Sound Waves

For the sake of simplicity we use P for change of pressure from normal (thus avoiding a bunch of D 's in the discussion). In a sound wave, P will have wave fluctuations, and for a pure tone traveling in the x direction we can write
P = P_msin(kx-w t), where P_m is the maximum P, the pressure amplitude.

The other variable we need to use is the displacement of molecules from their normal position, s. We can write a wave equation in terms of either s or P. To designate s at location x at time t, we write s(x,t). At a particular x the velocity and acceleration are
¶ s/¶ t and ¶ ²s/¶ t². These partial derivatives with respect to t, meaning for a fixed x.

To designate P for molecules whose home base is at x, we write P(x,t). Note the peculiar notation: it is not the pressure at x except when s is zero. This is to avoid having to write awkward stuff like P[x + s(x,t), t].

The plan will be to consider molecules which normally reside between x and x + D x, cross-sectional area A, hence volume AD x and the mass is r AD x, where r is the normal density.

(Later we shrink the D x to a very small value.) Before the derivation, let's relate P and s. If the pressure on a volume V is raised, its volume decreases: the relationship is P = -BD V/V, where B is a positive constant called the bulk modulus. The negative sign is because if P is +, D V is -. This is valid for all materials. Drop a rock into a lake and the rock decreases in volume as it sinks (OK, it is not noticeable for a rock, but it really does occur; it is just that for a rock, you would need high tech instrumentation to detect the D V. But trap an air bubble in an inverted open jar and you will notice its volume change a lot as you take it deeper into the lake.)

Now in terms of volume AD x: If s is the same at both ends, there is no volume change. The volume change is AD s. So P = -BD s/D x after canceling A, and considering very small D we have P = -B¶ s/¶ x.

Finally, consider S F = ma on the volume AD x. On the left we have force
A[P_o + P(x)], where P_o is the normal pressure, and on the right of the volume we have force
-A[P_o + P(x + D x)]. The mass is
r AD x, where r is the normal density, and the acceleration is
¶ ²s/¶ t². Hence we have
-¶ P/¶ x = r ¶ ²s/¶ t², after shrinking the D x. But if
P = -B¶ s/¶ x, then -¶ P/¶ x = B¶ ²s/¶ x². So finally,

B¶ ²s/¶ x² = r ¶ ²s/¶ t².

Let s = s_msin(kx-wt), differentiate twice with respect to x and twice with respect to t, plug into the above equation and get
Bk² = rw². Recall v = w/k. Hence we easily see that the velocity is v=(B/r )^1/2.

If the sound is traveling in a rod, the sides are unconstrained, and the appropriate constant is Young's modulus rather than the bulk modulus, but otherwise the derivation is the same.

Much of the time we are concerned with sound in air, so let's see what B is for air. You need a little calculus for this, also. It turns out that for a fast process like sound, too fast for significant heat transfer, PV ^g = constant, where g is the ratio of the specific heat at constant pressure to the specific heat at constant volume, and if this is Greek to you, you can find this in any thermodynamics book. For air, g is 1.4. Then d(PVg ) = 0, or
dPVg + P(g )Vg ^-1 = 0. Now divide by Vg and get
dP = -g PdV/V. Recalling dP = -BdV/V, we see that the bulk modulus is g P. But PV = nRT can be written P = r RT/M, where M is the molecular mass, so B/r = g RT/M, and

v = (g RT/M)^1/2.

At 0^o C = 273 K, v = 331 m/s. Then for any other T, v/331 = (T/273)^1/2. Another way is to note that (a+b)^1/2, with b<<a, is a^1/2 + b/(2a^1/2). So if T_c is the Celsius temperature,

v = [g R(273)/M + g RT_c/M]^1/2 = 331 m/s + 0.606T_c,

where the last term has units of m/s. We have used a = 331² and b = (331²/273)T_c.

If you are asked for the speed of sound, you can always say mach 1, and it is correct by definition (but don't blame me if it is marked wrong). Now undulate back to the main page on waves and stuff.

Other Main pages: mechanics,
fluid/heat/e&m,
quantum
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Do your drive-by shooting off at the mouth at fredrick.gram @ tri-c.edu (remove spaces).