Relation between frequency of vibrator generator and frequency of waves in a stretched string

Question

Imagine a vibration generator is attached to a stretched string. Will the frequency of the vibration generator be equal to the frequency of the waves produced in the stretched string?

Also, a similar follow up question, imagine I have a tuning fork of a specific frequency forming sound waves in an air column. Will the frequency of that tuning fork be the same as the frequency of the sound waves in the air column? Or will they be different?

Some people have said the frequencies of the waves of the string/of the sound waves will be equal to the vibration generator/tuning fork, respectively. But this doesn't make sense, because the frequency of standing waves changes with temperature. So if the temperature changes, won't the frequency of the standing waves also change, and thus become different from the frequency of the tuning fork/vibration generator? What am I getting wrong here?

Simple answers would be very appreciated, as I am only in Grade 11, and we do not need to go into excessive details about this topic in our syllabus.

Answer 1

The string vibrates with the frequency of the generator, for any frequency of the generator. Standing waves are formed only for a specific set of frequencies. If you vibrate the string and slowly increase the frequency, you will see that the string vibrates all the time but for some specific values of the frequency the specific patterns of standing waves (with nodes and antinodes) will form and then dissapear as the frequency is increased further and then at a higher frequency another pattern forms (with moe nodes). The frequencies at which standing waves are observed depend on the string properties (length, tension, thickness) and may depend on temperatue as well. You may have seen these called the fundamental frequency and the harmonics. Another ptoperty of these frequenies is that the transfer of energy from the generatot to the string. The amplitude of vibration is relatively small when the frequency is not one of the harmonics and inceases near the harmonics. Buu the vibration frequency is the same as the frequency of the generator for all frequencies. If you like to read more, this is an example of what is called "forced vibrations".

Answer 2

The frequencies of waves in the string and the air column will be the frequencies of the vibration generator and the tuning fork. But the waves will be of significant amplitude only when the generator or the fork is the same as a natural frequency of the string or column.

These natural frequencies are given by $f_n= nv_t/2L$, $f_n=v_l(2n+1)/4L$, ($n = 1, 2, 3 ...)$in which $v_t$ is the speed of transverse waves on the string and $v_l$ is the speed of longitudinal waves in air. [The air column is assumed to be a pipe open at one end and closed at the other.] It is these natural frequencies which are somewhat temperature-dependent because $v_t$ and $v_l$ are temperature-dependent. And the frequency of the vibration generator has to be set, or the frequency of the fork has to be chosen, so as to match the natural frequency of the string or column in order to ge the maximum amplitude of standing wave.