Simple Nature - Light and Matter

mathematical statement of the intuitive fact developed above, thatthe string accelerates so as to flatten out its curves.Before even bothering to look for solutions to this equation, wenote that it already proves the principle of superposition, becausethe derivative of a sum is the sum of the derivatives. Therefore thesum of any two solutions will also be a solution.Based on experiment, we expect that this equation will be satisfiedby any function y(x, t) that describes a pulse or wave patternmoving to the left or right at the correct speed v. In general, sucha function will be of the form y = f(x − vt) or y = f(x + vt), wheref is any function of one variable. Because of the chain rule, eachderivative with respect to time brings out a factor of v. Evaluatingthe 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 validsolution for any function f, provided that v is given by√Tv = .µ6.1.3 Sound and light wavesSound wavesThe phenomenon of sound is easily found to have all the characteristicswe expect from a wave phenomenon:- Sound waves obey superposition. Sounds do not knock othersounds out of the way when they collide, and we can hear more thanone sound at once if they both reach our ear simultaneously.- The medium does not move with the sound. Even standing infront of a titanic speaker playing earsplitting music, we do not feelthe slightest breeze.- The velocity of sound depends on the medium. Sound travelsfaster 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 handsa short distance from a large, flat wall, and the delay of the echo isno shorter for a louder clap.Although not all waves have a speed that is independent of theshape of the wave, and this property therefore is irrelevant to ourcollection of evidence that sound is a wave phenomenon, sound doesnevertheless have this property. For instance, the music in a largeconcert hall or stadium may take on the order of a second to reachsomeone seated in the nosebleed section, but we do not notice orSection 6.1 Free Waves 349