If a wave's third harmonic has a frequency of 24 Hz, what is its
natural (fundamental) frequency and what is the frequency of H6?

8 Hz, 48 Hz

Explanation:

The standing waves on a string (or inside a pipe, for instance) have different modes of vibrations, depending on how many segments of the string are vibrating.

The fundamental frequency of a standing wave is the frequency of the fundamental mode of vibration; then, the higher modes of vibration are called harmonics. The frequency of the n-th harmonic is given by

where

is the fundamental frequency

In this problem, we know that the wave's third harmonic has a frequency of

This means this is the frequency for n = 3. Therefore, we can find the fundamental frequency as:

Now we can also find the frequency of the 6-th harmonic using n = 6:

answer and explanation;

pythagoras' theorem

square root (-5^2+9^2)

square root (25+81)

square root (106)

angle is tan^-1(9/5)

neg x, pos y