phy1020.DVI

Chapter 10

Standing Waves

Supposes you attach one end of a string to a wall, and hold the other end in your hand. Now give your end a
quick “flip”, and you will see a wave pulse travel down to the wall, get inverted, and the reflected wave will
come back to you.
Now suppose you set up a continuouswave trainat your end. The waves will travel to the wall, get
inverted and reflected back toward you. On the way back, they will interfere with the waves coming in the
opposite direction, and you will get a complicated-looking jumble of interfering waves.
But suppose you time things just right, with just the right frequency, so that the returning reflected waves
interfere constructively with the waves coming the other way. In this case the waves all add together nicely,
and you get a pattern ofstanding waves. Standing wave patterns look like the patterns in Fig. 10.1; you’ll see
a set of “segments” vibrating up and down, where each segment is a half wavelength. At the points between
segments, the string does not move at all; these points are called thenodes. Halfway between the nodes are
the points of maximum displacement; these are theantinodes.
It’s important to realize that if you drive one end of the string with simple harmonic motion, you will,
in general, not get standing waves—you’ll get a jumbled mess at first, that will eventually settle into non-
standing waves that oscillate at the forcing frequency. Only at certain specific frequencies will you get
standing waves.
So what frequencies will give standing waves? That depends on whether the string is fixed at both ends,
or just one end, or if both ends are free.

10.1 Fixed or Free at Both Ends.

If the string is fixed at both ends and the ends are a distanceLapart, then you can see from examining
Fig. 10.1 that an integer number of segments have to fit into the distanceL. Since each segment is a half
wavelength, the condition for standing waves in this case is that an integer number of half-wavelengths must
fit into lengthL:

LDn

2

.nD1;2;3;4;:::/ (10.1)

Now since the wave speedvDf, we can substitute for and solve forfto find an expression for the
frequencies that give rise to standing waves:

fnDn

v

2L

.nD1;2;3;4;:::/ (10.2)

As shown in Fig. 10.1, there is a sequence of standing waves, one pattern for each integernD1;2;3;4;:::.
The standing wavef 1 is called thefirst harmonic; the next one (f 2 ) is called thesecond harmonic, and so