do not cancel each other out completely because of the phase change. Instead, they form a new resultant vector with a smaller amplitude, creating the nodes and antinodes of the standing wave.

In essence, the vector sum of the two waves creates a new wave with a different amplitude, but the same frequency and wavelength as the original waves. This is why a standing wave is still formed even though the waves are opposites in terms of their direction of propagation.

I hope this explanation helps to clarify the concept for you. Remember, understanding the underlying principles and mechanisms of a concept is crucial for success in physics. Best of luck on your exam!