Physics 6 Tutorial 6  Mathematical Treatment of Waves
Contents 
In Waves Tutorials 1 and 2, we used a graphical approach to describe progressive waves. As an extension, we modelled a sinusoidal wave, primarily to show how sine waves can be drawn. In this tutorial, we are going to take a more mathematical approach to study waves.
The simplest type of wave is called a sine wave. This is because the displacement varies with the sine of the time. The equation is shown below:
The terms are:
x  the displacement (m);
A  the amplitude (m) which is the maximum displacement;
w  the angular velocity (rad s^{1});
t  the time (s).
Note that x is often used for the displacement in waves. The code s can be used as well. The symbol w is omega, a Greek lower case letter long 'o' (ō). It represents the frequency of the wave, and is linked to the frequency by the equation:
So we can also write:
Be careful about how you input
wt
into your calculator:

Velocity of the particles that form the wave can be worked out using the gradient of the sine wave. In calculus notation, we can write:
Maths Window The derivatives of trigonometrical functions are as follows:
And
We won't look at the tangent functions here.
If there is a constant being processed by the function, then the derivative is multiplied by the constant. In this case, we'll make the constant B. Therefore:
And:

For acceleration, we can write:
The minus sign tells us that the acceleration is towards the average level (zero displacement).
A wave has an amplitude of 0.25 m and a frequency of 6.0 Hz. Calculate: a. The angular velocity; b. The displacement at 1.1 s; c. The velocity at 1.1 s; d. The acceleration at 1.1 s. 
Phase tells us the relative displacement of two points on a wave. In Waves Tutorial 1 we saw the equation for the phase difference between two points in a wave:
The phase angle is in radians.
When you use radians, you must ensure that your calculator is set to radians. The easiest way to lose marks is put your angle in radians while you are set to degrees…
It is up to you to make sure how to work your calculator. 
A wave of wavelength 3.5 m has two points that are 20 cm apart. Calculate the phase difference. 
Phase Difference
Consider two surfers, P and Q on a water wave. The water wave has a wavelength, l, and an amplitude A. This is a displacementdistance graph.
The wave is travelling from right to left. Surfer P is further up the wave than his mate, Q. This means that P leads Q. Or we can say that Q is behind P; Q lags P.
On a displacementtime graph at a certain time, t, this looks like:
On the displacement time graph, surfer P is at angle q. The amplitude of the wave at angle q is x, where:
Surfer Q is at angle (q  f). The amplitude of the wave here is:
We know that:
q = wt
So we can rewrite this expression as:
y = A sin (wt  f)
Or:
y = A sin (2pft  f)
From above, we see that:
So it doesn't take a genius to write:
The expression tidies up to give:
Let's put some numbers in.
The surfers in the argument above are riding a wave that has a wavelength of 50 m, amplitude 2.0 m and is travelling at a speed of 3.5 m s^{1}. At time t = 0, Surfer P is about to ride the wave as shown in the diagram:
Surfer Q is 6.5 m behind Surfer P.
(a) Show that the period of the wave is about 14 s. (b) Calculate the displacement x of Surfer P after 2.5 s. (c) Calculate the displacement y of Surfer Q after 2.5 s.

We have looked at how waves can superpose. See Waves 3. In this case, we will look at how two waves add using their formula. Normally this would be highly tedious, but a spreadsheet makes light work of it. We can use an Excel Spreadsheet to model the superposition of two waves. In this case we have used the basic relationship for amplitude:
For Wave 1, the numerical parameters are:
Amplitude = 0.20 m;
Frequency = 0.5 Hz.
For Wave 2, the numerical parameters are:
Amplitude = 0.20 m;
Frequency = 0.55 Hz.
A screenshot shows the spreadsheet data:
The formula in Cell 6 is:
=$D$2*SIN($J$2*A6)
The dollar signs ($) are to lock the reference cells. The * sign is the multiplication operator for Excel. The graph shows the two waves:
And the resultant wave is:
Remember that the displacement is a vector. If the two displacements have the same sign, the result is reinforcement or constructive interference. If the two displacements have opposite signs, the result is cancellation or destructive interference.
If we extend the time axis, we get:
In this picture we see that there is a periodic rise from a minimum of 0 to a maximum of 0.4 m and back to 0. These are called beats.
What is the period of the beats? What is the frequency of the beats? 
The effect is most noticeable if the two (sound) waves have a frequency that is almost the same. The beat frequency is simply the difference of the two frequencies:
You can download the Excel Spreadsheet used to produce the graphs above in the links at the top of the page. Try changing the frequencies and amplitudes to see what happens.
Show that the beat frequency you worked out in Question 4 is consistent with the data that are given in the Excel Spreadsheet model.

The sound of a musical instrument is determined by:
The pitch or frequency of the note;
The volume or amplitude of the note;
The quality or waveform made by the instrument.
The pitch is NOT the volume. 
Pure sine waves are very dull to listen to. The waveform of an instrument is quite complex. Musical instruments have their own characteristics that depend on the waveform that they produce. For example:
Image from Yuval Nov
The complex waveforms shown in the diagram are actually made up of lots of sine waves that are superposing.
The factors that produce the distinctive sounds are complex and may even vary between instruments of the same type. For example, the Stradivarius violin has a particular tone that is much sought after by professional classical musicians over other violins. It is not just the way the soundbox and other characteristics of the instrument are made; the tone can be influenced by things like the grain of the wood, or even the varnish used by the craftsman. Antonio Stradivari (1644  1737) had his own techniques that were used in his family, which are not known, and cannot be copied. His instruments are priceless.
One of the most important features of the sound of an instrument is the production of the harmonics. If the note A_{3} (A below Middle C) is played, its fundamental frequency is 220 Hz. Its harmonics are shown in the table:
Harmonic 
Frequency / Hz 
1 
220 
2 
440 
3 
660 
4 
880 
5 
1100 
6 
1320 
Each frequency is 220 Hz apart. Beats of 220 Hz are also set up, which the ear can pick up. If the fundamental frequency is removed by filtering, it is still possible for a listener to "deduce" the fundamental frequency. This is called the missing fundamental effect.
Oldfashioned police whistles have two separate frequencies that are close to each other. These produce beats, which gave the whistles their characteristic warble.
Woodwind and brass instruments use beats to give a strong sense of pitch. Two flutes played together make beats, and this gives the impression of a third instrument ("a trio of two flutes"). A brass instrument player can hum another frequency to make a third tone. This is called multiphonics.
This is a mathematical technique that is applied to complex waveforms, first worked out by the French mathematician JeanBaptiste Joseph Fourier (1768  1830). The idea is that a complex waveform can be broken down or decomposed into its component sine waves. This is done using advanced and complex mathematical functions. Consider a bass guitar playing the note A_{2} (55 Hz). The waveform looks like this:
Image by Fourier1789  Wikimedia Commons
Once Fourier Analysis (or Transform) is applied, we can see the components that make up the complex wave:
Image by Fourier1789  Wikimedia Commons
What do you notice about the peaks? 
A Fourier transform in reverse can be used to synthesise musical sounds. This enables musicians to connect a synthesiser box to an instrument like a guitar to make a range of musical (and nonmusical sounds). You can buy keyboards at low cost that have a variety of synthesised musical sounds that you can play.
Fourier analysis can be applied to a range of oscillating systems, for example, the processes within a large chemical plant. If there are several linked steps in a continuous process, and one process is slowed down, then the others will be as well. If we try to speed the first process up to compensate, we get a build up of the intermediate product. This means that the next stage needs to be speeded up, and so on. The plant starts to oscillate, which can lead to problems (including the risk of an explosion). The oscillations can be complex, but can be more easily understood using a Fourier analysis. The interpretation of the analysis enables chemical engineers to see where the oscillations start, and apply measures to keep the oscillations to a minimum.