Superposition of waves
Dear Dr. Stern:
For two electromagnetic waves of different frequencies (and wavelengths) and amplitude traveling in the same direction, I have noticed they combine their amplitudes when they superimpose. However, I am also noticing an increase in frequency. Am I correct to think that the two frequencies also combine in some manner so that the resulting wave is the combined frequency of the two waves?
Or am I perhaps misinterpreting observations in that the increased frequency likely has a different cause not necessarily due to the superimposing? I suppose the situation is similar to a radio wave interacting with a visible light wave.
Reply
Your question can be answered at several levels. On the simplest level--two long wave trains of different well-defined frequencies--no, they do not combine, and indeed, can be separated by suitable filters.
Two waves at SLIGHTLY different frequencies will exhibit beats--a modulation in overall intensity as they slowly get in phase and out of phase again. A freight railroad runs about 2 miles from my home, and I can clearly hear beats when it is pulled by multiple diesel locomotives whose engines run at slightly different speeds. Twin engine airplanes also sometimes sound beats. If your detector smoothes over the rapid variations, it extracts the beat frequency.
Things get more complicated when wave trains are finite and changing, as in radio waves which carry signals, say music encoded as AM or FM. When such waves are analyzed by frequency (without regard to phase--to where valleys and peaks fall), they always cover a finite "bandwidth" of frequency, around the main one of the carrier signals. Sometimes your radio will receive a mix of two stations, a sign that their bandwidths overlap.
The question continues:
Dear Dr. Stern:
I appreciated your response. I was using a simple copper wire antenna to receive a consistent frequency radio wave, and I was receiving a frequency slightly higher at periodic times, when I was introducing a different frequency electromagnetic wave.
Initially I thought that the waves may be combining, but based on your e-mail, could I conclude superposition instead? In other words, the antenna was receiving both frequencies simultaneously, which appeared as one larger frequency on the oscilloscope.
Is my conclusion correct? Is it possible for the antenna to receive different frequencies from different types of electromagnetic waves simultaneously resulting in a larger frequency reading? I used a radio receiver antenna in place of the oscilloscope antenna, and at the periodic times I could hear choppiness or pulses with the earpiece that coincided with amplitude increases on an oscilloscope attached to the radio receiver. Could the pulses be the beats to which you were referring in that possibly what I was observing was two radio waves of slightly different frequencies? Thus, the oscilloscope attached to the simple copper antenna was giving me the readings of the radio waves and the other electromagnetic wave was not being recorded.
Reply
Dear Chris
Your questions go somewhat beyond my experience. I suggest you look up "The Radio Amateur Handbook" which most amateurs (and some libraries) have, and perhaps discuss them with a knowledgeable amateur.
In general, superposing signals of one frequency does not generate a higher frequency stable enough to be observed on an oscilloscope. The sidebands I mentioned give irregular wave trains, to which a filter can respond, but not a 'scope.
Beats occur between very close frequencies. If one diesel train engine emits a low growl at 100 cycles/sec and one next to it gives 101 cycles, the combined intensity will rise and fall with a 1-second period. I believe however you can get beats if a radio wave from a transmitter reaches you by two different paths, and the path difference slowly changes (ionosphere rising, say). That may explain the "choppiness" you observe.
The Sun and Seasons
After looking at the page on "#1. Stargazers and Sunwatchers" I have a question related the sun's position on the horizon. The page talks about how the sun sets at different places on the horizon depending on the time of year, and my question is about the statement that "during the Winter Solstice the sun is as far to the south as it can be" at any one location.
I'm assuming that this means that at different locations the sun can set further to the south or closer to west. If this is what was implied, then do you have an understanding on the relative direction the sun can set on the horizon at the equator or the poles or about mid-way between the two? And most importantly, what is the cause for this?
Reply:
Dear Jason
You should read on! Your question is answered in section #2 "The Path of the Sun, the Ecliptic" and section #3, "Seasons of the Year." Right in front in section #2 you will see a schematic view of the path of the Sun at the summer solstice, at equinox and at the winter solstice.
Why the different paths? You must understand that because of the rotation of the Earth, all objects of the sky seem embedded in a huge "celestial sphere" which seems to rotate around us, in a tad under 24 hours (that is discussed in section #1a). Some objects seem to migrate slowly around that sphere: the Sun, the Moon and the planets. Most stars don't migrate, however, and form the same patterns night after night, the "constellations" of the sky.
The Sun seems to move in a huge circle, one circuit per year. If that circle were the celestial equator--halfway between the "poles" of the sky, the points around which the sphere seems to turn--then day after day the Sun would follow the same path, of which exactly 180 degrees are visible (during the other half, at night, it is below the horizon), and all days would be 12 hours long.
Actually, the circle it follows is inclined to the equator by 23.5 degrees, because the axis of the Earth is not perpendicular to the plane of its orbit around the Sun ("the plane of the ecliptic"), but is 23.5 degrees off that perpendicular. So the Sun is half the year south of the ecliptic, and half the year north of it. When it is north of it, it is summer--each day (in the US) it covers more than 180 degrees, and days are longer than 12 hours. When it is south of it, its daily path above the horizon is less than 180 degrees long, and days are shorter than 12 hours, as they are now in December.
The location of sunrise and sunset on the horizon varies over the year, but the ANGLE between the path of the Sun and the horizon depends on your latitude. On the equator it is 90 degrees, and sunrise wanders south or north in the same way as it does in the US. At the north pole, on the other hand, the angle is ZERO. At this location, the celestial equator is along the horizon. In half the year when the Sun is north of the equator, you will see it 24 hours a day, and it NEVER SETS. That is the long polar day, and a sun-observing telescope at the South Pole Research Station has taken advantage of this. If the Sun is south of the equator, in the part of the sky you do not see, it NEVER RISES and you have the famous polar night, lasting nearly half a year. Regions near the pole have similar seasons.