Simple Nature - Light and Matter

mathematical statement of the intuitive fact developed above, that
the string accelerates so as to flatten out its curves.

Before even bothering to look for solutions to this equation, we
note that it already proves the principle of superposition, because
the derivative of a sum is the sum of the derivatives. Therefore the
sum of any two solutions will also be a solution.

Based on experiment, we expect that this equation will be sat-
isfied by any functiony(x,t) that describes a pulse or wave pattern
moving to the left or right at the correct speedv. In general, such
a function will be of the formy=f(x−vt) ory=f(x+vt), where
f is any function of one variable. Because of the chain rule, each
derivative with respect to time brings out a factor ofv. Evaluating
the second derivatives on both sides of the equation gives

(±v)^2 f′′=

T

μ

f′′.

Squaring gets rid of the sign, and we find that we have a valid
solution for any functionf, provided thatvis given by

v=

√

T

μ

.

6.1.3 Sound and light waves

Sound waves

The phenomenon of sound is easily found to have all the char-
acteristics we expect from a wave phenomenon:

• Sound waves obey superposition. Sounds do not knock other
  sounds out of the way when they collide, and we can hear more than
  one sound at once if they both reach our ear simultaneously.


• The medium does not move with the sound. Even standing in
  front of a titanic speaker playing earsplitting music, we do not feel
  the slightest breeze.


• The velocity of sound depends on the medium. Sound travels
  faster in helium than in air, and faster in water than in helium.
  Putting more energy into the wave makes it more intense, not faster.
  For example, you can easily detect an echo when you clap your hands
  a short distance from a large, flat wall, and the delay of the echo is
  no shorter for a louder clap.

Although not all waves have a speed that is independent of the
shape of the wave, and this property therefore is irrelevant to our
collection of evidence that sound is a wave phenomenon, sound does
nevertheless have this property. For instance, the music in a large
concert hall or stadium may take on the order of a second to reach
someone seated in the nosebleed section, but we do not notice or

Section 6.1 Free waves 363