A function generator drives a mechanical driver, which is loosely attached to an elastic cord near one end. The end of the cord near the driver is tied to a clamp. The other end goes over a pulley, and a large mass hanger hangs from the bottom of it to provide tension, which you can adjust by adding more masses. With this apparatus you can demonstrate standing waves on the cord, which should be visible with or without the optional strobe light. The adjustable tension allows you to vary the natural frequency of the cord.
Below is a photograph of the system in operation with the strobe light. For purposes of photography, the apparatus is set up so that the cord is backlit by the strobe. You can also shine the strobe on the cord to light it directly.
This demonstration is one of several whose purpose is to show standing waves. (See 40.72 -- Standing waves on tubing, 40.78 -- Standing waves on a wire circle, 40.60 -- Transverse wave machine with free or fixed end, 44.33 -- Speaker over pipes, 44.36 -- Speaker over water-filled tube and 44.39 -- Rubens flame tube.) A standing wave occurs when a periodic wave traveling through a medium is reflected back on itself in such a way as to create alternating regions of constructive interference and destructive interference, forming, respectively, antinodes and nodes. (The addition of two waves this way is also called superposition.) This depends on the reflected wave being in the proper phase relation with the original wave, which is determined by the speed of the wave in the particular medium, the length of the medium, the frequency of oscillation, and whether the reflection retains its original phase, as at an open end, or is inverted, as at a closed end. (The effect of the last constraint also depends on whether the end from which the wave originates behaves as an open or a closed end.)
Mathematically, 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, as noted above, 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, as noted above, and they also are spaced one-half wavelength apart.
As shown in the explanation for demonstration 40.54 -- Waves on cord, tubing, spring, the speed of a wave along a string is v = √(T/μ), where T is the tension in the cord and μ is the mass of the string per unit length. Both ends of the cord are fixed, so a reflection at either end is inverted, so in order for a standing wave to form in this apparatus, it must have a node at either end of the cord, 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 cord, or nλ/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, or (n/2l)√(T/μ), 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.
The length of the cord from the clamp to the top of the pulley is about 1.57 m, and μ for the relaxed cord is 1.7 × 10-3 kg/m. With only the mass hanger, which is 1 kg, the tension is 9.81 N. The wave speed is √(9.81 kg·m/s2/1.7 × 10-3) m, or 76 m/s. The wavelength of the fundamental is twice the string length, or 3.1 m, which gives the frequency of the fundamental as (76 m/s)/(3.1 m), or 24 Hz, which is the actual frequency that excites the fundamental in this apparatus when only the mass hanger is attached.
For a given tension, you can vary the frequency to obtain standing waves for the higher harmonics. You can also add more mass to increase the tension and, thus, the frequencies of the fundamental and higher harmonics. If you use the strobe, below about 33 Hz, you can use the EXT INPUT/MED INTENSITY setting. For all frequencies above this, you must use the EXT INPUT/LOW INTENSITY setting. A table of frequencies for various tensions, calculated as above for the tension of 9.8 N, is below:
An interesting thing to do is to set a mass equal to 3 kg on the mass hanger to make the tension 39 N, tune the fundamental, and then remove the three kilograms to reduce the tension to 9.8 N. The four-fold reduction in tension halves the frequency of the fundamental, so the cord is now being driven at the second harmonic. When you remove the mass, then, the standing wave pattern instantly changes from that of the fundamental to that of the second harmonic.
References:
1) Resnick, Robert and Halliday, David. Physics, Part One, Third Edition (New York: John Wiley and Sons, 1977), pp. 417, 420-424.
