Virtual Science Ltd
+44 1249 736656
Stationary Waves
Introduction
A wave is an oscillation of something that transfers energy along its path of propagation without transferring any matter. Think of a wave produced by flicking the end of a long rope. A wave will travel along the rope, but no part of the rope is actually moving in the direction of the wave; the parts of the rope are simply going up and down.
If we tied the far end of the rope to a fixed point and oscillated the near end, we would set up a pattern of moving waves on the rope. This is a transverse wave, where the direction of the wave motion is at right angles to the disturbance, which is up and down. This is also like the wave motion in the sea, where the water is going up and down and a surfer can ride the crest of a wave and be propelled forward in the direction of propagation of the wave. The distance between successive crests is called the wavelength. The number of complete waves per second is called the frequency, measured in hertz (Hz). One Hz is one cycle (complete wave) per second. The wavelength is often represented by the Greek letter lambda (λ) and the frequency by v. There is a simple relationship between the speed of the wave, its frequency, and its wavelength:
v = fλ
This is easiest to understand if you just consider the case where the frequency is one cycle per second. In this case, the wave must propagate exactly one wavelength each second.
The kind of waves we are considering here are called mechanical waves, as something physical is moving. This is different from electromagnetic waves, which can travel through a vacuum and don’t involve the motion of anything physical. In our experiment, we are going to use an apparatus first proposed by John Shive of Bell Laboratories in 1959, which he called the mechanical wave machine. This consisted of wires along which the waves travel, not as a vertical displacement but rather as a twisting of the wire. Along the wire are placed horizontal pendulums; these have the effect of slowing the wave motion, allowing the wave propagation to be seen more easily.
We will use a particularly simple version of this where we use a central tape instead of a wire, which is attached to fifty horizontal pendulums. Note that in this apparatus, it is the twisting of the tape that propagates the wave.
The objective
The aim is to find the relationship between the tension and the speed of propagation of the wave.
The apparatus
-
The mechanical wave machine
-
A frequency generator
-
An electromechanical pusher (this is used to drive the pendulum at the start of the wave machine)
If you want to make your own wave machine, use a slightly more than 2-meter length of duct tape, onto which you place barbeque skewer sticks 5 cm apart. Put a layer of duct tape over the top to make the sticks secure. Skewer fruit pastilles (or any other soft sweet) onto the ends of the skewers. Then suspend with one end firmly fixed and the other tied to a string that goes over a pulley to a tensioning weight. If the sticks are not all horizontal when at rest, adjust the sweets along the sticks to obtain the right balance.
The variables
The independent variable is the tension in the string and the tape, and the dependent variable is the speed of the wave.
The Physics
The important relationship is that between frequency, wavelength, and speed—which, as previously discussed, is given by:
v = fλ
We will use this to calculate the speed of the wave, by observing the wavelength of the wave and the frequency as set on the generator. We do this by observing ‘standing’ waves. When a wave reaches the fixed end, it is then reflected back along its original path. It then combines with the original wave, either strengthening or weakening it at various positions. At particular frequencies it will produce a standing wave—one that looks perfectly stationary. Figure 26.2 shows what a standing wave of five half-wavelengths looks like.
The tape, string, or wire is not actually stationary; it is oscillating between the continuous line shown here and the broken one, so it looks like it is at rest. The significance of the standing wave for this experiment is that we can count the number of half-wavelengths and hence the number of full wavelengths and then, using the frequency of the generator, we can calculate the speed of the wave.
We can find the theoretical speed of a wave on a string using Newton’s second law fairly straightforwardly. Consider a segment of the string of length Δs at the top of the propagating wave, as shown here:
Although this is moving, it is moving at a constant speed, so Newton’s second law is still applicable. There is some radius of curvature at this point, say R. The acceleration of this segment towards O is given by v2/R.
Now consider the forces acting on the segment: T is the tension in the string and the force downwards on the segment. F is the sum of the vertical components of the tensions; as shown here:
Resolving vertically, the equation of motion is:
2Tsin(θ) = mv²/R
As θ is small, sin(θ) can be approximated by θ. The mass of the segment is given by its mass per unit length (μ) times its length:
m = Δs μ
Where Δs = 2Rθ. Substituting into the previous equation gives us the mass as:
m = 2Rθ μ
Substituting this into the equation of motion yields:
2Tθ = 2Rθ μ v²/R
Which can be simplified by canceling identical terms to:
T = μ v²
Which can be rearranged to give the speed as:
V = √(T/ μ)
The method
Switch the frequency generator on and slowly increase the frequency until you see a standing wave like that shown here?
This shows five half-wavelengths. You should note the frequency at this point. Repeat using different weights, recording the number of half-wavelengths and the frequency for each. You need to take great care to spot the best point for the standing waves for each reading.
The Video
Watch a video of the Stationary Waves e-practical here.
This shows how to use it and how to collect the data.
The e-Practical
Perform the experiment yourself, collect your own data, make mistakes and be able to correct them. The e-practical requires that your browser can run WebGL 2 (usually found on Windows browsers, safari on iOS, and various Mobile browsers, test with https://get.webgl.org/webgl2/). This link is for students and evaluation only, schools should purchase a site licence.
All the e-practicals will run on devices as small as a mobile 'phone. However, the best experience is on a PC using a mouse which gives very precise control, but if you are limited to a small device, consider using a stylus or a blue tooth mouse.
The frequency generator can be switched on by clicking the on/off button. Use the Frequency knob to adjust the frequency either by using the mouse wheel or by dragging when using the on-screen controls. The wave machine is 2 m in length. The weights start at 0.5 kg. You can increase and decrease the weights by using the mouse wheel or by clicking when using on-screen controls. Each of the additional weights is 0.25 kg.
The Results
Create a table showing your results like this:
Plot the speed squared on the Y-axis and the weight on the X-axis and you should see a linear relationship, showing that the square of the speed of propagation is proportional to the tension.
Further Discussion
Can you deduce from the Physics what the speeds you inferred from your observations would be if each pendulum was twice the weight?
This section is adapted from material developed by Dr Robert Lucas and is related to the book High School and Undergraduate Physics Practicals, published by CRC Press.