Do two mechanical transverse waves traveling in the same medium have the same speed whatever the source might be? If the answer to this question is yes, can I generalize and say "all waves of the same EXACT category traveling in same medium have the same wave speed" and by the same exact category I meant to specify the exact type of the wave, that is to state its both kinds; mechanical and traverse, or mechanical and longitudinal. That being said, I can't say that two mechanical waves one is transverse while the other is longitudinal have the same speed in the same medium, they could have the same speed though, but its not a general statement, is that right?
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The behavior depends on both the medium and the type of wave. In general, the speed of the wave (c) depnds on the frequency of the wave. The relationship between speed and frequency (or frequency and wave-number) is called dispersion relation. If the dispersion relation is not the trivial c=constant it is said that the medium is dispersive or that we have dispersion of that wave in the given medium. Othervise, there is no dispersion. Both transversal and longitudinal bulk waves in solids are technically dispersive but for low frequencies (including audio frequencies) the speed is practically constant, at least for typical crystalline solids. But there are other types of waves in solids like the surface waves called Lamb waves where the dispersion may be significant even at low frequencies. So, I suppose that the answer to your question is kind of "no" in general, it's not enough to specify the type of wave and the polarization (longitudinal or transversal). However, for some range of frequencies, which include ultrasound used in usual industrial nondestructive testing (5-50 MHz or about there), the dispersion is practically zero and you will find, for a given material, just one speed for each one of the two bulk waves (longitudinal and transversal or shear wave).
Edit: an example of table with sound velocities. I belive that "Extensional" refers to surface waves (Lamb waves). https://www.engineeringtoolbox.com/sound-speed-solids-d_713.html
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$\begingroup$ So no matter the source of the waves, if the medium is non dispersive, I will inevitably find one speed for all kinds of mechanical waves in that medium, right? $\endgroup$– JackCommented Sep 7, 2023 at 4:26
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$\begingroup$ No, each kind has it's own speed. Unless by "kind" you mean frequency. $\endgroup$– nasuCommented Sep 7, 2023 at 4:30
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$\begingroup$ What did you mean by this "the dispersion is practically zero and you will find, for a given material, just one speed for each one of the two bulk waves (longitudinal and transversal or shear wave)" $\endgroup$– JackCommented Sep 7, 2023 at 4:38
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$\begingroup$ By kind I meant longitudinal and transverse. $\endgroup$– JackCommented Sep 7, 2023 at 4:39
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$\begingroup$ I added a link in my answer. You can see one speed for each type but you can also see the note on top of the table. If you want to actually see how much the speed depends on frequency (if it does) you will have to look up research papers for a specific material. $\endgroup$– nasuCommented Sep 7, 2023 at 12:21