How are the longitudinal waves of a C note different than a D note diagrammatically?

Question

In the same octave of an instrument, we hear that the sound produced are different. What cases this change? Which term tells us about how the different tones are produced?

Answer 1

How are the longitudinal waves of a C note different than a D note diagrammatically?

Notes with different pitches have different frequencies. For example: The C$_4$ note is defined to have a frequency of $262$ Hertz, while the D$_4$ note has a frequency of $294$ Hertz. (Hertz is the unit for measuring frequency, just meaning oscillations per second.)

So the diagrams (compression/rarefaction vs. time) of these notes look like this:

_{(diagrams created with Desmos calculator)}

But actually the description above is still an oversimplification, because it doesn't consider the timbre of the tones. This means, real tones from musical instruments are not pure sine waves as shown above, but instead come with some overtones added to it. For example: A D$_4$ tone contains not only the fundamental frequency ($294$ Hz), but also some overtone frequencies ($2\cdot 294$ Hz, $3\cdot 294$ Hz, $4\cdot 294$ Hz, ...). The following diagrams show how the D$_4$ tone from some different musical instruments may look like:

We perceive all of these still as a D$_4$ note, but with different timbres. From the characteristic amounts of overtones we can distinguish whether the tone is from a piano, a flute, a trumpet, or what ever.

Answer 2

The strings on a piano are such that standing waves are set up along their length when excited by striking them. Variables like density, tension, and length determine fundamental frequency or mode at which the string vibrates. As you climb up the scale within one octave, each string vibrates at a progressively higher frequency, i.e. the pitch changes by one whole scale degree between successive white keys. It is important to note that the piano's strings do not produce pure tones at the fundamental, rather, the string's motion consists of many harmonic overtones which contribute to the overall timbre of the piano. This is the reason that concert pitch (A 440Hz) sounds different from a piano than it does from the viola or guitar. If you do a fourrier analysis of the sound emitted by a piano you can get an idea of the individual modes and their amplitudes that contribute to the overall sound. You could then plot these on a graph of amplitude versus frequency to get a visual display or diagram of the harmonic differences between different notes of the piano.