professor Dave here let's discuss
standing waves explain waves in the
ocean can travel a great distance just
like sound waves but some are confined
to a specific region like if you shake a
taut rope with one end fixed in space
waves will travel down this rope and
then back again reflected at the
boundary some waves are confined between
two fixed boundaries like on a plucked
string where they will experience
repeated reflections at both ends of the
string resulting in a multitude of wave
cycles traveling in both directions if
this vibration is of a particular
frequency it will produce an
interference pattern that appears to be
a completely stationary wave these are
called transverse standing waves these
kinds of waves can only have particular
frequencies for strings of a given
length because they can only have
integer numbers of half wave lengths
since the waves must return to zero
amplitude at both boundaries if the
number of half wave lengths was not an
integer the wave couldn't exist this
means that the number of half wave
lengths in any standing wave must be
quantized meaning it can only exhibit
certain discrete values such as the set
of integers rather than any value from a
continuous spectrum and this idea of
quantization will be an important one
when we get to the modern physics course
standing waves contain nodes where there
is destructive interference and an
amplitude of zero as well as anti nodes
where the amplitude is at a maximum the
string will be completely stationary at
the nodes and the other sections if
vibrating rapidly enough will appear to
the human eye to create loops and more
nodes means more energy two dimensional
standing waves also exist which manifest
on a flat two-dimensional surface like a
drumhead
rather than a one-dimensional string on
a one-dimensional string the
ODEs are points but on a two-dimensional
surface the nodes are lines and curves
these can be radial nodes which are
circles of a particular radius and
angular nodes which are lines at
particular angles these nodes can be
visualized if we place sand on a
vibrating surface the sand gets pushed
over to the node to by the vibrations
and this helps us visualize the
two-dimensional standing wave a
three-dimensional standing wave would
have two dimensional planes as nodes but
we will have to use our imaginations for
that one interesting thing about
standing waves is that they combine to
produce all of the consonant intervals
in music that sound good to our ears if
we consider this half wavelength to be
the first harmonic with a frequency of
f1 then doubling this frequency to 2 f1
will produce a standing wave with one
full wavelength and will give us the
second harmonic which in music would be
perceived as an octave tripling the
original frequency to three f1 will give
us the third harmonic which in music is
called a fifth this will continue to
give the rest of the consonant intervals
so the reason that certain combinations
of notes sound Pleasant is no random
matter it is rooted in the harmonics of
standing waves this concept of standing
waves also represents our first
encounter with quantization given that
each harmonic has an integer number of
half wave lengths and quantization will
be a very important concept in the
modern physics course as we will later
see standing waves are at the heart of a
number of quantum phenomena that seemed
to completely defies logic such as the
quantization of energy of an electron
but we aren't quite there yet so let's
check comprehension
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