The third special case of solutions to the wave equation is that of *standing waves*. They are especially apropos to waves on a string fixed
at one or both ends. There are two ways to find these solutions from
the solutions above. A harmonic wave travelling to the right and
hitting the end of the string (which is fixed), it has no choice but to
reflect. This is because the *energy* in the string cannot just
disappear, and if the end point is fixed no work can be done by it on
the peg to which it is attached. The reflected wave has to have the
same amplitude and frequency as the incoming wave. What does the *sum* of the incoming and reflected wave look like in this special case?

Suppose one adds two harmonic waves with equal amplitudes, the same
wavelengths and frequencies, but that are travelling in *opposite*
directions:

(121) | |||

(122) | |||

(123) |

(where we give the standing wave the arbitrary amplitude ). Since all the solutions above are independent of the phase, a second useful way to write stationary waves is:

(124) |

In this solution a sinusoidal form oscillates harmonically up and down,
but the solution has some very important new properties. For one, it is
always *zero* when for all possible :

(125) |

(126) |

(127) |

(128) |

*Only waves with these wavelengths* and their associated frequencies
can persist on a string of length fixed at both ends so that

(129) |

It is also possible to stretch a string so that it is fixed at one end
but so that the *other* end is *free to move* - to slide up and
down without friction on a greased rod, for example. In this case,
instead of having a node at the free end (where the wave itself
vanishes), it is pretty easy to see that the *slope* of the wave at
the end has to vanish. This is because if the slope were not zero, the
terminating rod would be pulling up or down on the string, contradicting
our requirement that the rod be frictionless and not *able* to pull
the string up or down, only directly to the left or right due to
tension.

The slope of a sine wave is zero only when the sine wave itself is a
maximum or minimum, so that the wave on a string free at an end must
have an *antinode* (maximum magnitude of its amplitude) at the free
end. Using the same standing wave form we derived above, we see that:

(130) |

(131) |

There is a second way to obtain the standing wave solutions that
particularly exhibits the relationship between waves and harmonic
oscillators. One assumes that the solution can be written as
the *product* of a fuction in alone and a second function in
alone:

(132) |

(133) | |||

(134) | |||

(135) |

where the last line follows because the second line equations a function of (only) to a function of (only) so that both terms must equal a constant. This is then the two equations:

(136) |

(137) |

From this we see that:

(138) |

(139) |

(140) |