BACKGROUND AND THEORY

Sound is the general term we apply to the propagation of vibratory disturbances through matter. In this experiment, we shall concern ourselves with sound waves in the air.

We are familiar with the fact that sound waves move through the air at somewhat over 300 meters per second, and that loud events can be heard for every long distances; yet it is true that the total distance travelled by any sample of air in carrying the sound is so small as to be usually microscopic.

A physical picture of what happens to the individual molecules may be seen in the following quite idealized diagram.

Figure 1

Each row of dots represents air molecules, which are assumed to be evenly spaced (meaning that they have uniform density) in unperturbed air, as represented for time t₀ in row 1. In row 2, at t₁, a sound wave, having the shape of a rather sudden pulse, has appeared. Its effect is to force some of the molecules ahead along its direction of travel; those near the center of the pulse have been displaced farthest, which those ahead and behind displaced less. (Arrows in the diagram have lengths representing the amount of displacement of the particular molecules over which they are drawn). Note that the result of this displacement is to crowd some molecules together, thus increasing their density, while moving others farther apart and so, lowering their density.

As the wave moves past, the molecules that were displaced now fall back to where they were, and new samples of air farther ahead are given movement. Having this basic picture in hand, we must now establish a quantitative description of the process.

II. MATHEMATICAL REVIEW

We recall first that if some variable (e.g., velocity (v) or density ()) can be associated with a wave motion, its variation with space and time is such as to satisfy the wave equation:

=

where c is the speed of the wave. For sound,

c² =

where B is the bulk modules of elasticity and is the density. In sea-level air at 20deg.C, c = 343 meters/sec.

Any function of the form

f(x ct)

satisfies (1), as can be seen by direct substitution, and one may see by inspection that this function represents a traveling wave: let x ct, and f become a constant, indicating that at this speed (c) one is moving along with the disturbance.

The most convenient function for mathematical analysis, as well as the most important in practice, is the trigonometic, or "sine" wave:

f(x,t) = or

where the sine and cosine functions are equivalent except for a shift of origin.

We employ here the quantities k and c, defined by

k = 2/,

where is the wavelength, or distance between successive maxima of the wave, and

= 2,

where is the wave frequency in cycles per second or hertz, as seen by a stationary observer past whom the wave is moving. The speed of the wave is then given by

c = .

III. MASS CONSERVATION

To further develop the relations we need, we employ the notion of mass conservation, which says simply that the total rate of mass increase inside a region is equal to the total rate of mass inflow through its boundaries. For the one-dimensional problem, we consider the region between x and x + x, bounded by two infinite planes normal to the x - axis.

Figure 2

If n designates the molecule number density, we see that the number of molecules between the planes, per unit area of the planes is

N = nx .

The principle of mass conservation (or particle conservation here) then states that

= = flow in - flow out .

But, the flow, or flux, of particles in number per unit area per second is the product of density times average velocity; so,

= (nv)_x - (nv)_x+x .

From the definition of the derivative, we have that

(nv)_x+x - (nv)_x = = ( + )x .

Now, we insert the fact that for ordinary sound, n fluctuates by only an infinitesimal fraction of its mean value, while v changes by a very large fraction of its mean, which is near zero. This means that

>> ,

so that we can neglect the second term in the parenthesis and finally get

= 0 .

Since the mass density is = nm,with m the mean molecular mass (which is constant), we get the more usual form of the one dimensional partial differential equation of mass conservation:

= 0 .

Note that we have changed to the partial derivative notation to show that when a derivative is taken, other variables are held constant.

Consider now how and v will be related to each other in a simple monochromatic (single frequency) sound wave. First, in accordance with our assumption that the average density is only slightly perturbed, we write its distribution as

= ₀(1 + sin(kx-t)), where << 1.

Figure 3

Then

= - ₀cos(kx-t) .

Secondly, we must assume that the velocity of the particles also varies sinusoidally with the same frequency and wavelength, but perhaps shifted in phase with respect to ; i.e.,

v = v₀sin(kx-t+),

where shifts the wave maxima of and v with respect to one another.

Differentiating (6) with respect to x, we get

= kv₀cos(kx-t+) .

Now, (5) and (7) will exactly satisfy (3), the mass conservation equation, provided we set

= 0 and v₀ = = c .

These say, then, that

1. The density and gas velocity are in phase with each other, i.e., they reach their maxima together in space and time, and

2. The maximum speed of the gas particles is only a fraction of the wave speed c .

Lastly, consider the displacement, s, of the gas from equilibrium as the wave passes. Since

v = ,

we can immediately conclude that

s(x,t) = cos(kx-t) .

Displacement is thus seen to be out of phase with density and velocity by /2 radian, meaning that it is zero when the other two are maximum, and vice versa.

IV. REFLECTION AND RESONANCE

Suppose a sine wave is incident from the right upon a rigid wall at x = 0. Consider now the displacement function, s, as the wave variable

Figure 4

Since it is moving leftward, it has the form

s = s₀sin(kx+t) .

Now, we must contend with a boundary condition at the wall, which is that exactly at the surface (x = 0) no displacement is possible. Nature takes care of this by generating, at the wall, a second wave of equal amplitude that propagates rightward:

Figure 5

Thus, the sum of the two wave amplitudes becomes

s = s₀sin(kx+t) + s₀sin(kx-t) .

Since sin (a+b) = sin(a)cos(b) + cos(a)sin(b) we get

s = 2s₀sin(kx)cos(t) .

From this new form, we see that the composite of the incident and reflected waves is a standing wave that is always zero at x = 0, and so, satisifies the boundary condition that s(0,t) = 0 for all t. It has the further property, moreover, that s(x_n,t)=0 for

x_n = = , n = 1, 2, ....

Figure 6 Each curve represents s at a different time.

The condition of resonance is achieved if we place a second wall at some position x = a, such that a = x_n.

Figure 7

Waves are reflected back and forth between the walls, always satisfying the s = 0 boundary condition at both surfaces.

It is important to realize that the only condition permitting multiple back and forth reflection of a wave between two walls is (from above)

k_na = n,

or

_n = n = 1, 2, 3, ....

The corresponding resonant frequency is then

_n = = .

Waves of any other length will quickly cancel themselves out to zero amplitude after one or two passes.

The density in this situation behaves, according to what we have seen, in an out-of-phase (90 degree) manner, that is a cosine function. For convenience, we define a function (x,t) which represents the fluctuating part of the total density, from which it follows (eq. 4) that

= ₀ (kx t) where ₀ = ₀ .

We then have,

(x,t) = 2₀cos(kx)cos(t) .

Notice here that density () has a maximum at the wall, but that, as a function of time, and s reach their maxima together, and go through zero at the same instant.

Figure 8

Since

v = ,

we get

v = 2v₀sin(kx)sin(t) .

This shows that velocity must vanish at the walls, and that it vanishes everywhere when s and are maximum. But this is a familiar result: in any oscillation, the velocity is zero at the time of maximum displacement, and is maximum just when the displacement swings back through zero.

V. TWO-DIMENSIONAL RESONANCE

Our next step toward the real situation we will study in this experiment is to look at a geometry consisting of four walls that together make a rectangular enclosure. What rules govern the reflections of waves within this box?

Figure 9

It is easy to conclude intuitively and correctly that resonance will occur here if waves are moving parallel to x, in which case the resonance condition we derived previously applies, e.g.,

ka = n, n = 1, 2, ....

Similarily, if the waves were propagating along y, the condition would be

kb = n, n = 1, 2, ....

Now, however, we can consider the additional possibility that resonance might occur for waves moving in off-axis directions.

Figure 10

To study this problem, we will resort to a more mathematically formal approach than we have employed so far. After our answers have been derived, it will be shown how they correspond to a fairly simple physical picture.

We begin with the wave equation, but now written in a form correct for two dimensions.

=

where again, c² = B/ is the square of the wave speed.

Since this equation contains derivatives with respect to x, y, and t, it follows that is a function of these three variables, and perhaps more. Let us assume, then that

= (x,y,t) .

Now, in looking for a function that satisifies the wave equation, the simplest procedure is to try out particularly simple trial forms. If one such trial works, then we know also that it can be achieved in a real experiment.

We can be guided by the fact that we are looking for resonances. We have seen in the one-dimensional case that resonant solutions are not of the form

f(x ct),

but are, rather, of the form

f(x) h(t) where f(x) = ₀cos(kx) and h(t) = cos(t)

i.e., they are standing waves.

Let us then try, for the two dimensional problem, a solution

(x,y,t) = f(x)g(y)h(t)

where is assumed to be the product of three functions, each of which depends on one variable alone. This is a very special, restrictive assumption, justified only by the fact that it works!

When this is substituted into the wave equation, we get

= .

Now, divide each term by fgh

= .

Notice now that each term in the equation depends on a different variable and on that variable alone. The only way the equation can remain true while x, y, and t are independently varied is for each term to be equal to a constant. Thus,

= Af,

= Bg,

= Ch.

One may recognize each of these simple differential equations as having an oscillating solution. For example, eq. (16) is satisfied by

f(x) = cos(k_xx), if we set A = -k_x².

Similarily,

g(y) = cos(k_yy); B = -k_y²,

h(t) = cos(t); C = -² .

Since eq. (16) is just A+B = C, we get, upon substituting,

k_x²+k_y² = .

Let us now return to eq. (14), and substitute our trigonometric functions for f, g, and h:

= ₀cos(k_xx)cos(k_yy)cos(t)

(we have multiplied by an arbitrary ₀, which does not invalidate the solution).

At this point, we may argue that k_x and k_y are now found independently subject to the kind of "quantifying" rule we found for one-dimensional resonance:

Let k_xa = l, l = 0, 1, 2, ....

and k_yb = m, m=0, 1, 2 ....

Here, we have allowed l and m to be zero. For example, if m = 0, then eq. 20 becomes

= ₀cos(k_xx)cos(t)

which is exactly the one-dimensional resonance of Section IV, eq. 12, and which, we surmised at the head of this section, could exist in our rectangular box. Similarly, if we set l = 0, we get resonance along y.

Together, however, l and m 0 produce a quite new resonant condition. To restate eq. (19),

k_x²+k_y² = .

Therefore

= ,

or,

= since =2

i.e.,

_lm = .

Note that when, for example, m = 0, we get

_l0 = ,

which is the same formula given in the previous section.

The resonances characterized by simultaneously non-zero l and m result from waves propagating diagonally across the box in such a way that after four reflections they once again repeat themselves.

Each resonant configuration of waves, corresponding to a particular l and m, is called a normal mode, or eigenmode of the system. The associated frequencies _lm are called eigenfrequencies.

Expressions for velocity and displacement for the two-dimensional box will be somewhat more complicated because these are vector quantities, and so, have separate x and y components.

The mass conservation equation is now a vector expression:

= 0

where , the divergence of the vector velocity is:

= .

Since = ₀cos(k_xx)cos(k_yy)cos(t), it follows (you can verify this by substitution in eq. (23)) that the velocity components are

v_x = v₀sin(k_xx)cos(k_yy)sin(t)

and

v_y = v₀cos(k_xx)sin(k_yy)sin(t) .

To assist in visualizing these patterns of density and velocity, we include here some diagrams depicting their distribution.

Figure 11

The left-hand diagram is for l = m = 1 at t = 0. The shaded areas represent a density maximum, and clear areas are for density minimum.

The right-hand diagram shows the pattern of velocities at t = /2 or one quarter-cycle later than the time of density maxima. Note that:

1. Flow at the walls is only parallel to the walls.
2. Flow is always away from (previously) high density regions toward low density regions.
3. Flow vanishes at the point where four regions come together.

The result of this flow pattern will be to reverse the signs of the density extrema when the velocity again goes to zero another quarter cycle later. Then the whole process reverses and brings us back to where we began.

Some more typical normal modes of the box are:

Figure 12

VI. THREE DIMENSIONS

Any real physical system is three-dimensional, and so, to make our discussion complete, so that it relates to actual problems, we must make our resonant box three-dimensional.

The mathematical procedure is nearly obvious. We add a third term to the spatial derivatives on the left-hand side of eq. 13 (the wave equation) and through the same "separation of variables" technique, arrive at a wave function that is a product of three spatial parts and a time part; e.g.,

(x,y,z,t) = ₀cos(k_xx)cos(k_yy)cos(k_zz)cos(t) .

The eigenfrequencies now have three quantum numbers l, m, and n, where

_lmn =

where d is the box dimension along the z axis.

A typical eigenmode, in this case l = 1, m= 2, n = 0, or in usual notation, the 120 mode, could be diagrammed as follows:

Figure 13

Here, the shaded rectangular cells are at density maxima, and the light cells at minima. One half-cycle later in time, the shadings will have reversed. One quarter cycle later in time than the diagram, all density will be uniform, but there will be a maximum velocity at the cell interfaces, directed normal to each interface, and away from the previously high density.

VII. COUPLING TO NORMAL MODES

If we wish to set up a condition of oscillation in a resonant system, we usually must inject energy at some local point, and in such a way that the displacement which is forced at the point agrees in direction and frequency with the displacement the desired mode would itself produce there.

For example, in our present experiment, the resonant system is a hollow rectangular box, and we excite sound waves within it by placing a small loudspeaker at various positions in the interior. The loudspeaker (or transducer) forces the air near it to oscillate back and forth along the direction of its axis of symmetry. Any normal mode whose displacement at the speaker position is not zero, and not perpendicular to the speaker axis, will be excited when its particular eigenfrequency is applied to the speaker.

Let us consider some examples. Suppose we place the speaker at the exact center of our rectangular box, with its axis aligned along x .

Figure 14

Now, we represent three different modes, M_lm0, and inquire as to the ability of the driver to excite them:

Figure 15

Since (effectively eq. 24)

v_x = v₀sin(k_xx)cos(k_yy)cos(k_zz)sin(t),

we get for the three modes:

(100): v_x = v₀ ;

(110): v_x = v₀ ;

(200): v_x = v₀ .

The speaker is at x = a/2, y = b/2

so, at this center position,

(100): v_x = v₀sin(t)

(110): v_x = 0

(200): v_x = 0 .

Of these three modes, only (100) can be excited. However, you may verify that if the speaker is moved off-center along y, (110) will be excited, and if it is shifted along x, (200) will respond.

The effect of turning the speaker axis into different directions (not necessarily along the coordinate axes) can also be inferred in the above manner. We leave some such exercises to the experiment itself.

Finally, we must consider the problem of detecting the presence of energy in various modes. This is done, not surprisingly, by placing a device sensitive to air motion at selected points in the cavity and measuring its electrical (or other) output. This device is, of course, a microphone, and for that purpose in this experiment, we employ a second small loudspeaker identical to the one used as a driver.

The rule for placement of the microphone is simple: energy in a mode can be detected if the microphone is placed so that the mode velocity is not zero at its position and not perpendicular to its axis.