READ: Sections 13.2, 7, 8; EXAMPLES: 7, 8, 9; PROBLEMS:
12.73, 13.45, 51, 54, 67
Review Kinds ofWaves.
Superposition is the property of waves in which two are added together,
y (x,t) = y1(x,t) + y2(x,t)
See problems 72 and 73 in Chapter 12: wave pulses moving in opposite directions
on a string are combined.
One example of superposition is the production of beats. The two waves
which are added together have slightly different frequency and wavelength. As
these waves combine, they are sometimes "in phase", and sometimes "out of phase".
In sound waves, the result is alternating loud and soft, with frequency which
depends on the frequencies of the two combining waves:
Click here to see some animations which demonstrate
Another example of superposition is the production of standing waves, which are produced when two waves of the same frequency and wavelength move in opposite directions. Click here for a demonstration of standing waves.
For standing waves on a string, each end is a node (point where there is no displacement or motion). The length L of the string must equal an integer number of half wavelengths: L = n / 2.
- = 2L/ n
For standing sound waves in a pipe which is open at both ends, each end is (approximately) an antinode (point where displacement of air molecules is maximum). The length L of the pipe must equal an integer number of half wavelengths. This is referred to as an open pipe. If one end is closed, there must be a node at that end, and the length L of the pipe equals an odd integer number of quarter wavelengths. This is referred to as a closed pipe.
- fn = v/ = n v/ 2L
- fn = n v/ 4L, n = 1, 3, 5, ...