Given a standing wave in the form of a length of string attached to two oscillators, is there a relationship between the number of antinodes present and the maximum displacement of the antinodes? Demonstrations have shown that increasing the frequency of the oscillations to reach the nth harmonic and increase the number of antinodes reduces the amplitude of the antinodes; however, I have yet to find a quantitative relationship.
I assume the demonstrations use an elastic string fixed at both ends. To set up a first harmonic standing wave the string must stretch to produce the extra length needed to form a single sine wave shape for the standing wave. To form two sine waves of the same amplitude as the single sine wave it would need to stretch further--and so forth for the higher harmonics. The increasing stretching causes a reduction in amplitude of the antinodes in a practical demonstration with a fixed string length and a fixed initial string tension.
The answer: An equation would involve string mass, Hooks constant (probably a nonlinear version), string length, oscillation frequency, initial string tension, and more.