Section 1-11 - Waves and Sound

Speed of Mechanical Waves

Derivation for the specific case of a chain of oscillators:

Let x be the equilibrium position of each mass.  The masses are separated at equilibrium by distance D.  Let y(x, t) be the displacement of each mass from its proper position.
Consider Newton's Second Law for the mass at position x + D.
F_x+D = ma_x+D = m d²y/dt²|_x+D

NB Here I use delta to indicate a partial derivative.

The force is due to the springs on either side of the mass (Hooke's Law), but the amount each spring is stretched or compressed depends on the locations of both of the masses:
F_x+D = k[y(x+2D, t) - y(x+D, t)] - k[y(x+D, t) - y(x, t)]

Combine these expressions, divide each side by m, factor the k, and multiply the right side by D²/D²:

d²y/dt²|_x+D = [D² k/m]  [[y(x+2D, t) - y(x+D, t)]/D - [y(x+D, t) - y(x, t)]/D]/D
Now, let's define some new quantities:
K_EFF = k/N is the effective spring constant for N of the oscillators.
L = ND is the length of the chain of N oscillators.
M = Nm is the mass of N oscillators.
So,
d²y/dt²|_x+D = [K_EFFL²/M]  [[y(x+2D, t) - y(x+D, t)]/D - [y(x+D, t) - y(x, t)]/D]/D

Now, let's smooth the system out by taking a limit D-> 0 by letting N go to infinity while D and m go to zero:

lim _{D-> 0} d²y/dt²|_x+D = [K_EFFL²/M]  lim _{D-> 0} [[y(x+2D, t) - y(x+D, t)]/D - [y(x+D, t) - y(x, t)]/D]/D

The expression in red then becomes the derivative of y wrt x, evaluated at x+D, and the blue expression is the derivative of y at x.  The left side simply becomes the second derivative at x.
d²y/dt²|_x = [K_EFFL²/M]  lim _{D-> 0} [dy/dx|_x+D - dy/dx|_x]/D

The limit that remains is the derivative wrt x of the derivative of y, or in other words, the second derivative of y wrt x evaluated at x:

d²y/dt²|_x = [K_EFFL²/M]  d²y/dx|²_x

This then results in a version of the wave equation:
d²y/dx² = [M/K_EFFL²] d²y/dt².

We'll come back to this result in a while.

Properties of a Wave

• the frequency (f or sometimes n) counts how many oscillations each piece of material experiences each second, or alternatively, how many peaks pass by an observer each second.
• the amplitude (A) describes the maximum deviation from equilibrium.  This is easy to visualize in the example above where A refers to the maximum displacement, but it can also refer to the maximum excess of pressure over average atmospheric as in sound, or the maximum electric field strength as in light, et c.
• the speed (discussed above).
• the wavelength (l) measures the physical distance between correponding points on adjacent waves (e.g., from peak to peak).

There is a relationship among  f, l, and v, which we can deduce from the 'railcar analogy.'

Suppose that a train with cars of length L passes you at speed v.  You count N cars in time t.  The distance traveled by the train in that time is NL.  The speed of the train is v = d/t = (NL)/t = (N/t)L.  We recognize (N/t) as the frequency f and L as the analog of l, so v = fl.

Sinusoidal Waves

Although we won't prove it, sinusoidal waves are described mathematically by the expression:
y(x,t) = A sin(2px/l ^-/₊ 2pf t + f),
where f is a phase angle which allows us to change the function to cosine or to some combination of sine and cosine.  This is sometimes written this way:
y(x,t) = A sin(kx ^-/₊ wt + f).
The quantity k above is the magnitude of the wave vector, which is then 2p/l. In both cases, the negative sign indicates a wave moving the the right (positive x) and the plus sign indicates that the wave is moving toward the left.

Let's check this out.  If we choose a particular time, then the function looks like this (here, I'm combining the now constant time term with the phase angle and renaming the combination as a new phase angle):
y(x) = Asin(2px/l + f'),
that is, a snap shot of the wave at that instant.  We see that the wave has a sinusoidal shape with regard to position, x, and that as x increases by one wavelength, the function advances by one cycle.
On the other hand, if we look at what's happening at a particular location x (again, I'll include the now constant spatial term into a new phase angle), the function looks like this:
y(t) = Asin(wt +f"),
that is, that piece of the medium at that particular location undergoes simple harmonic motion.  These results are at least consistent with our notions about the wave.

We see that the wave has a combined spatial and temporal dependence, since both x and t occur in the argument of the sine function.  How does this correspond to a moving wave?  The crest (for example) of the wave will occur when the argument of the sine function equals p/2 radians (or p/2 + 2np, n an interger, depending on exactly which peak we want to consider):
kx - wt + f = p/2.
The position x at which this peak occurs is then given by
x = [p/2 - f]/k+ [w/k]t,
so that, as time progresses, the location x of the crest must become more positive, that is, the crest moves to the right.  If the sign between the two terms had been positive instead, then the value for x would have to become more negative as t increases, so that the crest moves to the left.

Now, let's see if our sinusoidal wave function is a solution to the wave equation:

 y(x,t) = A sin(2px/l ^-/₊ 2pf t + f),

dy/dx = [A2p/l] cos(2px/l ^-/₊ 2pf t + f)
d²y/dx² = -A[2p/l]² sin(2px/l ^-/₊ 2pf t + f)
l² d²y/dx² = -A[2p]² sin(2px/l ^-/₊ 2pf t + f)

dy/dt = ^-/₊ [A 2pf] cos(2px/l ^-/₊ 2pf t + f)
d²y/dt² = -A[2pf]² sin(2px/l ^-/₊ 2pf t + f)
f  ^-2 d²y/dt² = -A[2p]² sin(2px/l ^-/₊ 2pf t + f)

Equating the last two results:
l² d²y/dx² = f  ^-2 d²y/dt²
d²y/dx² = [lf]  ^-2 d²y/dt²
You may remember that lf is the wave's velocity, so our wave equation becomes:
d²y/dx² = [v] ^-2 d²y/dt²
So, the constant of proportionality is the square of the wave's velocity.  This will be true for any type of wave we study.  Is it restricted to waves that are sine or cosine functions?  No.  We will make use of a technique called Fourier decomposition and argue that any well behaved wave shape can be written as a sum of sine and cosine terms, each of which independently solves this wave equation.

Non-Sinusoidal Waves

Or, a square wave is the sum of waves of the form A_n sin(nwt) with n = 1, 3, 5, ... and the amplitude A_n decreasing as 1/n.

Or, a triangular wave is the sum of waves of the form A_n sin(nwt) with n = 1, 3, 5, ... and the amplitude A_n decreasing as 1/n².
Go to the Excel workbook again and try some combinations of harmonics to see if you can duplicate these waveforms on the Fourier Demo worksheet.  Enter the amplitudes of the various harmonics and observe the resulting wave shape.

To wrap it all up, the functions describing waves, in one dimension at least, generally are solutions to the wave equation:
d²y/dx² = [v] ^-2 d²y/dt².

Special case - tranverse waves on a string

Reflections

I have no animation for this yet, but download this Excel workbook, go to the 'reflection demo', choose 'Enable Macros,' and press 'Ctrl Z' to advance the simulations.  The dark line represents the actual string, the coloured lines the real and imaginary pulses on it.  Reset the demo by pressing 'Esc' and typing '0' in the black box.

Now, instead of considering the two extreme cases (completely fixed or completely free ends), think about what would happen if the string were tied to another string.  In all cases, we would expect that some of the wave would continue down the second string with the same orientation (and frequency) as the original wave; this is called the transmitted wave.  We also notice that there is a reflected wave, the orientation (and size) of which depends on whether (and by how much) the second string is 'heavier' or 'lighter' than the first.  The quantity used to measure the difficulty of a wave in passing through some medium is called the impedance, Z.  If Z₂>Z₁, the reflected wave is inverted; if Z₂<Z₁, the reflected wave is upright.  This is a general result, even though the exact values of the impedances are calculated in different ways for different media.  In the specific example of transverse waves on a string, we have that (asserted without proof)
Z = [Tm]^1/2.
In that case, we see that our original examples correspond to Z = 0 (end of string loose, so m₂ = 0) and Z = infinity (end of string tied to wall, so m₂ = infinity).

What would happen if the two media had the same impedance?

The impedances also tell us how much energy is reflected and how much is transmitted:
Some other day....

Standing Waves

In our class demonstration, we used reflected waves as the second wave.  But in systems with finite length, there will be many reflections from each end which will have to be added to determine the overall shape of the string.  We started with a string fixed at each end, and we excited the waves with different frequencies.  In some cases, we noted that the waves all added up to a random pattern, eventually canceling out.  In other cases, we saw that the intial and reflected waved added to produce a standing wave.  What conditions need to be met to do this?  We could do a very mathematical derivation of this, but it is just as correct to base our investigation on the experimental data.

String fixed both ends:
We saw a series of patterns like those shown below as we increased the frequency at which the system was excited.  The lines indicate the limits of the oscillations of the string (the envelope).

We noticed that in each case, the length of the string L was a positive integer multiple of half of the wavelength,
L = nl/2, n = 1, 2, 3, ....
We can convert that to frequency by remembering that f = v/l, so that
f _n = nv/2L.
We do the demonstration again with one end free, and find patterns like these:

such that L = nl/4, n = 1, 3, 5, ..., which then becomes f _n = nv/4L.
Lastly, we image what might happen if we could have a string free at each end. This might seem impossible, but remember that we can use the concept for a number of other systems which do meet this criterion, for example, sound waves in a tube open at each end where the air is free to vibrate back and forth.

We see that the results here are identical to the case of fixed both ends, just with the positions of the nodes and anti-nodes exchanged.

We see that, unlike for a single oscillator with a single natural frequency, we here have a system with many natural frequencies:

Remember that, even though we derived these results for transverse waves on a string, the results are valid for other system.  For example, consider a stopped organ pipe, which means that it is open at one end and closed off at the other.  At the open end, air is free to vibrate, while at the closed end, no vibration is possible because of the stopper.  This pipe will support standing waves of the form f _n = nv/4L, n  = 1, 3, 5, ....  How are these frequencies produced?  In an organ , air is pumped into the pipe against a sharp edge, which produces all frequencies.  However, those frequencies which do not correspond to the favoured frequencies reflect back and forth in the pipe and, on average, cancel themselves.  But the few special frequencies re-inforce one another and produce standing waves.  These frequencies are often referred to as the harmonics of the system.  On occasion, they are referred to as the fundamental (n = 1) and the overtones (n > 1).

Now here's a question.  How can a listener distinguish different musical instruments which are playing the same note?  For example, an oboe and a clarinet are both essentially cylindrical tubes, closed at one end and open at the other, and so they produce the same sequence of harmonics, f_n = nv/4L.  The answer is that they each put slightly different amounts of energy into the different harmonics, and it is that distribution that your brain remembers and labels as one instrument or the other (see below).
FIGURE

Also, it is possible to suppress certain harmonics.  For example, we managed to suppress all odd harmonics on a string fixed at both ends simply by grabbing the middle of the string, thus forcing a node to form there; any harmonics which do not have a node there do not form.  We also investigated the suppression of harmonics in an aluminum rod; by grasping the rod at different points, we could make the sound the rod makes when struck different.  This technique is used in the forked fingerings of woodwinds, and in the 'nodal vents' used in some reproduction natural trumpets.

Intensity

An alternate way of expressing intensity is in units of decibels.  The decibel scale is is logarithmic, and thus follows more closely the actual size of the signal sent from human ear to human brain.  The bel is named for Alexander Graham Bell, who was not, as one might suppose, American, but rather a Scot-born Canadian working in Boston.  A reference intensity I_o is defined as 10^-12 wts/m², which corresponds roughly to the quietest sound a normal human can hear.  The intensity to be converted is compared to this standard, and the log base ten is taken of the ratio.  This gives the number of bels, so the number of decibels (dB) must be ten times more:
b = 10 log₁₀[I/I_o].
Let's try some examples:
Suppose that the sound that one professor produces (at a given distance) has an intensity of 10^-7 wts/m².  How many dB does this correspond to?

Note: we assume that we can simply add the intensities of individual waves.  This works so  long as those individual waves are incoherent.  When waves are coherent, a more difficult approach is necessary (which we'll investigate in the second semestre).  Please, no comments about how professors are always incoherent.

Here are the intensity levels of some common situations:

Situation	Intensity	Intensity level
Threshold of Hearing	10-12 W/m2	0 dB
Library Reading Room	10-9 W/m2	30 dB
Conversation	10-6 W/m2	60 dB
Vacuum Cleaner	10-3 W/m2	90 dB
Rock Concert	10-1 W/m2	110 dB
Thunder	10 W/m2	130 dB

Of course, these values depend on the distance between source and listener.  Prolonged exposure to sounds above 90dB will cause permanent damage, and exposure to sounds over 110dB will be painful.  Here are some helpful hints: ALWAYS wear ear protection in noisy situations, such as lawn mowing or vacuuming and on up.  If you must wear headphones to listen to music, place them just in front of your ears, not right over them.

Beats

Now, we'll make use of a trig identity, (which we'll prove below):
sin a + sin b = 2 sin[(a+b)/2] cos[(a-b)/2],
so that
Y_total =  2 A_o cos(2p[(f₁- f₂)/2]t) sin(2p[(f₁+ f₂)/2]t).
From this we see that the frequency of oscillation is the average of the two original frequencies, but also that the amplitude of the oscillation is modulated by an envelope with a frequency of | f₁ -  f₂|/2.  What will be heard, though is a pulsing or beating in the amplitude with twice that frequency; this is known in fact as the beat frequency:
 f_Beat =  | f₁ - f₂|.

This can be (and often is) used as a method for tuning pianos and other such instruments.  Once a 'C' string is tuned to the correct pitch, it and the 'G' a fifth above it above are struck simultaneously.  The third harmonic of the C and the second harmonic of the G are the same note, and so the G string's tension is adjusted until no beats are heard between those two harmonics (or in other tuning schemes, a certain number of beats per second should be heard, but that's a whole 'nother story...).

Doppler Effect

Consider a source moving at speed v_Source toward (approaching) a stationary observer (or listener, if you insist).  Instead of having the source emit a sinusoidal wave, let's assume that it emits pulses; we can later correlate these pulses to the peaks of a sinusoidal wave, if necessary.  Let the frequency of the pulse emitted by the source be f_o, and the time between the emission of pulses be T_o = 1/ f_o.
Here at t = 0, the source releases a pulse, which then travels to the right at speed v_Sound.

Now, let's look at the locations of everything a time t = T_o later:

Here, at a time T_o later, Pulse 1 has traveled a distance
d_Sound = v_Sound T_o,
while the source has traveled a distance
d_Source = v_Source T_o,
at which point it emits Pulse 2.
Now, the wavelength that the observer will measure is the distance between the two pulses:
l' = d_Sound - d _Source
l' = v_Sound T_o -  v_Source T_o.
Now we remember that, in general,  f  = 1/T and that fl = v_wave, and so this last expression can be re-written as
v_Sound/ f ' = v_Sound/f_o - v_Source/ f_o,
which get re-arranged to be
 f ' =  f _o [v_Sound/(v_Sound - v_Source)].
Now, if the source had instead been moving the other way (receding), those two distances would have had to have been added, changing the red minus sign to a plus sign:
 f ' =  f _o [v_Sound/(v_Sound + v_Source)].

Now, suppose instead that it were the observer moving toward (approaching) the stationary source at speed v_Observer.  Once again, let the source emit pulses at an interval of T_o.  Let t = 0 when Pulse 1 arrives at the observer:

The second pulse arrives at the observer at time T', during which interval the pulse has traveled distance (to the right)
d_Sound = v_Sound T'
and the observer has traveled distance (to the left)
d_Observer = v_Observer T'.
T' is now the time between pulses, as heard by the observer.
The sum of these distances is the old wavelength, l_o:
l_o = d_Sound + d_Observer = v_Sound T' + v_Observer T'.
Once again remembering that, in general,  f  = 1/T and that  f l = v_wave, we substiute to obtain
v_Sound/ f _o  = v_Sound/ f ' + v_Observer/ f '.
This, we re-arrange to obtain
 f ' =  f _o [v_Sound + v_Observer]/v_Sound.
Once again, if the observer had been receding from the source, there would have been a sign reversal (blue) to
 f ' =  f _o [v_Sound - v_Observer]/v_Sound.

Now, we can combine all these relationships, if we're careful.  First, we need to define better the terms 'approach' and 'recede.' 'Approach' is to head in the direction of the other object, regardless of whether the distance between the objects is becoming smaller or not, and 'recede' means to head in the opposite direction of the other object, whether the distance between is increasing or not.  Then
f' = f_o [v_Sound +/- v_Observer]/[v_Sound -/+ v_Source],
where the upper sign is used if that object is approaching and the lower sign if that object is receding.

What exactly would one do if there were wind?

What happens if the source travels more quickly than sound?  Consider these diagrams:

The upper left figure shows the locations of the crests of three waves emitted by a source such as a jet while it is stationary.  The upper right figure shows the the same when the source is moving to the right at about 0.5 the speed of sound; note that the wavelengths will be shorter for listeners in the path of the source, but longer for listeners from which the source is receding.  When the source reaches the speed of the wave in that medium, a bow shock wave is generated; this is most easily seen when generated by a boat, but recent photos of jets breaking the 'sound barrier' have caught these shock waves as they condensed water vapour in the air.  Once the speed of the source exceeds the speed of the wave in that medium, the crests of all waves co-incide to produce a giant shock wave (red line); for jets, this results in the familiar sonic boom.

Some random notes:
1) The reason the Concorde was so quiet (to the passengers anyway, not to those living on the flight path) during supersonic flight is that the the noise from the engines could not keep up with the cabin; one could hear only the noise transmitted through the body of the plane.
2) The apex angle (2q) of the cone formed by the shockwave depends on the speed of the source:

The wave shown was generated at the instant that the source was at its centre.  In time interval t, the source moved a distance v_sourcet and the sound moved outward a distance v_soundt.  Consider the right triangle formed in the diagram.  We see that
sinq = v_soundt/v_sourcet = v_sound/v_source.
The inverse of this ratio is referred to as the Mach number.  The official speed record of any jet aircraft is about Mach 3, set by a Soviet fighter.  There are rumours however that the US's SR-71 has hit Mach 5.  The Space Shuttle enters earth's atmosphere at about Mach 25.

Consider a rectangular block of a solid, length L and cross-sectional area A.  The material can be modelled by small balls of mass m (representing the atoms) connected by bonds represented by springs with stiffness k, and relaxed length l_o.  We can justify this last statement by examining the typical potential energy curve of these bonds:

The atoms would 'like' to arrange themselves so that the potential energy is a minimum, at r = l_o.  At that point, the shape of the curve is approximately parabolic, and so the system follows Hooke's law,
F₁ = - 2k(r-l_o) = - 2k Dl,
where F₁ approximately represents the x-component of the force on one atom from two adjacent atoms (the problem is actually more complicated than this, even, since the amount of stretching or compression of any spring depends on the positions of both of the masses to which it's connected.).  Now, we remember that the Young's modulus is given experimentally and macroscopically by
Y = -(F/A)/(DL/L),
where F/A is the pressure exerted on each end of the material, and DL/L is the resulting fractional change in length.  We assert that DL/L = Dl/l_o, since we assume that any change of the dimensions (during equilibrium compression or stretching ) is evenly divided among all of the bonds.  Also, the pressure F/A is the same for the whole face of the metal as for the area corresponding to one atom, F₁/l_o².  So,
Y = -(F/A)/(DL/L) = -(F₁/l_o²)/(Dl/l_o) = -(F₁/Dl)/l_o = +2k/l_o.
In the same way, the density can be found with the mass of one atom and the volume that atom occupies on average:
r = m/l_o³.
We know also from macroscopic experimentation that the speed of sound in a solid is given by
v = [Y/r]^1/2
So, we then show that
v = [Y/r]^1/2 = [2k/l_o/m/l_o³]^1/2 = [2]^1/2 l_o [k/m]^1/2
So, as asserted, the speed of propagation is proportional to the frequency of the individual oscillators:
v ~ w_o.
Does this make sense?  In general, v = d/t = l_o/(0.11T).  Yes, the disturbance travels one spring length in an appreciable fraction of the mass's oscillation period.

Trig Identity