Transverse Waves: UNIZOR.COM - Physics4Teens - Waves
Channel: Zor Shekhtman on UNIZOR Education
Date: 2021-09-12

UNIZOR.COM - Physics through a Looking Glass of Mathematician

Notes to a video lecture on http://www.unizor.com

Transverse Wave

We are familiar with longitudinal waves, like sound waves in the air.
Their defining characteristic is that molecules of air (the medium) are oscillating along the direction of the wave propagation, which, in turn, causing oscillation of pressure at any point along the direction of wave propagation.

Consider a different type of waves.
Take a long rope by one end, stretching its length on the floor. Make a quick up and down movement of the rope's end that you hold.
The result will be a wave propagating along the rope, but the elements of rope will move up and down, perpendicularly to the direction of waves propagation.

(open this picture in a new tab of your browser by clicking the right button of a mouse to better see details)

These waves, when the elements of medium (a rope in our example) are moving perpendicularly to a direction of waves propagation, are called transverse.

Such elementary characteristics of transverse waves as crest, trough, wavelength and amplitude are clearly defined on the picture above.

Some other examples of transverse waves are strings of violin or any other string musical instrument.

Interesting waves are those on the surface of water. They seem to be transverse, but, actually, the movement of water molecules is more complex and constitutes an elliptical kind of motion in two directions - up and down perpendicularly to waves propagation and back and forth along this direction.

Our first problem in analyzing transverse waves is to come up with a model that resembles the real thing (like waves on a rope), but yielding to some analytical approach.

Let's model a rope as a set of very small elements that have certain mass and connected by very short weightless links - sort of a long necklace of beads.

Every bead on this necklace is a point-object of mass m, every link between beads is a solid weightless rod of length r.
Both mass of an individual bead m and length of each link r are, presumably, very small. In theory, it would be appropriate to assume them to be infinitesimally small.

Even this model is too complex to analyze. Let's start with a simpler case of only two beads linked by a solid weightless rod.
Our purpose is not to present a complete analytical picture of waves, using this model, but to demonstrate that waves exist and that they propagate.

Consider the following details of this model.
Two identical point-objects α and β of mass m each on the coordinate plane with no friction are connected with a solid weightless rod of length r.

Let's assume that α, initially, is at coordinates (0,−A), where A is some positive number and β is at coordinates (a, −A).
So, both are at level y=−A, separated by a horizontal rod of length r from x=0 for α to x=a for β.

We will analyze what happens if we move the point-object α up and down along the vertical Y-axis (perpendicular to X-axis), according to some periodic oscillations, like
yα(t) = −A·cos(ω·t)
where
A is an amplitude of oscillations,
ω is angular frequency,
t is time.
Incidentally, for an angular frequency ω the period of oscillation is T=2π/ω.
So, the oscillations of point α can be described as
yα(t) = −A·cos(2π·t / T)

Point α in this model moves along the Y-axis between y=−A and y=A.
It's speed is
y'α(t) = A·ω·sin(ω·t) = A·(2π/T)·sin(2π·t/T)

The period of oscillations is, as we noted above
T = 2π/ω