Of Particular Significance

# Chapter 10, Endnote 3

• Quote: In a sense, frequency and amplitude keep out of each other’s way. This carries over to our perceptions: frequency determines pitch without affecting loudness, while amplitude determines loudness without affecting pitch.

• Endnote: These statements are true as long as the amplitude is small enough. When the amplitude is very large, the independence of frequency and amplitude may fail. For instance, overblowing a flute or a recorder pushes the frequency (and the pitch) up slightly.

In first-year physics, the example is given of a ball on a idealized spring. I’ve discussed this in detail in this article, but I’ll do a quick summary here.

If the spring is unstretched, the ball will be stationary. If the ball is moved a distance x away from this position, the spring will pull it back with a force F that is proportional to x:

$F = - \kappa x$

where κ is a constant that tells us how strong the spring is. Subjected to this force, the ball will bounce back in forth, its motion a sine or cosine wave over time

$x = A \cos(2 \pi f t)$

where A is the amplitude of the motion, chosen by us and dependent on how far we moved the ball before releasing it, and f is the resonant frequency of the motion, determined by the system itself:

$f = \sqrt{\kappa/m}$

where m is the mass of the ball. Notice the frequency is independent of the amplitude.

But realistically, no spring gives a force law with F exactly proportional to x. Let’s take something a bit more realistic. Let’s still keep it symmetric: if we compress the spring by a distance x, we get the same force as we would find if we stretched the string by the same distance x, except pointing in the opposite direction. The simplest smooth force law that we can get, beyond the one we started with, is then

$F = - (\kappa x + \lambda x^3)$

As I’ll justify in a moment, the resulting motion is now only an approximate cosine, and its frequency is now

$f = \sqrt{\frac{\kappa + \frac{3}{4}\lambda A^2}{m}}$

which, as you see, depends on the amplitude A. If A is very small, then the second term is very small also, and we recover the result we had before, for which the frequency is independent of the amplitude. But as the amplitude A becomes larger, the frequency f starts to depend upon it.

This feature is also true of a pendulum. Due to the fact that the pendulum bob moves on a circle, the force on the pendulum the force is slightly smaller than for an ideal spring (i.e. the quantity λ is negative), and so the frequency becomes slightly lower as the amplitude increases. This effect has to be accounted for in pendulum clocks.

Now, if you’d like to get a sense for why this shift in the frequency happens, here’s a crude estimate, using a bit of math and physics, the more or less reproduces the result just stated. (A full calculation can be done in various ways, but to do it correctly requires some additional work.)

We use Newton’s second law

$F = m a = m\frac{d^2 x}{dt^2} = - \kappa x - \lambda x^3$

and make a guess that the motion is still nearly sinusoidal, and that the frequency of this motion is only slightly shifted from what it was before:

$x \approx A \cos(2 \pi f t)$

where we assume A is not too large and f is not too far from the original resonant frequency.

We will also make an approximation that, since the motion is nearly sinusoidal, we can take some averages:

$x^3 \approx 3 x (\overline{ x^2})$

where by

$\overline{ x^2 } =\frac{1}{2} A^2$

we mean the average value of x2 during the near-sinusoidal cycles (using the fact that the average value of the square of a cosine, over a full cycle, is 1/2.) This gives us

$m\frac{d^2 x}{dt^2} \approx -m f^2 A = - \kappa A + \frac{3}{2} \lambda A^3$

Dividing by (m A) gives

$f^2 = \kappa/m + \frac{3}{2}\frac{ \lambda A^2 }{ m}$

which is almost right; there is a factor of 2 off in the second term. [??]

Despite this imperfection, which can be fixed by doing the calculation more carefully, this expression does capture the correct idea: the frequency now does depend on the amplitude squared, and is independent of the amplitude only when the amplitude is very small.

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