Standing waves on tubing

A variable-speed motor drives a crank and lever assembly, which causes one end of a long piece of rubber tubing to oscillate up and down. This periodic motion sends a wave down the tubing, which is reflected at the other end of the tubing, which is fixed. If you adjust the motor speed appropriately, the reflected wave and the outgoing wave combine to produce a standing wave. You can use a strobe light to “freeze” the motion of the tubing, but the nodes and antinodes are clear enough that the standing wave patterns are fairly easy to see without a strobe light.

The apparatus shown above consists of a variable-speed motor drive, which turns a crank that operates a lever, to one end of which is attached one end of a length of latex rubber tubing. The motor drive sets the rubber tubing oscillating, sending a transverse wave down it in the vertical plane. The other end of the tubing is fixed, which causes the wave to be inverted upon reflection from it. (You can show this with demonstration 40.60 -- Transverse wave machine with free or fixed end, or with 40.54 -- Waves on cord, tubing, spring. When you send a pulse down the wave machine with the opposite end clamped, or down the cord, tubing or spring, the reflected pulse comes back inverted.) As the outgoing and reflected wave pass through each other, the displacement of the tubing at any point is the alegebraic sum of the diplacement of the individual waves. This is called superposition or interference. At the fixed end, where the tubing cannot move, the displacements of the two waves always sum to zero. The end at the motor drive is constrained by the lever, and this end also acts as a fixed end. At points between the ends, the displacement of the tubing is whatever the sum of the displacements in the outgoing and reflected waves is. At certain drive frequencies, the superposition of the two waves produces a pattern of alternating regions in which the displacement of the tubing is a maximum, called antinodes, and points where the displacement is zero, called nodes. Such a pattern is called a standing wave.

We may express the initial, outgoing, wave as y1 = ym sin (kx - ωt), where y is the vertical displacement, m stands for “maximum,” k is the wave number, which equals 2π/λ, ω is the angular frequency (in rad/s), and t is time in seconds. The reflected wave, then, is y2 = ym sin (kx + ωt), and their sum is

y = ym sin (kx - ωt) + ym sin (kx + ωt)

If we use the trigonometric equation for the sum of the sines of two angles (sin B + sin C = 2 sin (1/2)(B + C) cos (1/2)(C - B)), this becomes

y = 2ym sin kx cos ωt

We see that the amplitude of this wave has maximum values for kx = π/2, 3π/2, 5π/2, etc., which correspond to x = λ/4, 3λ/4, 5λ/4, etc. These places where the amplitude is maximum are called antinodes, and they are spaced one-half wavelength apart. The amplitude is minimum when it is zero, which occurs for kx = π, 2π, 3π, etc., which correspond to x = λ/2, λ, 3λ/2, etc. These places where the amplitude is zero are called nodes, and they also are spaced one-half wavelength apart.

In order for a standing wave to form in this apparatus, it must have a node at either end of the rubber tubing, with any number of nodes in between (including zero). For this to happen, an integral number of half wavelengths must fit between the ends of the tubing, or /2 = l, where n = 1, 2, 3, . . . Since λ = v/ν, the frequency at which we must drive the system must satisfy the equation ν = nv/2l, where n = 1, 2, 3, . . . These frequencies correspond to the natural oscillating frequencies of the system, and if we drive the system at any of these frequencies, it oscillates with relatively large amplitude. This phenomenon is called resonance. For the rubber tubing, the speed of the wave, v, equals √(T/μ), so the frequencies at which resonance occurs and we obtain a standing wave are ν = (n/2l)√(T/μ).

The lowest frequency for which we obtain a standing wave is called the fundamental or first harmonic, for which n = 1, and its standing wave pattern has a node at each end of the tubing and one antinode in the middle. For n = 2, we obtain the second harmonic, which has a node at each end of the tubing, one node in the center, and two antinodes midway between the center and each end. The photograph above shows the apparatus operating at the second harmonic. Each successive harmonic has one more node and one more antinode than the one below it. The third harmonic has four nodes and three antinodes, the fourth harmonic has five nodes and four antinodes, etc.

With this apparatus you should be able to dial the frequency from the fundamental up to the fourth harmonic. If you dial up the frequency first and then switch the motor on, you may be able to obtain the fifth harmonic.

References:

1) Resnick, Robert and Halliday, David. Physics, Part One, Third Edition (New York: John Wiley and Sons, 1977), pp. 417, 420-424.